
The flat earth and heliocentric models are opposites in almost every way. Is it really that hard to tell which is correct?
The answer is NO. The earth is a sphere. It's obvious by the apparent positions of the celestial objects in the sky as observed from earth. By simple observation you can determine conclusively that the earth is round and that a flat earth is impossible.
One of the simplest examples illustrating this is Polaris. See why here:
http://debunkingflatearth.blogspot.com/2016/02/debunkingflatearthhowpolarisproves.html (http://debunkingflatearth.blogspot.com/2016/02/debunkingflatearthhowpolarisproves.html)

The flat earth and heliocentric models are opposites in almost every way. Is it really that hard to tell which is correct?
The answer is NO. The earth is a sphere. It's obvious by the apparent positions of the celestial objects in the sky as observed from earth. By simple observation you can determine conclusively that the earth is round and that a flat earth is impossible.
One of the simplest examples illustrating this is Polaris. See why here:
http://debunkingflatearth.blogspot.com/2016/02/debunkingflatearthhowpolarisproves.html (http://debunkingflatearth.blogspot.com/2016/02/debunkingflatearthhowpolarisproves.html)
Yes, but did you look up the wiki?
You must realise how much information can be gained there, but please don't try to swallow it all or you might choke!
For example:
from: http://www.sacredtexts.com/earth/za/za37.htm (http://www.sacredtexts.com/earth/za/za37.htm)
DECLINATION OF THE POLE STAR
Another phenomenon supposed to prove rotundity, is thought to be the fact that Polaris, or the north polar star sinks to the horizon as the traveler approaches the equator, on passing which it becomes invisible. This is a conclusion fully as premature and illogical as that involved in the several cases already alluded to. It is an ordinary effect of perspective for an object to appear lower and lower as the observer goes farther and farther away from it. Let any one try the experiment of looking at a lighthouse, church spire, monument, gas lamp, or other elevated object, from a distance of only a few yards, and notice the angle at which it is observed. On going farther away, the angle under which it is seen will diminish, and the object will appear lower and lower as the distance of the observer increases, until, at a certain point, the line of sight to the object, and the apparently uprising surface of the earth upon or over which it stands, will converge to the angle which constitutes the "vanishing point" or the horizon; beyond which it will be invisible. What can be more common than the observation that, standing at one end of a long row of lampposts, those nearest to us seem to be the highest; and those farthest away the lowest; whilst, as we move along towards the opposite end of the series, those which we approach seem to get higher, and those we are leaving behind appear to gradually become lower.
This lowering of the pole star as we recede southwards; and the rising of the stars in the south as we approach them, is the necessary result of the everywhere visible law of perspective operating between the eyeline of the observer, the object observed, and the plane surface upon which he stands; and has no connection with or relation whatever to the supposed rotundity of the earth.
Ergo, when I stand outside and look into the skies, the star constellations I do not see are simply invisible past the vanishing point, beyond my perspective. When I travel south I am moving to a new location, changing my perspective, rising up a completely different set stars.
A bit wordy!
So, it's all "perspective"! Mind you it does seem strange the each 111.1 km we move south from the North Pole the elevation of Polaris decreases by (almost) exactly 1°, finally reaching 0° (+ a bit due to refraction) at the equator. So close is this that it has been used for navigation for many centuries!
Perspective simply cannot make an object at least 5,000 km high disappear below the horizon when we move only about 10,000 km away!
This is not possible unless the light bends in some peculiar way. More than about 0.5° is not likely under normal conditions!
One big problem with quoting "Earth not a Globe" is the sheer volume of material to wade through! Rowbotham states categorically, though takes lots of words to say it, that the "South Pole Star" (Sigma Octantis) and the Southern Cross (Crux) cannot be seen from all longitudes in the southern hemisphere!
Another thing is certain, that from and within the equator the north pole star, and the constellations Ursa Major, Ursa Minor, and many others, can be seen from every meridian simultaneously; whereas in the south, from the equator, neither the socalled south pole star, nor the remarkable constellation of the Southern Cross, can be seen simultaneously from every meridian, showing that all the constellations of the south–pole star included–sweep over a great southern arc and across the meridian, from their rise in the evening to their setting in the morning. But if the earth is a globe, Sigma Octantis a south pole star, and the Southern Cross a southern circumpolar constellation, they would all be visible at the same time from every longitude on the same latitude, as is the case with the northern pole star and the northern circumpolar constellations. Such, however, is strangely not the case; Sir James Clarke Ross did not see it until he was 8° south of the equator, and in longitude 30° W.
Saying "Sir James Clarke Ross did not see it until he was 8°" (it being the Southern Cross) means nothing as we all (including Rowbotham) know full well that the Southern Cross is not at the South Celestial Pole, but some 30° away. In other words while the South Celestial Pole is visible every night everywhere (baring obstructions) over the whole of the Southern Hemisphere, the Southern Cross is only visible at all times south of 30° Latitude!
Yes, Rowbotham can never be accused of using one word when he can get away with 10!

The flat earth and heliocentric models are opposites in almost every way. Is it really that hard to tell which is correct?
The answer is NO. The earth is a sphere. It's obvious by the apparent positions of the celestial objects in the sky as observed from earth. By simple observation you can determine conclusively that the earth is round and that a flat earth is impossible.
One of the simplest examples illustrating this is Polaris. See why here:
http://debunkingflatearth.blogspot.com/2016/02/debunkingflatearthhowpolarisproves.html (http://debunkingflatearth.blogspot.com/2016/02/debunkingflatearthhowpolarisproves.html)
Yes, but did you look up the wiki?
You must realise how much information can be gained there, but please don't try to swallow it all or you might choke!
For example:
from: http://www.sacredtexts.com/earth/za/za37.htm (http://www.sacredtexts.com/earth/za/za37.htm)
DECLINATION OF THE POLE STAR
Another phenomenon supposed to prove rotundity, is thought to be the fact that Polaris, or the north polar star sinks to the horizon as the traveler approaches the equator, on passing which it becomes invisible. This is a conclusion fully as premature and illogical as that involved in the several cases already alluded to. It is an ordinary effect of perspective for an object to appear lower and lower as the observer goes farther and farther away from it. Let any one try the experiment of looking at a lighthouse, church spire, monument, gas lamp, or other elevated object, from a distance of only a few yards, and notice the angle at which it is observed. On going farther away, the angle under which it is seen will diminish, and the object will appear lower and lower as the distance of the observer increases, until, at a certain point, the line of sight to the object, and the apparently uprising surface of the earth upon or over which it stands, will converge to the angle which constitutes the "vanishing point" or the horizon; beyond which it will be invisible. What can be more common than the observation that, standing at one end of a long row of lampposts, those nearest to us seem to be the highest; and those farthest away the lowest; whilst, as we move along towards the opposite end of the series, those which we approach seem to get higher, and those we are leaving behind appear to gradually become lower.
This lowering of the pole star as we recede southwards; and the rising of the stars in the south as we approach them, is the necessary result of the everywhere visible law of perspective operating between the eyeline of the observer, the object observed, and the plane surface upon which he stands; and has no connection with or relation whatever to the supposed rotundity of the earth.
Ergo, when I stand outside and look into the skies, the star constellations I do not see are simply invisible past the vanishing point, beyond my perspective. When I travel south I am moving to a new location, changing my perspective, rising up a completely different set stars.
A bit wordy!
So, it's all "perspective"! Mind you it does seem strange the each 111.1 km we move south from the North Pole the elevation of Polaris decreases by (almost) exactly 1°, finally reaching 0° (+ a bit due to refraction) at the equator. So close is this that it has been used for navigation for many centuries!
Perspective simply cannot make an object at least 5,000 km high disappear below the horizon when we move only about 10,000 km away!
This is not possible unless the light bends in some peculiar way. More than about 0.5° is not likely under normal conditions!
One big problem with quoting "Earth not a Globe" is the sheer volume of material to wade through! Rowbotham states categorically, though takes lots of words to say it, that the "South Pole Star" (Sigma Octantis) and the Southern Cross (Crux) cannot be seen from all longitudes in the southern hemisphere!
Another thing is certain, that from and within the equator the north pole star, and the constellations Ursa Major, Ursa Minor, and many others, can be seen from every meridian simultaneously; whereas in the south, from the equator, neither the socalled south pole star, nor the remarkable constellation of the Southern Cross, can be seen simultaneously from every meridian, showing that all the constellations of the south–pole star included–sweep over a great southern arc and across the meridian, from their rise in the evening to their setting in the morning. But if the earth is a globe, Sigma Octantis a south pole star, and the Southern Cross a southern circumpolar constellation, they would all be visible at the same time from every longitude on the same latitude, as is the case with the northern pole star and the northern circumpolar constellations. Such, however, is strangely not the case; Sir James Clarke Ross did not see it until he was 8° south of the equator, and in longitude 30° W.
Saying "Sir James Clarke Ross did not see it until he was 8°" (it being the Southern Cross) means nothing as we all (including Rowbotham) know full well that the Southern Cross is not at the South Celestial Pole, but some 30° away. In other words while the South Celestial Pole is visible every night everywhere (baring obstructions) over the whole of the Southern Hemisphere, the Southern Cross is only visible at all times south of 30° Latitude!
Yes, Rowbotham can never be accused of using one word when he can get away with 10!
Thanks for the response, yes, "perspective" is an inadequate explanation and was actually addressed in the blog post I linked to. You quoted, "It is an ordinary effect of perspective for an object to appear lower and lower as the observer goes farther and farther away from it".
The question is how much lower and at what rate will the object appear to descend? Can this be determined?
Yes. It can. "Perspective" can be worked out using trigonometry. It's done all the time in real world applications. FEers don't seem to understand this.
The illustration below, which was shown in the blog, shows precisely the effect of perspective on the apparent position of Polaris above a flat plane.
(https://www.filesanywhere.com/FS/M.aspx?v=8c7068865aa6a1b1b09f)
As you can see, the effect of flat earth "perspective" cannot account for what is observed in reality.

This is actually the same question as "If the sun is disappearing to perspective, shouldn't it slow down as it approaches the horizon?" question, for which the answer is here: http://wiki.tfes.org/Constant_Speed_of_the_Sun
Q. If the sun is disappearing to perspective, shouldn't it slow down as it approaches the horizon?
A. The sun moves constant speed into the horizon at sunset because it is at such a height that already beyond the apex of perspective lines. It has maximized the possible broadness of the lines of perspective in relation to the earth. It is intersecting the earth at a very broad angle.
It's widely observable that overhead receding bodies move at a more constant pace into the horizon the higher they are. For an example imagine that someone is flying a Cessna into the distance at an illegal altitude of 700 feet. He seems to zoom by pretty fast when he is flies over your head, only slowing down when he is off in the far distance.
Now consider what happens when a jet flies over your head at 45,000 feet. At that altitude a jet appears to move very slowly across the sky, despite that the jet is moving much faster than the Cessna. With greater altitude the plane seems to move more consistently across the sky. It does not zoom by overhead, only seeming to slow when in the far distance.
When a body increases its altitude it broadens its perspective lines in relation to the earth and the observer, and thus appears to move slower and at a more constant pace into the horizon. In FET the stars and celestial bodies are at such a great height that they have maximized the perspective lines. They are descending into the horizon at a consistent or near consistent velocity. As consequence they do not slow down in the distance by any significant degree, and hence the stars do not appear to change configuration and build up in the distance, nor does the sun or moon appear to slow as they approach the horizon.
(http://wiki.tfes.org/images/f/f6/Perspective_speed.png)
The rate of descent of two bodies at different altitudes is more constant because it take a lot longer for a high altitude body to reach the horizon than it does for a low altitude body. The higher a body is, the broader its perspective lines, the longer and more constantly it will appear to approach the horizon to the observer.
I plan on rewriting this article at some point, but you get the idea.

This is actually the same question as "If the sun is disappearing to perspective, shouldn't it slow down as it approaches the horizon?" question, for which the answer is here: http://wiki.tfes.org/Constant_Speed_of_the_Sun
Q. If the sun is disappearing to perspective, shouldn't it slow down as it approaches the horizon?
A. The sun moves constant speed into the horizon at sunset because it is at such a height that already beyond the apex of perspective lines. It has maximized the possible broadness of the lines of perspective in relation to the earth. It is intersecting the earth at a very broad angle.
It's widely observable that overhead receding bodies move at a more constant pace into the horizon the higher they are. For an example imagine that someone is flying a Cessna into the distance at an illegal altitude of 700 feet. He seems to zoom by pretty fast when he is flies over your head, only slowing down when he is off in the far distance.
Now consider what happens when a jet flies over your head at 45,000 feet. At that altitude a jet appears to move very slowly across the sky, despite that the jet is moving much faster than the Cessna. With greater altitude the plane seems to move more consistently across the sky. It does not zoom by overhead, only seeming to slow when in the far distance.
When a body increases its altitude it broadens its perspective lines in relation to the earth and the observer, and thus appears to move slower and at a more constant pace into the horizon. In FET the stars and celestial bodies are at such a great height that they have maximized the perspective lines. They are descending into the horizon at a consistent or near consistent velocity. As consequence they do not slow down in the distance by any significant degree, and hence the stars do not appear to change configuration and build up in the distance, nor does the sun or moon appear to slow as they approach the horizon.
(http://wiki.tfes.org/images/f/f6/Perspective_speed.png)
The rate of descent of two bodies at different altitudes is more constant because it take a lot longer for a high altitude body to reach the horizon than it does for a low altitude body. The higher a body is, the broader its perspective lines, the longer and more constantly it will appear to approach the horizon to the observer.
I plan on rewriting this article at some point, but you get the idea.
This is pure nonsense. The illustration I posted is to scale. The angles and distances are right there in front of you. I'll post it again below. Notice that the distance needed to see a change in altitude from 20° to 10° (9,040 miles) is greater than the distance needed for polaris to drop from 90° to 20° (8,557 miles). If the diagram were to continue, the distance needed for Polaris to drop from 10° to 5° is more than the distance needed for it to drop from 90° to 10°, about 17,835 miles (a total of 35,433 miles from 90° to 5°). To see Polaris at 0°, the distance needed is infinity.
It would therefore be impossible to see the apparent altitude of any celestial object drop at a constant rate due to perspective if it was moving away at a constant speed. You can draw it out and measure the angles for yourself if you like, or just use an online right triangle calculator.
Triangles don't lie.
(https://www.filesanywhere.com/FS/M.aspx?v=8c70678d619cb478aca7)

This is actually the same question as "If the sun is disappearing to perspective, shouldn't it slow down as it approaches the horizon?" question, for which the answer is here: http://wiki.tfes.org/Constant_Speed_of_the_Sun
Q. If the sun is disappearing to perspective, shouldn't it slow down as it approaches the horizon?
A. The sun moves constant speed into the horizon at sunset because it is at such a height that already beyond the apex of perspective lines. It has maximized the possible broadness of the lines of perspective in relation to the earth. It is intersecting the earth at a very broad angle.
It's widely observable that overhead receding bodies move at a more constant pace into the horizon the higher they are. For an example imagine that someone is flying a Cessna into the distance at an illegal altitude of 700 feet. He seems to zoom by pretty fast when he is flies over your head, only slowing down when he is off in the far distance.
Now consider what happens when a jet flies over your head at 45,000 feet. At that altitude a jet appears to move very slowly across the sky, despite that the jet is moving much faster than the Cessna. With greater altitude the plane seems to move more consistently across the sky. It does not zoom by overhead, only seeming to slow when in the far distance.
When a body increases its altitude it broadens its perspective lines in relation to the earth and the observer, and thus appears to move slower and at a more constant pace into the horizon. In FET the stars and celestial bodies are at such a great height that they have maximized the perspective lines. They are descending into the horizon at a consistent or near consistent velocity. As consequence they do not slow down in the distance by any significant degree, and hence the stars do not appear to change configuration and build up in the distance, nor does the sun or moon appear to slow as they approach the horizon.
(http://wiki.tfes.org/images/f/f6/Perspective_speed.png)
The rate of descent of two bodies at different altitudes is more constant because it take a lot longer for a high altitude body to reach the horizon than it does for a low altitude body. The higher a body is, the broader its perspective lines, the longer and more constantly it will appear to approach the horizon to the observer.
I plan on rewriting this article at some point, but you get the idea.
The article IMHO is an example of one of the biggest problems the FES needs to overcome. I will just compare illustrations. One has data like distances, what you should observe from different points. The other only provides angles, does not provide any distances/heights or display what you should observe at different times and/or places.

This is pure nonsense. The illustration I posted is to scale. The angles and distances are right there in front of you. I'll post it again below. Notice that the distance needed to see a change in altitude from 20° to 10° (9,040 miles) is greater than the distance needed for polaris to drop from 90° to 20° (8,557 miles). If the diagram were to continue, the distance needed for Polaris to drop from 10° to 5° is more than the distance needed for it to drop from 90° to 10°, about 17,835 miles (a total of 35,433 miles from 90° to 5°). To see Polaris at 0°, the distance needed is infinity.
It would therefore be impossible to see the apparent altitude of any celestial object drop at a constant rate due to perspective if it was moving away at a constant speed. You can draw it out and measure the angles for yourself if you like, or just use an online right triangle calculator.
Triangles don't lie.
(https://www.filesanywhere.com/FS/M.aspx?v=8c70678d619cb478aca7)
Under traditional perspective it is also impossible for the sun to ever set. However, Samuel Birley Rowbotham teaches us in Earth Not a Globe that we must adopt our concept of perspective from real world experience and observations, not some mathematical concept.

Under traditional perspective it is also impossible for the sun to ever set. However, Samuel Birley Rowbotham teaches us in Earth Not a Globe that we must adopt our concept of perspective from real world experience and observations, not some mathematical concept.
Oh, I am so sorry, but now we must all accept the "bible" of all Flat Earthers!
For myself, I find all this bendy light, objects magnified 4 time due to the atmosphere and other absurdities a bit hard to swallow.
I would think that, even it Flat Earth circles, there could have been a few advances since 1885.
May a map that did not grossly distort half the earth, and all of it to a lesser extent would be a big advance in 230 years.
Oh yes, we have the BiPolar map that distorts everything, and somehow need teleportation (via aether I guess) to circumnavigate either on the equator of on longitude 0 deg.
Yes, big advances! One step forward and two steps back. Rowbotham I might understand with the lack of knowledge of polar regions in thw late 1800's, but for people in the 21st century, it is completely mystifying! Stuck in a time warp?
It almost sickens me to read a lot of Rowbotham and the 100 Proofs! I suppose if you start with a biased view of history and the world around you it might be understandable.

It absolutely makes more sense to base science off of the observed and experienced rather than the theoretical and hypothetical. What was provided was a model based on little more than an idea of how things should work under the theories of art school perspective and geometry, not how they actually work.
The Ancient Greeks made a lot of assumptions about the physical world when coming up with Geometry. A lot of the assumptions turned out to be mistakes. For one, circles do not actually exist, since the universe is quantized, and any such related math is inaccurate. If one were to trace a line along all of the little pixilated plancks which make up the circumference of the most perfect "circle" in the universe one would find that pi is actually equal to 4, rather than the theoretical value of 3.14159...
I will be writing more on this topic of experience vs hypothesis in The 21st Century Edition of Earth Not a Globe, a modernized reboot of Earth Not a Globe by Samuel Birley Rowbotham, which we are working on in the Earth Not a Globe Workshop.

It absolutely makes more sense to base science off of the observed and experienced rather than the theoretical and hypothetical. What was provided was a model based on little more than an idea of how things should work under the theories of art school perspective and geometry, not how they actually work.
The Ancient Greeks made a lot of assumptions about the physical world when coming up with Geometry. A lot of the assumptions turned out to be mistakes. For one, circles do not actually exist, since the universe is quantized, and any such related math is inaccurate. If one were to trace a line along all of the little pixilated plancks which make up the circumference of the most perfect "circle" in the universe one would find that pi is actually equal to 4, rather than the theoretical value of 3.14159...
I will be writing more on this topic of experience vs hypothesis in The 21st Century Edition of Earth Not a Globe, a modernized reboot of Earth Not a Globe by Samuel Birley Rowbotham, which we are working on in the Earth Not a Globe Workshop.
Interesting thoughts about the differences between the physical and the theoretical world. I'll try to make this small experiment: I'll estimate the volume of an orange using pi= 4 or pi=3,1415, and then I'll submerge in water and see how much water it'll displace and compare the measured volume with the calculated ones. Does that make sense?

It absolutely makes more sense to base science off of the observed and experienced rather than the theoretical and hypothetical. What was provided was a model based on little more than an idea of how things should work under the theories of art school perspective and geometry, not how they actually work.
The Ancient Greeks made a lot of assumptions about the physical world when coming up with Geometry. A lot of the assumptions turned out to be mistakes. For one, circles do not actually exist, since the universe is quantized, and any such related math is inaccurate. If one were to trace a line along all of the little pixilated plancks which make up the circumference of the most perfect "circle" in the universe one would find that pi is actually equal to 4, rather than the theoretical value of 3.14159...
I will be writing more on this topic of experience vs hypothesis in The 21st Century Edition of Earth Not a Globe, a modernized reboot of Earth Not a Globe by Samuel Birley Rowbotham, which we are working on in the Earth Not a Globe Workshop.
Many Flat Earthers (including Gotham in "the other place) seem to think that in a short while the Flat Earth Hypothesis might be accepted by the world in general.
It is baffling at times to understand just how REers can go on and on expressing their beliefs without opening their eyes and seeing what is past their text books and out the door of their lab.
It is writings like yours that convinces me it will never happen. Your ideas haven't the faintest chance of acceptance when the only real observation seems to be "The Earth looks flat" and everything else has to be bent to suit that one observation.
I open my eyes and what do I see! The Earth looks flat  it does, it's big!
 On a clear day looking out to sea the skyhorizon interface is a sharp line (it is only about 5 km away!). On a flat earth it would have to fade into the distance with no distinct boundary.
 The sun appears to rise from behind the horizon and appears to set behind the horizon.
 The sun stays the same size as it arcs up and over the sky  it sometimes seems a bit bigger at sunrise and sunset.
 The sun always appears to be a disk, though sometimes a bit distorted at sunrise and sunset.
 Likewise the moon appears to rise from behind the horizon and appears to set behind the horizon.
 The moon stays the same size as it arcs up and over the sky  it sometimes seems a bit bigger at moonrise and moonset.
 The moon always appears to show the same face wherever it is in the sky. (And from wherever we observe it  have to travel for this observation).
 The full moon always appears to be a circle, though sometimes a bit distorted at moonrise and moonset.
Note that none of this is claimed as direct evidence of a rotating earth, but I believe is strong evidence of a Globe with a distant (far further than the earths size) sun and moon. So many of these points are "explained away" by TFES using "perspective", "bendy light" (massive refraction), extreme "magnification" by the atmosphere or simply ignored. These explanations are simply quoted with no justification at all!
I could go on about the direction of sunrise and sunset etc.
Of these, number (1) might indicate a flat earth, but then when we try to work out what the sun and moon are doing, we get into big trouble.
The Flat Earth movement just takes (1) and says "The earth is flat", then gets into terrible trouble explaining away all of the others with fanciful ideas of perspective, bending light, "celestial gears", universal acceleration (powered by "dark energy") and on and on.
But all the other points are far more simply explained on a Globe Earth, though not necessarily rotating.
There are more points you can see around every day (like the movement of the stars at night!) that are hard to explain on any flat earth model without resorting to nothing more than guesswork about strange things like celestial gears and aetheric whirlpools etc.
Even the problems with the stationary Globe earth were found in the past from observations made without modern instruments. Largely eyes and simple (though large) angle measuring equipment.
Honestly, I find that the Globe Earth conforms far better to the Zetetic approach than all the imagination and guesswork needed to support any Flat Earth model!
I could go on and on but this is enough for now!
On top of the, TFES simply has no accurate map of the earth! Nothing that shows correct distances (as have been surveyed over hundreds of years) and the correct shape (and dimensions) of continents.
There is not the slightest chance that the idea of a Flat Earth will ever be accepted without an accurate map!
BTW I measure (with a tape measure) th circumference of a metal lid of diameter 111.4 mm and it comes to 351 mm. When I divide that out I get a 3.15  (can't get 4 out of it) I'll take those Greeks over the rubbish Mathis puts out any day!

If one were to trace a line along all of the little pixilated plancks which make up the circumference of the most perfect "circle" in the universe one would find that pi is actually equal to 4, rather than the theoretical value of 3.14159...
In addition to being used to calculate the circumference of a circle, pi is also used to calculate the area of the same shape. In order for pi to be of use in calculations, its value needs to remain constant. Disagree with that if you will.
If we compare the area of a circle of radius 10 cm using both pi = 3.142 (to three decimal places) and pi = 4.000
10 x 10 x 3.142 = 314.2 cm^2
10 x 10 x 4.000 = 400.0 cm^2
The difference is an area of 85.8 cm^2.
I cannot visualise a circle with radius 10 cm and area 400 cm^2. If anyone can draw one, then please do. It might be useful to compare it (to scale) with a square of side length 20cm, as they have the same area.

It absolutely makes more sense to base science off of the observed and experienced rather than the theoretical and hypothetical. What was provided was a model based on little more than an idea of how things should work under the theories of art school perspective and geometry, not how they actually work.
The Ancient Greeks made a lot of assumptions about the physical world when coming up with Geometry. A lot of the assumptions turned out to be mistakes. For one, circles do not actually exist, since the universe is quantized, and any such related math is inaccurate. If one were to trace a line along all of the little pixilated plancks which make up the circumference of the most perfect "circle" in the universe one would find that pi is actually equal to 4, rather than the theoretical value of 3.14159...
I will be writing more on this topic of experience vs hypothesis in The 21st Century Edition of Earth Not a Globe, a modernized reboot of Earth Not a Globe by Samuel Birley Rowbotham, which we are working on in the Earth Not a Globe Workshop.
Interesting thoughts about the differences between the physical and the theoretical world. I'll try to make this small experiment: I'll estimate the volume of an orange using pi= 4 or pi=3,1415, and then I'll submerge in water and see how much water it'll displace and compare the measured volume with the calculated ones. Does that make sense?
That would assume the orange is perfectly round. It is not.
BTW I measure (with a tape measure) th circumference of a metal lid of diameter 111.4 mm and it comes to 351 mm. When I divide that out I get a 3.15  (can't get 4 out of it) I'll take those Greeks over the rubbish Mathis puts out any day!
You are assuming the circumference of a metal lid is perfectly round. It is not. If you were to actually trace in all of the imperfections of the circumference there would be additional length there.
If one were to trace a line along all of the little pixilated plancks which make up the circumference of the most perfect "circle" in the universe one would find that pi is actually equal to 4, rather than the theoretical value of 3.14159...
In addition to being used to calculate the circumference of a circle, pi is also used to calculate the area of the same shape. In order for pi to be of use in calculations, its value needs to remain constant. Disagree with that if you will.
If we compare the area of a circle of radius 10 cm using both pi = 3.142 (to three decimal places) and pi = 4.000
10 x 10 x 3.142 = 314.2 cm^2
10 x 10 x 4.000 = 400.0 cm^2
The difference is an area of 85.8 cm^2.
I cannot visualise a circle with radius 10 cm and area 400 cm^2. If anyone can draw one, then please do. It might be useful to compare it (to scale) with a square of side length 20cm, as they have the same area.
There is hidden area in the circumference of a nonperfect circle.
(http://www.intmath.com/blog/wpcontent/images/2015/03/piis4c.png)

It absolutely makes more sense to base science off of the observed and experienced rather than the theoretical and hypothetical. What was provided was a model based on little more than an idea of how things should work under the theories of art school perspective and geometry, not how they actually work.
The Ancient Greeks made a lot of assumptions about the physical world when coming up with Geometry. A lot of the assumptions turned out to be mistakes. For one, circles do not actually exist, since the universe is quantized, and any such related math is inaccurate. If one were to trace a line along all of the little pixilated plancks which make up the circumference of the most perfect "circle" in the universe one would find that pi is actually equal to 4, rather than the theoretical value of 3.14159...
I will be writing more on this topic of experience vs hypothesis in The 21st Century Edition of Earth Not a Globe, a modernized reboot of Earth Not a Globe by Samuel Birley Rowbotham, which we are working on in the Earth Not a Globe Workshop.
Interesting thoughts about the differences between the physical and the theoretical world. I'll try to make this small experiment: I'll estimate the volume of an orange using pi= 4 or pi=3,1415, and then I'll submerge in water and see how much water it'll displace and compare the measured volume with the calculated ones. Does that make sense?
That would assume the orange is perfectly round. It is not.
BTW I measure (with a tape measure) th circumference of a metal lid of diameter 111.4 mm and it comes to 351 mm. When I divide that out I get a 3.15  (can't get 4 out of it) I'll take those Greeks over the rubbish Mathis puts out any day!
You are assuming the circumference of a metal lid is perfectly round. It is not. If you were to actually trace in all of the imperfections of the circumference there would be additional length there.
If one were to trace a line along all of the little pixilated plancks which make up the circumference of the most perfect "circle" in the universe one would find that pi is actually equal to 4, rather than the theoretical value of 3.14159...
In addition to being used to calculate the circumference of a circle, pi is also used to calculate the area of the same shape. In order for pi to be of use in calculations, its value needs to remain constant. Disagree with that if you will.
If we compare the area of a circle of radius 10 cm using both pi = 3.142 (to three decimal places) and pi = 4.000
10 x 10 x 3.142 = 314.2 cm^2
10 x 10 x 4.000 = 400.0 cm^2
The difference is an area of 85.8 cm^2.
I cannot visualise a circle with radius 10 cm and area 400 cm^2. If anyone can draw one, then please do. It might be useful to compare it (to scale) with a square of side length 20cm, as they have the same area.
There is hidden area in the circumference of a nonperfect circle.
(http://www.intmath.com/blog/wpcontent/images/2015/03/piis4c.png)
Thanks Tom, now if you can show the most refined "circle" next to a square with side length 2 x radius [edited], we can see whether they have the same area.
In fact if you can (apologies, I cannot), overlay the two figures to show the difference in area. I wager the "circle" will fit inside the square with room to spare.
If that is the case, then where is the missing area from the "circle"? Very well hidden indeed!

Thanks Tom, now if you can show the most refined "circle" next to a square with side length 2 x diameter, we can see whether they have the same area.
In fact if you can (apologies, I cannot), overlay the two figures to show the difference in area. I wager the "circle" will fit inside the square with room to spare.
If that is the case, then where is the missing area from the "circle"? Very well hidden indeed!
Your logic needs a little work.
It is possible to draw a snake curled up inside of a square, with the length of that snake being longer than the circumference of the square it exists within.
A circle which contains curls of hidden area can easily fit in a square with the same circumference/perimiter.

I'm talking about area, not circumference.
The area of the "circle" you describe with radius "r" is significantly less than the area of a square with side length "2r".
I raised this issue re the value of pi being constant. If pi is the same value for all calculations, the calculation of the area of a circle demonstrates that it cannot be 4.

I'm talking about area, not circumference.
The area of the "circle" you describe with radius "r" is significantly less than the area of a square with side length "2r".
I raised this issue re the value of pi being constant. If pi is the same value for all calculations, the calculation of the area of a circle demonstrates that it cannot be 4.
Why would to total area within the two shapes need to be the same?
Would the total area within a square and a triangle need to be the same if they have an identical perimeter?

I'm talking about area, not circumference.
The area of the "circle" you describe with radius "r" is significantly less than the area of a square with side length "2r".
I raised this issue re the value of pi being constant. If pi is the same value for all calculations, the calculation of the area of a circle demonstrates that it cannot be 4.
Why would to total area within the two shapes need to be the same?
Would the total area within a square and a triangle need to be the same if they have an identical perimeter?
I see what you mean Tom. You are saying that even if pi = 4, a circle of radius "r" and a square of side length "2r" do not have to have the same area.
We must disagree on how to calculate the area of a circle. I use area = pi x r^2
Using my calculation, inputting pi as 4, then a circle with radius 10 cm has an area of 4 x 10 x 10 = 400 cm^2.
The area of a square with side length 2r is (2r)^2 = 20 x 20 = 400 cm^2.
So that is why I say, if pi = 4, then a circle and a square as described above will have the same area.
I don't see what triangles have to do with this, specifically.

(http://www.intmath.com/blog/wpcontent/images/2015/03/piis4c.png)
And obviously the length of the "Black Perimeter" never approaches the circumference, so saying this proves π = 4 is completely fallacious!.
If you are going to be that fussy, why do you assume those "plancks" are orthogonal? Maybe they should be in random directions. You can carry it on from there!
Mind you if I was using a method like that to find the circumference I would want the length of my "increments" to approach the length of the circle segments more and more closely as the length segments approach zero (Easier to write a mathematical expression than put it into words!). Clearly that is not true in your case.
I think I'll stick to the real world, where we are not trying to count plancks, but measure a distance with a finite resolution!
Anyone knows that the perimeter of say a country with its fractal like coastline continues to increase indefinitely as we decrease the "ruler" length.
So, being more akin to an engineer and not a mathematician (and less still a philosopher) I did not "count plancks" around the circular lid, but measured length with a steel tape!
Mind you sometimes I think that certain people regard the Flat/Globe Earth question as a matter of opinion. It is clearly (whatever John Davis says) in practical terms one or the other.
I am sure there are some on these forums who have been trained at diverting attention away from the real issues!

It absolutely makes more sense to base science off of the observed and experienced rather than the theoretical and hypothetical. What was provided was a model based on little more than an idea of how things should work under the theories of art school perspective and geometry, not how they actually work.
The Ancient Greeks made a lot of assumptions about the physical world when coming up with Geometry. A lot of the assumptions turned out to be mistakes. For one, circles do not actually exist, since the universe is quantized, and any such related math is inaccurate. If one were to trace a line along all of the little pixilated plancks which make up the circumference of the most perfect "circle" in the universe one would find that pi is actually equal to 4, rather than the theoretical value of 3.14159...
I will be writing more on this topic of experience vs hypothesis in The 21st Century Edition of Earth Not a Globe, a modernized reboot of Earth Not a Globe by Samuel Birley Rowbotham, which we are working on in the Earth Not a Globe Workshop.
Interesting thoughts about the differences between the physical and the theoretical world. I'll try to make this small experiment: I'll estimate the volume of an orange using pi= 4 or pi=3,1415, and then I'll submerge in water and see how much water it'll displace and compare the measured volume with the calculated ones. Does that make sense?
That would assume the orange is perfectly round. It is not.
Exactly Tom! That's why I choose an orange  because it's not perfectly round. I'll have water to fill up all of it's 'imperfections' so I can measure it's 'real' volume, and thereby estimate pi (which will be larger than 3,1415xx because of the orange's imperfections).
What's wrong with the experiment?

you're not wrong that the c/d ratio of real, physical circles is not pi. pi isn't a real constant.
you're very wrong that this means that pi is a constant and that that constant is 4. you're wronger to imply that 4 is a better approximation of c/d for real circles than pi. your 'proof' is even more wronger. the 'crinkled up' perimeter of the square is never going to actually 'straighten' up in a way that gets closer and closer to the perimeter of the circle. no matter how much you zoom in, it will never appear to approximate the perimeter of a circle. if you were to keep zooming in on the circle's perimeter, you'll only ever see this, no matter how much you zoom in:
(http://i.imgur.com/apmublm.png?1)
except it wouldn't look so shitty since i presume nature to be way better at ms paint than i am.

Exactly Tom! That's why I choose an orange  because it's not perfectly round. I'll have water to fill up all of it's 'imperfections' so I can measure it's 'real' volume, and thereby estimate pi (which will be larger than 3,1415xx because of the orange's imperfections).
What's wrong with the experiment?
Well, it's really the same experiment Daguerrohype is proposing. He seems to think that two shapes with the same perimeter should have the same total area within those shapes. He is wrong. I brought up the example of a triangle and a square with the same perimeters having different total areas within those shapes.

Exactly Tom! That's why I choose an orange  because it's not perfectly round. I'll have water to fill up all of it's 'imperfections' so I can measure it's 'real' volume, and thereby estimate pi (which will be larger than 3,1415xx because of the orange's imperfections).
What's wrong with the experiment?
Well, it's really the same experiment Daguerrohype is proposing. He seems to think that two shapes with the same parameter should have the same total area within those shapes. He is wrong. I brought up the example of a triangle and a square with the same parameter having different total areas within those shapes.
How would YOU best estimate the volume of e.g. an orange?

Exactly Tom! That's why I choose an orange  because it's not perfectly round. I'll have water to fill up all of it's 'imperfections' so I can measure it's 'real' volume, and thereby estimate pi (which will be larger than 3,1415xx because of the orange's imperfections).
What's wrong with the experiment?
Well, it's really the same experiment Daguerrohype is proposing. He seems to think that two shapes with the same parameter should have the same total area within those shapes. He is wrong. I brought up the example of a triangle and a square with the same parameter having different total areas within those shapes.
How would YOU best estimate the volume of e.g. an orange?
Your experiment is irrelevant. Surface area is not related to interior area in any meaningful with polygons. See the square vs triangle with identical perimeter example above.

Exactly Tom! That's why I choose an orange  because it's not perfectly round. I'll have water to fill up all of it's 'imperfections' so I can measure it's 'real' volume, and thereby estimate pi (which will be larger than 3,1415xx because of the orange's imperfections).
What's wrong with the experiment?
Well, it's really the same experiment Daguerrohype is proposing. He seems to think that two shapes with the same parameter should have the same total area within those shapes. He is wrong. I brought up the example of a triangle and a square with the same parameter having different total areas within those shapes.
How would YOU best estimate the volume of e.g. an orange?
Your experiment is irrelevant. Surface area is not related to interior area in any meaningful with polygons. See the square vs triangle with identical perimeter example above.
I'm just asking you a very simple question. How would you estimate the volume of an orange?

And what does all this rubbish on π = 4 have to do with the topic?
or with the OP:
The flat earth and heliocentric models are opposites in almost every way. Is it really that hard to tell which is correct?
The answer is NO. The earth is a sphere. It's obvious by the apparent positions of the celestial objects in the sky as observed from earth. By simple observation you can determine conclusively that the earth is round and that a flat earth is impossible.
One of the simplest examples illustrating this is Polaris. See why here:
http://debunkingflatearth.blogspot.com/2016/02/debunkingflatearthhowpolarisproves.html (http://debunkingflatearth.blogspot.com/2016/02/debunkingflatearthhowpolarisproves.html)

here's an easy way to demonstrate why this proof is unsound. let's try see if we can approximate the length of a line segment with sine waves of increasingly smaller amplitudes.
consider the following sine function, y=4sinx, and let's restrict the domain from x=0 to x=6.28.
(http://i.imgur.com/xBFZ45K.png?1)
it's obvious just from looking at the graph that the length of the sine wave is greater than the length of the domain (6.28 units). and, mathdoingrobots confirm that the length of that line is ~17.628 (https://www.wolframalpha.com/input/?lk=3&i=arc+length+of+y%3D4sin(x)+from+x%3D0+to+2pi). it's also obvious that if we want to approximate the length of the domain, then we must decrease the amplitude.
next we're going to add more sine functions to the graph. the pi=4 proof demands that the perimeter of the square remain constant by changing its shape in a specific way. likewise, we're going to keep the length of the sine wave constant while we decrease its amplitude. the only way to do that is to increase its period proportionally. in other words, if we decrease the amplitude by half, then we must increase the period by half. if i'm not making sense, just check out the following graph. this is y=4sinx, y=2sin(2x), y=sin(4x), y=.5sin(8x), all from x=0 to x=6.28
(http://i.imgur.com/oZSNsL7.png?1)
if you plug all those formulae into the mathwizardrobot, it will confirm that they all have the same length, ~17.628. but now we have a problem. as you can see, we can keep iterating and the sine wave will get smaller and smaller and smaller and smaller until it appears to be approximating the length the line, but since ~17.628 != 6.28, we know that it never does.
in fact, this notion of keeping the length of the sine wave constant by only letting amplitude vary inversely proportional to period is exactly what your proof does. just look at the corners. each time they "fold" the perimeter in the corners, they're doing it in a specific way that keeps the length the same, halves the amplitude, and doubles the period. graphing the absolute values of the same sine functions from before illustrates this. each iteration, starting with purple, has half the amplitude and double the period of the previous iteration, but their lengths are all the same. it might appear that they would approximate the length of a line as the amplitude approaches zero, but it never does, and they never do.
(http://i.imgur.com/I57kZHX.png?1)
/total thread derailment

Exactly Tom! That's why I choose an orange  because it's not perfectly round. I'll have water to fill up all of it's 'imperfections' so I can measure it's 'real' volume, and thereby estimate pi (which will be larger than 3,1415xx because of the orange's imperfections).
What's wrong with the experiment?
Well, it's really the same experiment Daguerrohype is proposing. He seems to think that two shapes with the same perimeter should have the same total area within those shapes. He is wrong. I brought up the example of a triangle and a square with the same perimeters having different total areas within those shapes.
Referring to Daguerrohype, you say "He seems to think that two shapes with the same perimeter should have the same total area within those shapes." His actual statement was:
If we compare the area of a circle of radius 10 cm using both pi = 3.142 (to three decimal places) and pi = 4.000
10 x 10 x 3.142 = 314.2 cm^2
10 x 10 x 4.000 = 400.0 cm^2
The difference is an area of 85.8 cm^2.
I cannot visualise a circle with radius 10 cm and area 400 cm^2. If anyone can draw one, then please do. It might be useful to compare it (to scale) with a square of side length 20cm, as they have the same area.
Where he said that if we use the correct (to 3 places) value of π=3.142, we get the area of 314.2 cm^{2} but if we take π=4.000, we get of 400.0 cm^{2}.
From what I can see what Daguerrohype actually says is quite correct. The circle and square only have the same perimeter and area if YOU insist on using YOUR stupid value of π=4. The sooner you can forget this "matter of definition" the better for everybody!
As I have tried to point out your so called proof is completely fallacious.
It's not a matter of definitions or maths it is simply wrong!

you're not wrong that the c/d ratio of real, physical circles is not pi. pi isn't a real constant.
you're very wrong that this means that pi is a constant and that that constant is 4. you're wronger to imply that 4 is a better approximation of c/d for real circles than pi. your 'proof' is even more wronger. the 'crinkled up' perimeter of the square is never going to actually 'straighten' up in a way that gets closer and closer to the perimeter of the circle. no matter how much you zoom in, it will never appear to approximate the perimeter of a circle. if you were to keep zooming in on the circle's perimeter, you'll only ever see this, no matter how much you zoom in:
(http://i.imgur.com/apmublm.png?1)
except it wouldn't look so shitty since i presume nature to be way better at ms paint than i am.
That's not really my point. My point is that there is no such thing as a circle in the universe. So therefore pi != 3.14159...
How would YOU best estimate the volume of e.g. an orange?
We could do the water experiment to get the volume. But the way you used it as a proof is irrelevant. Different shapes with the same perimeter don't all have the same interior area.

Exactly Tom! That's why I choose an orange  because it's not perfectly round. I'll have water to fill up all of it's 'imperfections' so I can measure it's 'real' volume, and thereby estimate pi (which will be larger than 3,1415xx because of the orange's imperfections).
What's wrong with the experiment?
Well, it's really the same experiment Daguerrohype is proposing. He seems to think that two shapes with the same perimeter should have the same total area within those shapes. He is wrong. I brought up the example of a triangle and a square with the same perimeters having different total areas within those shapes.
Referring to Daguerrohype, you say "He seems to think that two shapes with the same perimeter should have the same total area within those shapes." His actual statement was:
If we compare the area of a circle of radius 10 cm using both pi = 3.142 (to three decimal places) and pi = 4.000
10 x 10 x 3.142 = 314.2 cm^2
10 x 10 x 4.000 = 400.0 cm^2
The difference is an area of 85.8 cm^2.
I cannot visualise a circle with radius 10 cm and area 400 cm^2. If anyone can draw one, then please do. It might be useful to compare it (to scale) with a square of side length 20cm, as they have the same area.
Where he said that if we use the correct (to 3 places) value of π=3.142, we get the area of 314.2 cm^{2} but if we take π=4.000, we get of 400.0 cm^{2}.
From what I can see what Daguerrohype actually says is quite correct. The circle and square only have the same perimeter and area if YOU insist on using YOUR stupid value of π=4. The sooner you can forget this "matter of definition" the better for everybody!
As I have tried to point out your so called proof is completely fallacious.
It's not a matter of definitions or maths it is simply wrong!
I was referring to this statement:
Thanks Tom, now if you can show the most refined "circle" next to a square with side length 2 x radius [edited], we can see whether they have the same area.
In fact if you can (apologies, I cannot), overlay the two figures to show the difference in area. I wager the "circle" will fit inside the square with room to spare.
If that is the case, then where is the missing area from the "circle"? Very well hidden indeed!
This doesn't really make sense, since the interior area within different shapes of the same perimeter are not related.
As far as the statement you quoted, an attempt to calculate the area of a circle using A = pi*r^2 with 4 in the place of pi, the error in the logic is that circles do not exist. You can't use traditional circle math to calculate that. The shape is not a circle, and the correct way to calculate the area is to use a method of calculating the area within a polygon.

And how many angels can dance on the head of a pin?
Makes as much sense to me! But maybe I'm just too practical!

And how many angels can dance on the head of a pin?
Makes as much sense to me! But maybe I'm just too practical!
that depends
on how you define
an angel
since no such thing exists in the real world
(so you can make up your own terms :D)

And how many angels can dance on the head of a pin?
Makes as much sense to me! But maybe I'm just too practical!
that depends
on how you define
an angel
since no such thing exists in the real world
(so you can make up your own terms :D)
Yes,
since as far as we know no such thing exists in the "physical" world.
But, the expression has long been used as a description of the ultimate pointless discauusion.

A lot of the assumptions turned out to be mistakes. For one, circles do not actually exist, since the universe is quantized, and any such related math is inaccurate. If one were to trace a line along all of the little pixilated plancks which make up the circumference of the most perfect "circle" in the universe one would find that pi is actually equal to 4, rather than the theoretical value of 3.14159...
Wow. So you think the whole world has been underestimating the circumference and area of every circle ever, by about 25 percent? That the dimensions of a 55 gallon drum, calculated based on 3.14159, are wrong? It actually contains closer to 70 gallons, but nobody in a very moneydriven industry has noticed they are shipping more oil than they thought? That every round tower ever built has required more bricks than it should have, because the circumference was about 25% bigger than the math said, but not one single mason or architect in history ever noticed they consistently needed 25% more bricks? That the commonly available measuring tool known as a Pi Tape http://www.amazon.com/LufkinW606PDExecutiveDiameterEngineers/dp/B0002JT2AI/ref=sr_1_1?ie=UTF8&qid=1455477163&sr=81&keywords=pi+tape+measure (http://www.amazon.com/LufkinW606PDExecutiveDiameterEngineers/dp/B0002JT2AI/ref=sr_1_1?ie=UTF8&qid=1455477163&sr=81&keywords=pi+tape+measure), which gives you the diameter of a round object you wrap it around, is off by 25%? I have used a Pi Tape, me, myself, personally, and guess what: it wasn't 25% off, not ever, not even close!

That's not really my point. My point is that there is no such thing as a circle in the universe. So therefore pi != 3.14159...
... the error in the logic is that circles do not exist.
I would contend that you are wrong on this. While perfect circles may not actually exist in the physical world, as abstract mathematical concepts they absolutely do exist. What's more, these abstract conceptual circles have useful real world applications, albeit implicitly as approximations.

Wow. So you think the whole world has been underestimating the circumference and area of every circle ever, by about 25 percent? That the dimensions of a 55 gallon drum, calculated based on 3.14159, are wrong? It actually contains closer to 70 gallons, but nobody in a very moneydriven industry has noticed they are shipping more oil than they thought?
Using pi to calculate area does not make any sense if the shape is a polygon, since unlike a circle, the perimeter of a polygon is not related to its area.
Please refer to my example above of a triangle and a square with identical perimeters having different interior areas.

Wow. So you think the whole world has been underestimating the circumference and area of every circle ever, by about 25 percent? That the dimensions of a 55 gallon drum, calculated based on 3.14159, are wrong? It actually contains closer to 70 gallons, but nobody in a very moneydriven industry has noticed they are shipping more oil than they thought?
Using pi to calculate area does not make any sense if the shape is a polygon, since unlike a circle, the perimeter of a polygon is not related to its area.
Please refer to my example above of a triangle and a square with identical perimeters having different interior areas.
I fail to see any polygon mentioned in the post you were answering!
In Australia these drums are 44 (imperial) gallons and are cylindrical, so π would be very relevant.
So I can't make sense of your statement "Using pi to calculate area does not make any sense if the shape is a polygon"!

Using pi to calculate area does not make any sense if the shape is a polygon, since unlike a circle, the perimeter of a polygon is not related to its area.
As rabinoz points out, I'm not talking about a polygon. I'm talking about a circle. If you're talking about polygons in reference to your own earlier statement that
circles do not actually exist, since the universe is quantized, and any such related math is inaccurate
well, even a jagged notperfect circle does in fact follow the calculations of Pi to enough decimal places to be accurate enough to satisfy the needs of engineering, science, and math.
Please refer to my example above of a triangle and a square with identical perimeters having different interior areas.
I don't think anybody is disputing the difference between triangles and squares. Just don't see how it relates to circles, or Pi. But, since you've brought them up: why did you choose a square to use as your perimeter shape, why not a triangle? Same principle, right? Only....collapsing corners forever on a triangle gives you an answer of about 5.19. Why not some other regular polygon? Maybe a hexagon? That would give about 3.46 instead of 4 for Pi. In fact, the CORRECT way to approach an accurate value for the perimeter of a circle is to take polygons of ever more and more sides, no collapsing corners. For each polygon, calculate their perimeter, and do this over and over. If you do, you will find the value approaching the true circumference of the circle. I did this once as a student, in high school, when it became time to move from rote memorization of Pi and Pi R Squared into understanding of those concepts.

Ok Tom, here's a question. How would you go about finding the area of a circle? What formula do you prefer?

Ok Tom, here's a question. How would you go about finding the area of a circle? What formula do you prefer?
Or, as a good Zetetic SHOULD do, just go find a round thing in your house and measure it for yourself! For example, I happen to have a bunch of gallon paint cans handy. Diameter: 6 and 9/16 inches, circumference 20 and 3/4 inches. Ratio of those two numbers: 3.16 Slightly bigger than Pi due to imprecise measurement no doubt, but had Pi actually been 4 and the diameter accurate, the circumference would be 26 and 1/4 inches. I promise you I did not make a 5 and 1/2 inch measurement error in the circumference. Or maybe I got the circumference right, but botched the diameter measurement. With a 20 and 3/4 inch circumference and a Pi of 4, the diameter 'should' have been 5 and 3/16 inches. Again, I know I didn't make an error of 1 and 3/8 inches.

I'm completely lost as to why this became an abstract mathematics lecture. Does Polaris prove the earth is round or not?

Wow. So you think the whole world has been underestimating the circumference and area of every circle ever, by about 25 percent? That the dimensions of a 55 gallon drum, calculated based on 3.14159, are wrong? It actually contains closer to 70 gallons, but nobody in a very moneydriven industry has noticed they are shipping more oil than they thought?
Using pi to calculate area does not make any sense if the shape is a polygon, since unlike a circle, the perimeter of a polygon is not related to its area.
Please refer to my example above of a triangle and a square with identical perimeters having different interior areas.
I fail to see any polygon mentioned in the post you were answering!
In Australia these drums are 44 (imperial) gallons and are cylindrical, so π would be very relevant.
So I can't make sense of your statement "Using pi to calculate area does not make any sense if the shape is a polygon"!
The difference in area would be minuscule if you treated the cylinder as a perfect circle where pi = 3.14159.. or a really really pixilated circle one where pi = 4.
In calculating the area of a polygon, you cannot use A= pi * r^2 , as polygons do not have areas that are directly related to their perimeter. See the square vs triangle example. Polygons are not uniform shapes like circles are. The equation assumes a shape where perimeter is directly related to area.

well, even a jagged notperfect circle does in fact follow the calculations of Pi to enough decimal places to be accurate enough to satisfy the needs of engineering, science, and math.
Just look at the equation. Area = pi * radius * radius. It's assuming that the area is directly connected to the perimeter times the radius squared. You can't do that with a polygon.
Most of the additional length of the perimeter in the very pixilated circle in my example is in the very small steps at the edge of the circle, and does not add significant area to the whole of the object. Using that area equation just doesn't work, as it is a highly complex polygon and not a circle. The area of the object is obviously not the perimeter times the radius squared when much of the perimeter is so curled up at the edges like that.

"so much of the perimeter curled up at the edges", is it? If it's true of my circle, it's also true of your square. Those four "straight" lines you used to box in the circle? They are all kinds of jagged and crooked too, making each an unknown distance greater than the theoretical unit length. They might not even be equal lengths, for all we know. In fact, they could be infinitely long. We cannot know the true length of anything, including your collapsing corners box, and thus all geometry and trigonometry is useless.
Except...we know that in the real, physical world, geometry and trig are the opposite of useless. We know through experiments and observation that the objects we agree to call "circles" have a perimeter that measures 3.14159...... times the measured diameter of those objects. We use that number to calculate how much sheet metal it will take, when rolled into a cylinder, to create a drum of a desired diameter, and viola! The drum thus formed does indeed have the desired diameter! We use the same 3.14159..... times radius times radius to calculate how much area is enclosed by these objects we agree to call "circles" and when we check with (for example) liquid in a drum, guess what? There is as much liquid in the drum as the math said there would be! Whereas, if you take 4 as your value of Pi nd calculate the amount of sheet metal to use for a given diameter drum, when you build it your drum will have a diameter larger than you wanted.
BECAUSE PI ISN'T 4!!
Then when you do your volume calculation with the actual diameter and Pi=4, you will find your drum cannot hold the amount you calculated.
BECAUSE PI STILL ISN'T 4!!!
Didn't you do these very tests when you were a child in school? In my class we each were given a different length piece of construction paper. We measured its length, formed it into a circle by taping the ends, then measured its diameter. The whole class then reported their numbers, which the teacher wrote on the board. She then calculated the ratios, to demonstrate that every circle had the same ratio (give or take the measuring skill of children of course). We then filled the cylinders with a single layer of peas, and counted them as a rough measure of area. Again the numbers were called out to teacher, who applied Pi R squared to prove the rule, again subject to the imperfection of school children's construction.

you're not wrong that the c/d ratio of real, physical circles is not pi. pi isn't a real constant.
you're very wrong that this means that pi is a constant and that that constant is 4. you're wronger to imply that 4 is a better approximation of c/d for real circles than pi. your 'proof' is even more wronger. the 'crinkled up' perimeter of the square is never going to actually 'straighten' up in a way that gets closer and closer to the perimeter of the circle. no matter how much you zoom in, it will never appear to approximate the perimeter of a circle. if you were to keep zooming in on the circle's perimeter, you'll only ever see this, no matter how much you zoom in:
(http://i.imgur.com/apmublm.png?1)
except it wouldn't look so shitty since i presume nature to be way better at ms paint than i am.
That's not really my point. My point is that there is no such thing as a circle in the universe. So therefore pi != 3.14159...
i took your point to be that "if one were to trace a line along all of the little pixilated plancks which make up the circumference of the most perfect "circle" in the universe one would find that pi is actually equal to 4, rather than the theoretical value of 3.14159...." is this not right? my point is that the proof you offered to support your point is demonstrably incorrect.
also, why do you believe that space is discrete and not continuous? i think "it absolutely makes more sense to base science off of the observed and experienced rather than the theoretical and hypothetical," and there is no direct experimental evidence to suggest that space is discrete. i also don't get why you're more interested in the results of your thought experiment than you are with observed and experienced measurements of pi. you're being extremely pedantic.
but let's assume that it is. how would you go about "tracing a line" along all of the little pixelated plancks? more importantly, with what would you trace the line? wouldn't you have to make it with something also made out of little pixels? i don't know how to express this notion mathematically, but you can't unfold the little pixels into a line to measure their "length" in this manner. it's very similar to what i said above about how the perimeter of the square in your proof is never going to 'unfold' into a straight line, so it doesn't actually approximate the circle's diameter. you can't 'unfold' the perimeter of a planck unit.
here's a visual to try to get across what i'm saying in case i'm being unclear.
(http://i.imgur.com/tBCfJZs.png)(http://i.imgur.com/h4apMtw.png)
you're thinking of a pixelated circle like the image on the left, as if we could measure planck lengths with an even tinier piece of string or measuring tape or something. but if planck lengths are the smallest length, then we can't measure them with units smaller than planck lengths. the thing we use to measure the perimeter is also made of planck lengths and cannot be further subdivided, as on the right side.
i think maybe what i'm trying to get across is that you're thinking of a circle too much in terms of its circumference, and a circle should be thought of in terms of radius. it's the shape with a constant radius; or, it's the shape for which every point on its perimeter is equidistant from the center. making circle pixelated instead of continuous doesn't change anything. the definition of a pxelated circle would then be something like the shape for which each point on the perimeter is as equidistant as possible from the center. if you make such a shape, and if count the number of pixels composing the perimeter and the diameter, you'll get a ratio very close to 3.14159..., not 4. the only possible way to get 4 is to measure your discontinuous/pixelated circle with a continuous line; in other words, to subdivide the space that you're defining as indivisible. bad methodology.

. . . . . . . . . . . . . . . .
Don't let Tom Bishop bother you he is just trying to beat Jroa and TheEngineer at "the other place" in a competition for the champion derailer.
Jroa and TheEngineer do occasionally present an argument, so I am sure Tom Bishop wins hands down.
In the meantime π ≈ 3.141592653589793238462643383279502884197169399375105820974944592
roughly, see here for more (don't go there if you are not prepared to wait!):http://3.141592653589793238462643383279502884197169399375105820974944592.com/index314159.html (http://3.141592653589793238462643383279502884197169399375105820974944592.com/index314159.html).

what does pi, or the volume of an orange for that matter have to do with polaris? This thread seemed like a golden opportunity to show how ridiculous the accepted orbit wobble and spin of our earth simply doesn't align with what we actually observe in the heavens. Or that the advertised distance from earth of our own "star" has changed half a dozen times since the inception of the heliocentric theory, yet we are to believe science has the vaguest idea the proximity of stars in distant galaxies. It is their contension that they are trillions and trillions of miles away, the only way to account for the apparent lack of significanct stellar parallax on our spinning wobbling elliptical slingshot around a star 93,000,000 miles away (radial distance btw). Lucky us that polaris always hangs out above our north magnetic pole for our viewing convenience during this journey.

"so much of the perimeter curled up at the edges", is it? If it's true of my circle, it's also true of your square. Those four "straight" lines you used to box in the circle? They are all kinds of jagged and crooked too, making each an unknown distance greater than the theoretical unit length. They might not even be equal lengths, for all we know. In fact, they could be infinitely long. We cannot know the true length of anything, including your collapsing corners box, and thus all geometry and trigonometry is useless.
Except...we know that in the real, physical world, geometry and trig are the opposite of useless. We know through experiments and observation that the objects we agree to call "circles" have a perimeter that measures 3.14159...... times the measured diameter of those objects. We use that number to calculate how much sheet metal it will take, when rolled into a cylinder, to create a drum of a desired diameter, and viola! The drum thus formed does indeed have the desired diameter! We use the same 3.14159..... times radius times radius to calculate how much area is enclosed by these objects we agree to call "circles" and when we check with (for example) liquid in a drum, guess what? There is as much liquid in the drum as the math said there would be! Whereas, if you take 4 as your value of Pi nd calculate the amount of sheet metal to use for a given diameter drum, when you build it your drum will have a diameter larger than you wanted.
BECAUSE PI ISN'T 4!!
Then when you do your volume calculation with the actual diameter and Pi=4, you will find your drum cannot hold the amount you calculated.
BECAUSE PI STILL ISN'T 4!!!
Again, Area = Perimeter * Radius * Radius only works for a circle. Only a circle has the necessary uniform shape for that equation to work. It does not work for a square, a triangle, or a complex polygon that looks like a circle.
Didn't you do these very tests when you were a child in school? In my class we each were given a different length piece of construction paper. We measured its length, formed it into a circle by taping the ends, then measured its diameter. The whole class then reported their numbers, which the teacher wrote on the board. She then calculated the ratios, to demonstrate that every circle had the same ratio (give or take the measuring skill of children of course). We then filled the cylinders with a single layer of peas, and counted them as a rough measure of area. Again the numbers were called out to teacher, who applied Pi R squared to prove the rule, again subject to the imperfection of school children's construction.
That test is inaccurate, as you are not actually measuring the imperfections of the circumference. You are laying waypoints on a map to get a distance without considering the mountains.

also, why do you believe that space is discrete and not continuous? i think "it absolutely makes more sense to base science off of the observed and experienced rather than the theoretical and hypothetical," and there is no direct experimental evidence to suggest that space is discrete. i also don't get why you're more interested in the results of your thought experiment than you are with observed and experienced measurements of pi. you're being extremely pedantic.
There is experimental evidence. Space and time can be demonstrated to be quantized by simply walking from one end of your room through a door at the other end.
See: http://barang.sg/index.php?view=achilles&part=8
8. Is space quantized?
Men have long wondered if matter is infinitely divisible. For example, can you keep halving a piece of wood forever to obtain ever tinier pieces of wood? Nowadays, we know that the answer is no. Matter is atomic in nature and not infinitely divisible.
Surprisingly, the same question may be asked of space. (Not to mention time.)
Thus, when Zeno edges his points ever closer to the doorway, he tacitly assumes that the distance between a point and the doorway may be as small as one likes. This is to assume that space is infinitely divisible.
(http://barang.sg/caterpillar/achilles_doorway.jpg)
But if space, like matter, is atomic in nature, there will actually be a smallest distance beyond which we can divide space no further, as it were. And so there will be a limit to how close Zeno’s points can approach the doorway, meaning that his sequence of points must eventually come to a halt!
We can illustrate this concretely because contemporary physicists actually believe that space is atomic in nature, or quantized, as they like to say. (From the Latin term quantum, meaning amount.)
Thus, the “Planck length,” named after the German physicist Max Planck, is supposed to be the smallest quantum of length permitted by nature, so far as physicists know.
The underlying physics does not matter here (it has nothing to do with a tortoise), but the Planck length is about 1.6 × 1035 m, which is a miniscule distance indeed. Roughly speaking, it stands to an atom as an atom stands to the sun!
Let’s work with this number and see what happens.
Consider our table from before, where some relevant lines have been added. Notice that once Achilles reaches point 117, his distance to the doorway has become shorter than the Planck length.
Point Meters to doorway
1 1
2 1/2
3 1/4
.
.
. .
.
.
116 2.4 × 10^{35}
117 1.2 × 10^{35}
.
.
. .
.
.
But if the physicists are right, this distance is too short to be of spatial significance. This means that we cannot regard point 117 and the doorway as being two separate locations. Given the atomic nature of space, these locations must be regarded as being one and the same.
So the last point that Zeno can really lay down is point 116. And once Achilles reaches it, his next motion takes him to the doorway because there is no other location “in between” for him to occupy.
This sounds incredible, but if space is quantized, that is how it is.
Now, on the solution being considered, space must be quantized in this way because, as explained previously, Zeno’s sequence must contain a last point if Achilles is to “escape” to the doorway, and quantizing space seems to be the only way to ensure this. So even if the physicists had not yet discovered it, Zeno’s paradox already reveals the atomic nature of space.
Likewise, no pie can be infinitely sliced in the manner shown before because, beyond a certain point, the slices will be too thin to be spatially distinguishable and no further “slicing” can meaningfully occur.
This diagnosis, if correct, would be truly remarkable. It’s one thing to believe that space is quantized on detailed experimental grounds (like modern physicists), but quite another to deduce it simply by reflecting on whether a man can reach a doorway!
Does Zeno’s paradox really show that space must be quantized in this way?
Well, it really depends on the considerations of the previous section. In particular, is it really true that Achilles cannot reach the doorway unless Zeno’s sequence contains a last point for him to cross?
To answer this question, we must sharpen those considerations a little further. And we can do this by considering an unusual device known as Thomson’s lamp.
If you can make it through the door, it is a proof that space is quantized.
but let's assume that it is. how would you go about "tracing a line" along all of the little pixelated plancks? more importantly, with what would you trace the line? wouldn't you have to make it with something also made out of little pixels? i don't know how to express this notion mathematically, but you can't unfold the little pixels into a line to measure their "length" in this manner. it's very similar to what i said above about how the perimeter of the square in your proof is never going to 'unfold' into a straight line, so it doesn't actually approximate the circle's diameter. you can't 'unfold' the perimeter of a planck unit.
here's a visual to try to get across what i'm saying in case i'm being unclear.
(http://i.imgur.com/tBCfJZs.png)(http://i.imgur.com/h4apMtw.png)
you're thinking of a pixelated circle like the image on the left, as if we could measure planck lengths with an even tinier piece of string or measuring tape or something. but if planck lengths are the smallest length, then we can't measure them with units smaller than planck lengths. the thing we use to measure the perimeter is also made of planck lengths and cannot be further subdivided, as on the right side.
i think maybe what i'm trying to get across is that you're thinking of a circle too much in terms of its circumference, and a circle should be thought of in terms of radius. it's the shape with a constant radius; or, it's the shape for which every point on its perimeter is equidistant from the center. making circle pixelated instead of continuous doesn't change anything. the definition of a pxelated circle would then be something like the shape for which each point on the perimeter is as equidistant as possible from the center. if you make such a shape, and if count the number of pixels composing the perimeter and the diameter, you'll get a ratio very close to 3.14159..., not 4. the only possible way to get 4 is to measure your discontinuous/pixelated circle with a continuous line; in other words, to subdivide the space that you're defining as indivisible. bad methodology.
I see in the image on the right that you were able to trace plancks with other plancks to create a perimeter to identify the boundaries of a shape. No subdivision of space was required.

what does pi, or the volume of an orange for that matter have to do with polaris? This thread seemed like a golden opportunity to show how ridiculous the accepted orbit wobble and spin of our earth simply doesn't align with what we actually observe in the heavens. Or that the advertised distance from earth of our own "star" has changed half a dozen times since the inception of the heliocentric theory, yet we are to believe science has the vaguest idea the proximity of stars in distant galaxies. It is their contension that they are trillions and trillions of miles away, the only way to account for the apparent lack of significanct stellar parallax on our spinning wobbling elliptical slingshot around a star 93,000,000 miles away (radial distance btw). Lucky us that polaris always hangs out above our north magnetic pole for our viewing convenience during this journey.
It is rather common for hard questions to be ignored, given obtuse answers, or the thread be derailed here.
My guess with the limited knowledge I have about FE models is it has to do with celestial gears and the North Star is at the center of one of the gears. There could be another explanation involving aether and/or the firmament/dome.
The RE answer is easy to find with an internet search, which I suspect you already know.

It's not off topic. It is important for the topic to understand that the Geometry of the Ancient Greeks is simply wrong, and does not reflect reality. Zeno’s paradox alone leads to the conclusion that space is quantized, and therefore circles do not exist and pi is not 3.14159...
We see from experiment that we are able to walk through doors, and therefore we must design our science to make it possible to walk through doors, and not imagine some hypothetical construct imaging space and time as continuous. We must design our science from the observed and experienced, not idealistic theories.

It's not off topic. It is important for the topic to understand that the Geometry of the Ancient Greeks is simply wrong, and does not reflect reality. Zeno’s paradox alone leads to the conclusion that circles do not exist and therefore pi is not 3.14159...
You know something you might want tangle yourself in Greek paradoxes, but I'll stick to the mundane world where circles exist and pi is well defined.
Besides a distance (as is a circumference) is not measured by counting "plancks", but with a scale of finite resolution, be it a tape measure or even "counting" wavelengths of light.
By the way Zeno’s paradox is no paradox if looked at reasonably.

It's not off topic. It is important for the topic to understand that the Geometry of the Ancient Greeks is simply wrong, and does not reflect reality. Zeno’s paradox alone leads to the conclusion that space is quantized, and therefore circles do not exist and pi is not 3.14159...
We see from experiment that we are able to walk through doors, and therefore we must design our science to make it possible to walk through doors, and not imagine some hypothetical construct imaging space and time as continuous. We must design our science from the observed and experienced, not idealistic theories.
I just got back from taking my dog for a walk. Not only did I make it 1/2 way I made it to the end of the park and back to my boat.
During this walk my dog was able to catch up to me after lagging behind to smell different things. Reason being I was moving slower than her.
I threw a ball for her to chase and it traveled away from me and landed on the grass.
All observed and experienced by me.
Are you sure that the Zeno's paradoxes are not just thought experiments/exercises?
I am just asking since my experiences walking my dog surely seemed like reality.
Just like when I use 3.14 to determine things like the circumference of a circle and the answer being correct.

It's not off topic. It is important for the topic to understand that the Geometry of the Ancient Greeks is simply wrong, and does not reflect reality. Zeno’s paradox alone leads to the conclusion that space is quantized, and therefore circles do not exist and pi is not 3.14159...
We see from experiment that we are able to walk through doors, and therefore we must design our science to make it possible to walk through doors, and not imagine some hypothetical construct imaging space and time as continuous. We must design our science from the observed and experienced, not idealistic theories.
I just got back from taking my dog for a walk. Not only did I make it 1/2 way I made it to the end of the park and back to my boat.
During this walk my dog was able to catch up to me after lagging behind to smell different things. Reason being I was moving slower than her.
I threw a ball for her to chase and it traveled away from me and landed on the grass.
All observed and experienced by me.
Are you sure that the Zeno's paradoxes are not just thought experiments/exercises?
I am just asking since my experiences walking my dog surely seemed like reality.
Zeno's paradoxes are scathing criticisms of the theory that space and time are continuous. Since you were able to do all of those things, it is a proof that space and time is discrete.
Just like when I use 3.14 to determine things like the circumference of a circle and the answer being correct.
It's correct in the mathematical fantasy of the Ancient Greeks. Incorrect in reality.

It's not off topic. It is important for the topic to understand that the Geometry of the Ancient Greeks is simply wrong, and does not reflect reality. Zeno’s paradox alone leads to the conclusion that space is quantized, and therefore circles do not exist and pi is not 3.14159...
We see from experiment that we are able to walk through doors, and therefore we must design our science to make it possible to walk through doors, and not imagine some hypothetical construct imaging space and time as continuous. We must design our science from the observed and experienced, not idealistic theories.
I just got back from taking my dog for a walk. Not only did I make it 1/2 way I made it to the end of the park and back to my boat.
During this walk my dog was able to catch up to me after lagging behind to smell different things. Reason being I was moving slower than her.
I threw a ball for her to chase and it traveled away from me and landed on the grass.
All observed and experienced by me.
Are you sure that the Zeno's paradoxes are not just thought experiments/exercises?
I am just asking since my experiences walking my dog surely seemed like reality.
Zeno's paradoxes are scathing criticisms of the theory that space and time are continuous. Since you were able to do all of those things, it is a proof that space and time is discrete.
Just like when I use 3.14 to determine things like the circumference of a circle and the answer being correct.
It's correct in the mathematical fantasy of the Ancient Greeks. Incorrect in reality.
So I just measured the circumference and diameter of a circular can on my desk and obtained C = 167 mm, D = 53 mm.
Is this then not reality!??

So I just measured the circumference and diameter of a circular can on my desk and obtained C = 167 mm, D = 53 mm.
Is this then not reality!??
No, you drew a line through zig zags and came up with a figure that does not reflect reality.

So I just measured the circumference and diameter of a circular can on my desk and obtained C = 167 mm, D = 53 mm.
Is this then not reality!??
No, you drew a line through zig zags and came up with a figure that does not reflect reality.
Wait, so if I take a piece of string, measure it, wrap it around the bottom of a can, measure the diameter, apply Pi to the equation that gives me the circumference, and get the same result as my initial measurement, this is not reality either?

So I just measured the circumference and diameter of a circular can on my desk and obtained C = 167 mm, D = 53 mm.
Is this then not reality!??
No, you drew a line through zig zags and came up with a figure that does not reflect reality.
No, the can was round, at least to the Planck limit. Thus, pi = 3.14!

So I just measured the circumference and diameter of a circular can on my desk and obtained C = 167 mm, D = 53 mm.
Is this then not reality!??
No, you drew a line through zig zags and came up with a figure that does not reflect reality.
No, the can was round, at least to the Planck limit. Thus, pi = 3.14!
I dont completely follow, but I think the point Tom is trying to make is that circles don't exist in nature. It's obvious to everyone that the coca cola plant figured out a long time ago how to manufacture a can to be round.

So I just measured the circumference and diameter of a circular can on my desk and obtained C = 167 mm, D = 53 mm.
Is this then not reality!??
No, you drew a line through zig zags and came up with a figure that does not reflect reality.
No, the can was round, at least to the Planck limit. Thus, pi = 3.14!
I dont completely follow, but I think the point Tom is trying to make is that circles don't exist in nature. It's obvious to everyone that the coca cola plant figured out a long time ago how to manufacture a can to be round.
... so the 'real' circumference in nature will be longer than the perfect circle, and sometimes much longer than a perfect circle, because of all the imperfections made by nature, right? But then 'pi' would never be a constant (e.g. = 4)  it would just be bigger than 3,1415.
I made the silly experiment where I measured the volume of an orange by submerging it into water (and having the water filling all the imperfections in the surface of the orange). Then I backcalculated pi after I knew the 'true' volume, and it was damn close to 3,14. Just sayin'.
Now; can we get back to the interesting discussion of the findings of Polaris and the flat earth?
I'm sure there're many mathematical and medical fora with experts that would love to discuss Tom's new findings of pi and his cure for cancer.

This is pure nonsense. The illustration I posted is to scale. The angles and distances are right there in front of you. I'll post it again below. Notice that the distance needed to see a change in altitude from 20° to 10° (9,040 miles) is greater than the distance needed for polaris to drop from 90° to 20° (8,557 miles). If the diagram were to continue, the distance needed for Polaris to drop from 10° to 5° is more than the distance needed for it to drop from 90° to 10°, about 17,835 miles (a total of 35,433 miles from 90° to 5°). To see Polaris at 0°, the distance needed is infinity.
It would therefore be impossible to see the apparent altitude of any celestial object drop at a constant rate due to perspective if it was moving away at a constant speed. You can draw it out and measure the angles for yourself if you like, or just use an online right triangle calculator.
Triangles don't lie.
(https://www.filesanywhere.com/FS/M.aspx?v=8c70678d619cb478aca7)
Under traditional perspective it is also impossible for the sun to ever set. However, Samuel Birley Rowbotham teaches us in Earth Not a Globe that we must adopt our concept of perspective from real world experience and observations, not some mathematical concept.
I'm sorry, but this is just lame. We're talking about angles here, not "string theory ". Angles, like distances, are simply a way to quantify and/or describe the relationship physical objects have with one another in 3D space.
You say, "Samuel Birley Rowbotham teaches us in Earth Not a Globe that we must adopt our concept of perspective from real world experience and observations". It's terrific that you have such faith in the authority of your teacher, but he apparently has never really done much observing. You would do better to ask an architect, a navigator, a surveyor, or a cartographer, people who successfully use geometry everyday in real world observations. Applied mathematics like trigonometry were derived from pure observation and have been tried and tested for literally thousands of years. Trigonometry is not a theory. It works because it's true. Trigonometric relationships in the physical world are as certain as 2+2=4. They are as certain as any physical law. So if the celestial objects in your FE model do not obey physical laws then they must not be physical and so you need to stop pretending that Flat Earth Theory is not a religion, because that's exactly what you are saying.

Exactly. Zetetics claim to hold observation over theory, but then this nonsense about plancks, and PI = 4? Take a look around your home (AKA "conduct an observation"). Find an object that the rest of us would call 'round' and look at it. Does it appear to be round? If so, then the same philosophy that makes you say "The world LOOKS flat, I guess it must BE flat" should also lead you to the conclusion "This object LOOKS like a circle, I guess it must BE a circle". Wrap a string around the can, measure its length. "The strings MEASURES as if it were about 3.14 times the diameter, I guess it must BE 3.14 times the diameter." Repeat for other round objects. "Every round I object I MEASURE has a perimeter of 3.14 times its diameter, I guess ALL round objects exhibit that relationship"

... so the 'real' circumference in nature will be longer than the perfect circle, and sometimes much longer than a perfect circle, because of all the imperfections made by nature, right? But then 'pi' would never be a constant (e.g. = 4)  it would just be bigger than 3,1415.
The most perfect circle possible in a quantized universe would have a pi of 4. Most other circles may have slightly different values for pi, as they are less perfect, but that is mostly irrelevant to the discussion since the continuous universe of the Ancient Greeks also ignores imperfect circles. Any opposing model to the standard ancient one would assume the most perfect circle possible as well when coming up with a value for pi.
I'm sorry, but this is just lame. We're talking about angles here, not "string theory ". Angles, like distances, are simply a way to quantify and/or describe the relationship physical objects have with one another in 3D space.
You say, "Samuel Birley Rowbotham teaches us in Earth Not a Globe that we must adopt our concept of perspective from real world experience and observations". It's terrific that you have such faith in the authority of your teacher, but he apparently has never really done much observing. You would do better to ask an architect, a navigator, a surveyor, or a cartographer, people who successfully use geometry everyday in real world observations. Applied mathematics like trigonometry were derived from pure observation and have been tried and tested for literally thousands of years. Trigonometry is not a theory. It works because it's true. Trigonometric relationships in the physical world are as certain as 2+2=4. They are as certain as any physical law. So if your flat earth sun does not obey physical laws then it cannot be physical and so you need to stop pretending that Flat Earth Theory is not a religion, because that's exactly what you are saying.
Why are you trying to use unverified ancient geometry/trigonometry as a proof of anything?
The Ancient Greeks did not verify that circles actually exist, and they did not verify that perspective lines actually stretch into infinity as they theorized.
Exactly. Zetetics claim to hold observation over theory, but then this nonsense about plancks, and PI = 4? Take a look around your home (AKA "conduct an observation"). Find an object that the rest of us would call 'round' and look at it. Does it appear to be round? If so, then the same philosophy that makes you say "The world LOOKS flat, I guess it must BE flat" should also lead you to the conclusion "This object LOOKS like a circle, I guess it must BE a circle". Wrap a string around the can, measure its length. "The strings MEASURES as if it were about 3.14 times the diameter, I guess it must BE 3.14 times the diameter." Repeat for other round objects. "Every round I object I MEASURE has a perimeter of 3.14 times its diameter, I guess ALL round objects exhibit that relationship"
When I pick up a can it looks like an object. It looks like a shape of some sort. I can safely say that it is an object. Assigning words like "circle" or "cylinder" to that shape brings me into the continuous universe of the Ancient Greeks, which we are increasingly coming to find were full of baloney, and whose science can be disproven by a simple act of walking through a door.

... so the 'real' circumference in nature will be longer than the perfect circle, and sometimes much longer than a perfect circle, because of all the imperfections made by nature, right? But then 'pi' would never be a constant (e.g. = 4)  it would just be bigger than 3,1415.
The most perfect circle possible in a quantized universe would have a pi of 4. Most other circles may have slightly different values for pi, as they are less perfect, but that is mostly irrelevant to the discussion since the continuous universe of the Ancient Greeks also ignores imperfect circles. Any opposing model to the standard ancient one would assume the most perfect circle as well.
I'm sorry, but this is just lame. We're talking about angles here, not "string theory ". Angles, like distances, are simply a way to quantify and/or describe the relationship physical objects have with one another in 3D space.
You say, "Samuel Birley Rowbotham teaches us in Earth Not a Globe that we must adopt our concept of perspective from real world experience and observations". It's terrific that you have such faith in the authority of your teacher, but he apparently has never really done much observing. You would do better to ask an architect, a navigator, a surveyor, or a cartographer, people who successfully use geometry everyday in real world observations. Applied mathematics like trigonometry were derived from pure observation and have been tried and tested for literally thousands of years. Trigonometry is not a theory. It works because it's true. Trigonometric relationships in the physical world are as certain as 2+2=4. They are as certain as any physical law. So if your flat earth sun does not obey physical laws then it cannot be physical and so you need to stop pretending that Flat Earth Theory is not a religion, because that's exactly what you are saying.
Why are you trying to use unverified ancient geometry/trigonometry as a proof of anything?
The Ancient Greeks did not verify that circles actually exist, and they did not verify that perspective actually stretches into infinity as theorized.
It's at this point that I would normally assume that I'm being trolled. What a joke.
So not only is the earth flat, geometry and trigonometry, as have been used successfully for thousands of years for countless real world applications, are also wrong? Yes, you would have to believe that in order to defend your impossible flat earth, so it actually makes sense that you would claim such a thing. Is arithmetic also wrong? The ridiculous thing, though, is that you can test the accuracy of geometry yourself, any time, any where, on paper, or in three dimensions.
The wellknown relationship between apparent height and distance, for example, is used all the time for measuring large or distant objects like trees and mountains. You can try it with your house, your spouse, your basketball hoop, sasquatch, anything you can physically measure. For anyone interested, here's how to do it: http://www.instructables.com/id/Usingaclinometertomeasureheight/ (http://www.instructables.com/id/Usingaclinometertomeasureheight/)
(http://www.tiem.utk.edu/~gross/bioed/bealsmodules/triangle3.gif)
Here's how to make a climometer: http://www.wikihow.com/MakeaClinometer (http://www.wikihow.com/MakeaClinometer)
Here's a right triangle calculator: http://www.cleavebooks.co.uk/scol/calrtri.htm (http://www.cleavebooks.co.uk/scol/calrtri.htm)
Here's an online size calculator: http://sizecalc.com (http://sizecalc.com)

The Ancient Greek theory that perspective lines stretch infinitely into the distance and that the Vanishing Point is infinitely away from the observer isn't proven by measuring the height of a tree.

The Ancient Greek theory that perspective lines stretch infinitely into the distance and that the Vanishing Point is infinitely away from the observer isn't proven by measuring the height of a tree.
Here's what is proven by greek geometry and how it relates to trees and your comment:
We know the relationships between angles and sides of a triangle:
(http://www.csgnetwork.com/righttri.gif)
The longer the base of the triangle, the smaller the angle "A" will get:
(http://www.tiem.utk.edu/~gross/bioed/bealsmodules/triangle3.gif)
Angle "A" is the angle of elevation of the top of the tree as perceived by the viewer. The farther away, the smaller the tree will look and the closer to the horizon the top of the tree will appear. Imagine a star at the top of the tree. In order for that star to meet the horizon, angle "A" will have to equal 0 degrees. Do you know how long the base of the triangle will have to be for angle "A" to shrink to zero?
Answer: infinity.
It's simple mathematics.
(https://www.filesanywhere.com/FS/M.aspx?v=8c70688f5a626eb87266)

Do you know how long the base of the triangle will have to be for angle "A" to shrink to zero?
Answer: infinity.
It's simple mathematics.
There you go, using that Ancient Greek nonsense math where things are continuous and divide or stretch into infinities. You are assuming conclusions based on an Ancient Greek fantasy model where things are continuous, rather than an experience of the real world. Zeno put the theory of a continuous universe to bed (http://barang.sg/index.php?view=achilles&part=1).

Do you know how long the base of the triangle will have to be for angle "A" to shrink to zero?
Answer: infinity.
It's simple mathematics.
There you go, using that Ancient Greek nonsense math where things are continuous and divide or stretch into infinities. You are assuming conclusions based on an Ancient Greek fantasy model where things are continuous, rather than an experience of the real world. Zeno put the theory of a continuous universe to bed (http://barang.sg/index.php?view=achilles&part=8).
No. I'm using basic trigonometry to show you how perspective works. Like I said, anybody can test it for themselves at anytime in the real world using the information I posted above. You will always lose this debate against observable reality. Trigonometry is simply a way to quantify what we observe. It always works.

1) I really don't know where you get this idea from observations of the real world: "the most perfect circle possible in a quantized universe would have a pi of 4". Beyond the fact that your observations of the world do not contain "the most perfect circle possible in a quantized universe", if it did contain such a circle, you would only be able to observe and experiment upon it with real world tools. Remember, on the wiki it is claimed that a Zetetic "bases his conclusions on experimentation and observation rather than on an initial theory that is to be proved or disproved." So, let's do that. Let's abandon our "initial theory" from grade school that Pi has ANY fixed value, whether that be 4, or 3.14, or something else. Maybe every circle has a different ratio of perimeter to diameter, who knows? Let's find out! OBSERVE a round object (and please, let's not pretend we don't know perfectly well what that word "round" means in the common usage) and perform an EXPERIMENT upon it, measuring its circumference and diameter. Repeat with another round object. Go again, and again, and again. Do not speak of plancks, you cannot observe them and therefor they are the very definition of an "initial theory." And if there are no plancks, as far as you can tell, then the perimeter of each round object is exactly what you measure it to be. I have done this experiment as covered in a previous post, and I urge anyone still reading along with us to conduct the experiment yourself. No doubt you can find at least three different round objects in easy reach, go measure them.
2) You seem to hold Zeno as some sort of pinnacle of Greek science. Your statement "the Ancient Greeks...whose science can be disproven by a simple act of walking through a door" is clearly a reference to the earlier discussion of Zeno's Paradox. While it is true that the impact of his work is important and debated to this day, there is plenty more to Greek science than Zeno (who was more a philosopher than a scientist or mathematician anyway). The Greeks were the first to observe electricity and magnetism, for example. They made great strides in medicine for their time. Pythagorus, in addition to his famous work with triangles, developed early music theory with his study of vibration versus string length. For crying out loud, they nearly invented calculus!
3) You characterize the proofs offered for the commonly accepted value for Pi as "unverified ancient geometry/trigonometry", a phrase that elicited an actual LOL from me when I read it. Unverified? The Greeks built temples, some of which stand today, based on that geometry! So has every civilization since, including our own. As brainsandgravy above has posted, you can very easily verify it for yourself, any school worthy of the title will asign its geometry and trigonometry students homework or classroom activities doing exacly that, and I again urge any undecided readers to go out and perform those tests for their own edification.

Do you know how long the base of the triangle will have to be for angle "A" to shrink to zero?
Answer: infinity.
It's simple mathematics.
There you go, using that Ancient Greek nonsense math where things are continuous and divide or stretch into infinities. You are assuming conclusions based on an Ancient Greek fantasy model where things are continuous, rather than an experience of the real world. Zeno put the theory of a continuous universe to bed (http://barang.sg/index.php?view=achilles&part=8).
No. I'm using basic trigonometry to show you how perspective works. Like I said, anybody can test it for themselves at anytime in the real world using the information I posted above. You will always lose this debate against observable reality. Trigonometry is simply a way to quantify what we observe. It always works.
The math you are using is continuous and, therefore, wrong. We don't live in a continuous universe.
If you can disprove Zeno's Paradoxes which act as a disproof of that sort of math, you may use it as a rebuttal.
1) I really don't know where you get this idea from observations of the real world: "the most perfect circle possible in a quantized universe would have a pi of 4". Beyond the fact that your observations of the world do not contain "the most perfect circle possible in a quantized universe", if it did contain such a circle, you would only be able to observe and experiment upon it with real world tools. Remember, on the wiki it is claimed that a Zetetic "bases his conclusions on experimentation and observation rather than on an initial theory that is to be proved or disproved." So, let's do that.
The matter of whether the most perfect circle actually exists or not is rather immaterial to the discussion. Observation of perfect circles do not occur in the Ancient Greek continuous universe, either. But the value for pi in their universe assumes a perfect circle nonetheless. Using that as a baseline, any opposing model must also assume the most perfect circle for that value of pi for any relevant comparison.
It's not really a direct assertion that a most perfect circle exists, but rather the only way to compare a model to the Ancient Greek one which stupidly assumes that perfect circles exist.
Let's abandon our "initial theory" from grade school that Pi has ANY fixed value, whether that be 4, or 3.14, or something else. Maybe every circle has a different ratio of perimeter to diameter, who knows? Let's find out! OBSERVE a round object (and please, let's not pretend we don't know perfectly well what that word "round" means in the common usage) and perform an EXPERIMENT upon it, measuring its circumference and diameter. Repeat with another round object. Go again, and again, and again. Do not speak of plancks, you cannot observe them and therefor they are the very definition of an "initial theory." And if there are no plancks, as far as you can tell, then the perimeter of each round object is exactly what you measure it to be. I have done this experiment as covered in a previous post, and I urge anyone still reading along with us to conduct the experiment yourself. No doubt you can find at least three different round objects in easy reach, go measure them.
If I take an object like an orange, how am I supposed to measure the imperfections and dimples on the surface to get an accurate circumference? If I tied a string around it would only be a guess, since obviously, the string isn't going into the dimples.
The Greeks were the first to observe electricity and magnetism, for example. They made great strides in medicine for their time. Pythagorus, in addition to his famous work with triangles, developed early music theory with his study of vibration versus string length. For crying out loud, they nearly invented calculus!
The Ancient Greeks also believed that flies spontaneously generated from rotting meat.
3) You characterize the proofs offered for the commonly accepted value for Pi as "unverified ancient geometry/trigonometry", a phrase that elicited an actual LOL from me when I read it. Unverified? The Greeks built temples, some of which stand today, based on that geometry! So has every civilization since, including our own. As brainsandgravy above has posted, you can very easily verify it for yourself, any school worthy of the title will asign its geometry and trigonometry students homework or classroom activities doing exacly that, and I again urge any undecided readers to go out and perform those tests for their own edification.
A baby could build a temple out of blocks without knowing anything about the alleged correctness of Geometry.

Do you know how long the base of the triangle will have to be for angle "A" to shrink to zero?
Answer: infinity.
It's simple mathematics.
There you go, using that Ancient Greek nonsense math where things are continuous and divide or stretch into infinities. You are assuming conclusions based on an Ancient Greek fantasy model where things are continuous, rather than an experience of the real world. Zeno put the theory of a continuous universe to bed (http://barang.sg/index.php?view=achilles&part=8).
No. I'm using basic trigonometry to show you how perspective works. Like I said, anybody can test it for themselves at anytime in the real world using the information I posted above. You will always lose this debate against observable reality. Trigonometry is simply a way to quantify what we observe. It always works.
The math you are using is continuous and, therefore, wrong. We don't live in a continuous universe.
If you can disprove Zeno's Paradoxes which act as a disproof of that sort of math, you may use it as a rebuttal.
The math I am using is easily verifiable. I put it right there in front of you. It's triangles, the most basic trigonometry. Trigonometry has been tried, tested, verified, and validated for thousands of years. Instead of repeating your articles of faith over and over, I suggest you get a pen and paper or go out into the real world and test it yourself with some measurements and observations.
Avoid ignorance when it's easily avoidable.
Good luck.

If I take an object like an orange, how am I supposed to measure the imperfections and dimples on the surface to get an accurate circumference? If I tied a string around it would only be a guess, since obviously, the string isn't going into the dimples
Tell you what: do it anyway. Ignore the dimples and imperfections. Do the test as written, and do the math as specified. You will find that within the limits of your ability to perform the experiment, Pi will never come anywhere near 4.00, but instead will always be closer to 3 than to 4. In fact, throw away the orange and find something that appears smooth to an observer, like a soda can or drinking glass or water pipe. Do the test with that. At this point I am really wishing the example of an orange had never been put forth.
Taking your analogy of imperfections and dimples a little further: How far is it from one end of a basketball court to the other? At human scale, walking across the court with a measuring tape, it is 94 feet. To a microbe, who would have to travel up and down the imperfections in the floor's surface, it is probably twice that far, maybe more. And yet, in the real world, we can use the 94 foot dimension and do useful things with it. We can ignore the microbelevel imperfections and do real math with real number measured in the real world. That's all Pi is: a number useful in the real world. I can use 3.14159 to calculate how much material is needed to manufacture a cylindrical object of a desired diameter. I can use 3.14159 to calculate how big a cylinder needs to be in order to contain a specified amount of product. I can use 3.14159 to calculate the diameter of an object if I know that objects measurable perimeter, and having done so I can measure the object's diameter and find that it matches the calculated value. None of these tasks demand some kind of ideal, perfect to infinitely small scale circle; a circle that is good enough to be considered circular by a human observer is good enough.

Avoid ignorance when it's easily avoidable.
Now THAT should be a bumper sticker!

In which world are Zeno's paradoxes an "article of faith"? It must be an interesting place.
How do you feel about other elementary mathematical concepts? Are real numbers an "article of faith", too? Do you accept any other aspects of mathematics than "I like triangles"?

In which world are Zeno's paradoxes an "article of faith"? It must be an interesting place.
How do you feel about other elementary mathematical concepts? Are real numbers an "article of faith", too? Do you accept any other aspects of mathematics than "I like triangles"?
Zeno's paradox is a red herring. I'm talking about simple trigonometry. It's a statement of faith to claim that it is invalid. Actually, it's a statement of buffoonery. Trigonometric relationships are not theory in any way. They are fact. Again test it yourself.
If you're unclear how triangles relate to the topic of this thread, try this: see if you can identify any triangles in the illustration below:
(https://www.filesanywhere.com/FS/M.aspx?v=8c70678d619cb478aca7)

since the universe is quantized, and any such related math is inaccurate.
You don't know the universe is quantized, this is reaching. In fact, the very nature of quantum mechanics is that the universe appears to be both continuous and quantized, and one property is only favored based on your measuring method.
There is also a paradox contained in your statement since the notion of quantization relies on the very math you are claiming is inaccurate.

The math I am using is easily verifiable.
Please verify it then. Show us that two objects which recede infinitely into the distance on parallel lines will never touch.
I put it right there in front of you. It's triangles, the most basic trigonometry. Trigonometry has been tried, tested, verified, and validated for thousands of years.
Who validated that an overhead receding body on a flat surface will never set? You are clearly putting your faith in a mathematical model to tell us how perspective works at long distances.
In which world are Zeno's paradoxes an "article of faith"? It must be an interesting place.
How do you feel about other elementary mathematical concepts? Are real numbers an "article of faith", too? Do you accept any other aspects of mathematics than "I like triangles"?
Zeno's paradox is a red herring. I'm talking about simple trigonometry. It's a statement of faith to claim that it is invalid. Actually, it's a statement of buffoonery. Trigonometric relationships are not theory in any way. They are fact. Again test it yourself.
If you're unclear how triangles relate to the topic of this thread, try this: see if you can identify any triangles in the illustration below:
(https://www.filesanywhere.com/FS/M.aspx?v=8c70678d619cb478aca7)
The Greek math of Geometry and Trigonometry, with its hypothetical number lines and points in space, that Zeno is criticizing in his paradoxes, is really the same math you are trying to use here in your proofs. That type of continuous math really doesn't work, especially at extremes.
You said it yourself, under that model a star or sun will recede into infinity but never set. How do we know that? Under that model it is also impossible for a sun or a star to move any small distance across that number line at all.
It is erroneous to base conclusions for how the world should be at the large or the small scales that one cannot experience, on nothing more than an ancient mathematical model of a perfect universe.

. . . . . . . . . . . . . . . .
You said it yourself, under that model a star or sun will recede into infinity but never set. How do we know that? Under that model it is also impossible for a sun or a star to move any small distance across that number line at all.
It is erroneous to base conclusions for how the world should be at the large or the small scales that one cannot experience, on nothing more than an ancient mathematical model of a perfect universe.
I am curious about this bit "You said it yourself, under that model a star or sun will recede into infinity but never set."
All very well for you to ask "How do we know that?"
All the evidence that we have is that light travels in straight lines. Do you have anything to the contrary?
Other than that "bendy light" is needed to support the Flat Earth Hypothesis.
I do just wonder what astronomers think of this idea?
Also is seems an absolutely amazing coincidence that the supposed "magnification" in the "atmolayer" interface (or whatever) is precisely the amount needed to keep the sun (and moon) to exactly the size expected on the globe earth.
In other words what "Flat Earth theorists" have done is to tranfer the curvature on the earth into the curvature of light! Looks highly suspicious to me. I think poor old Occam has mislaid his razor!

. . . . . . . . . . . . . . . .
You said it yourself, under that model a star or sun will recede into infinity but never set. How do we know that? Under that model it is also impossible for a sun or a star to move any small distance across that number line at all.
It is erroneous to base conclusions for how the world should be at the large or the small scales that one cannot experience, on nothing more than an ancient mathematical model of a perfect universe.
I am curious about this bit "You said it yourself, under that model a star or sun will recede into infinity but never set."
All very well for you to ask "How do we know that?"
All the evidence that we have is that light travels in straight lines. Do you have anything to the contrary?
Other than that "bendy light" is needed to support the Flat Earth Hypothesis.
I do just wonder what astronomers think of this idea?
Also is seems an absolutely amazing coincidence that the supposed "magnification" in the "atmolayer" interface (or whatever) is precisely the amount needed to keep the sun (and moon) to exactly the size expected on the globe earth.
In other words what "Flat Earth theorists" have done is to tranfer the curvature on the earth into the curvature of light! Looks highly suspicious to me. I think poor old Occam has mislaid his razor!
Well, one thing I can speculate on is that if space is quantized, it is not possible for light to take all possible angles over very long distances. That is one example for why "sizing up" an ancient continuous model of a perfect universe is erroneous. We do not know about physics at larger scales.
It is incredibly short sighted to simply assume what will happen at all scales. We must begin from experience. Experience will tell us the truth independent of any particular model or theory. And if you cannot find an experience of two parallel lines or objects receding into infinity and never touching, then I am afraid you are delusional.

. . . . . . . . . . . . . . . .
You said it yourself, under that model a star or sun will recede into infinity but never set. How do we know that? Under that model it is also impossible for a sun or a star to move any small distance across that number line at all.
It is erroneous to base conclusions for how the world should be at the large or the small scales that one cannot experience, on nothing more than an ancient mathematical model of a perfect universe.
I am curious about this bit "You said it yourself, under that model a star or sun will recede into infinity but never set."
All very well for you to ask "How do we know that?"
All the evidence that we have is that light travels in straight lines. Do you have anything to the contrary?
Other than that "bendy light" is needed to support the Flat Earth Hypothesis.
I do just wonder what astronomers think of this idea?
Also is seems an absolutely amazing coincidence that the supposed "magnification" in the "atmolayer" interface (or whatever) is precisely the amount needed to keep the sun (and moon) to exactly the size expected on the globe earth.
In other words what "Flat Earth theorists" have done is to tranfer the curvature on the earth into the curvature of light! Looks highly suspicious to me. I think poor old Occam has mislaid his razor!
Well, one thing I can speculate on is that if space is quantized, it is not possible for light to take all possible angles over very long distances. That is one example for why "sizing up" an ancient continuous model of a perfect universe is erroneous. We do not know about physics at larger scales.
It is incredibly short sighted to simply assume what will happen at all scales. We must begin from experience. Experience will tell us the truth independent of any particular model or theory. And if you cannot find an experience of two parallel lines or objects receding into infinity and never touching, then I am afraid you are delusional.
Oh, come off it! How small is the Planck distance? About 1.6 x 10^{35} m or about 10^{20} times the size of a proton  utterly miniscule compared to even the wavelength of green light (5.10 x 10^{7} m)!
And you think that the directions of light might be quantised to an extent that might affect our observations.
In any case, what is it to you? Your whole universe is a tiny hemisphere 40,000 km in diameter and (I guess) 20,000 km high, so why are you bothering to discuss long distances!

Oh, come off it! How small is the Planck distance? About 1.6 x 10^{35} m or about 10^{20} times the size of a proton  utterly miniscule compared to even the wavelength of green light (5.10 x 10^{7} m)!
And you think that the directions of light might be quantised to an extent that might affect our observations.
Sure, plancks may be incredibly, incredibly dense. But have you ever heard of the inverse squared law?
(http://i.imgur.com/7dHWW27.gif)
Usually it deals with energy intensity, but we can also use it to see that the space to fill is increasing at an exponential rate away from the source.
In any case, what is it to you? Your whole universe is a tiny hemisphere 40,000 km in diameter and (I guess) 20,000 km high, so why are you bothering to discuss long distances!
It's only tiny if we assume the theory of an universe with distances of thousands light years. Speaking from human experience, not "theory", anyone traveling 25,000 miles would say that is a very long distance indeed.

It is incredibly short sighted to simply assume what will happen at all scales. We must begin from experience. Experience will tell us the truth independent of any particular model or theory. And if you cannot find an experience of two parallel lines or objects receding into infinity and never touching, then I am afraid you are delusional.
It is also shortsighted to assume that your experience is enough to perceive what would happen on the Planck scale. In fact, we can safely assert with no hope of counterexample, that you could not, without benefit of technological aid, perceive what is happening on that scale.
All this is somewhat moot anyway, since there is no one who would definitively say space is quantized. This is why there is even a search for spacetime foam. As far as we can tell, energy and matter sometimes exhibit quantum properties, but space appears to be continuous.

It is also shortsighted to assume that your experience is enough to perceive what would happen on the Planck scale. In fact, we can safely assert with no hope of counterexample, that you could not, without benefit of technological aid, perceive what is happening on that scale.
Zeno's Paradox experiments deal with how space and time work on the smallest scales.
All this is somewhat moot anyway, since there is no one who would definitively say space is quantized. This is why there is even a search for spacetime foam. As far as we can tell, energy and matter sometimes exhibit quantum properties, but space appears to be continuous.
It is my understanding that space and time does exhibit quantum properties, but it's easier to use continuous math for longer distances and bigger scales, so that is what is used.

It is also shortsighted to assume that your experience is enough to perceive what would happen on the Planck scale. In fact, we can safely assert with no hope of counterexample, that you could not, without benefit of technological aid, perceive what is happening on that scale.
Zeno's Paradox experiments deal with how space and time work on the smallest scales.
It is a thought experiment that may or may not have anything to do with the real world. Also, isn't it incredibly dishonest to use a classical philosopher to show that classical philosophy does not apply to the real world?
All this is somewhat moot anyway, since there is no one who would definitively say space is quantized. This is why there is even a search for spacetime foam. As far as we can tell, energy and matter sometimes exhibit quantum properties, but space appears to be continuous.
It is my understanding that space and time does exhibit quantum properties, but it's easier to use continuous math for longer distances, and so that is what is used.
Nope, quantum spacetime is just an idea with no empirical evidence. People are trying to find a way to see if foamy spacetime is true, but to no avail. You understanding is incorrect. Regardless, the waveparticle nature of the quantum world means that sometimes things appear quantum and sometimes they don't and this appears to be an actual property of matterenergy and not just an issue with uncertainty.

It is a thought experiment that may or may not have anything to do with the real world. Also, isn't it incredibly dishonest to use a classical philosopher to show that classical philosophy does not apply to the real world?
It's more dishonest to use classic philosophy which was disproven by a classic philosopher of the same era.
Nope, quantum spacetime is just an idea with no empirical evidence. People are trying to find a way to see if foamy spacetime is true, but to no avail. You understanding is incorrect. Regardless, the waveparticle nature of the quantum world means that sometimes things appear quantum and sometimes they don't and this appears to be an actual property of matterenergy and not just an issue with uncertainty.
Waveparticle duality of particle physics has nothing to do with whether space is quantized or not, as both particles and waves are above the resolution of plank length.

Regarding pi, 4 doesn't work. Looking at the image of the square on the first page, the first change in the perimeter reduces the area by a significant amount. It should be quite obvious why.
And you think that the directions of light might be quantised to an extent that might affect our observations.
In any case, what is it to you? Your whole universe is a tiny hemisphere 40,000 km in diameter and (I guess) 20,000 km high, so why are you bothering to discuss long distances!
This 'bendy perspective' would have to start working at about 3,000 miles (if the stars are a layer 3,000 miles high, give or take), as there would start being a measurable change their apparent spacing shortly after passing directly overhead.

It is a thought experiment that may or may not have anything to do with the real world. Also, isn't it incredibly dishonest to use a classical philosopher to show that classical philosophy does not apply to the real world?
It's more dishonest to use classic philosophy which was disproven by a classic philosopher of the same era.
So you agree you are being dishonest then?
Nope, quantum spacetime is just an idea with no empirical evidence. People are trying to find a way to see if foamy spacetime is true, but to no avail. You understanding is incorrect. Regardless, the waveparticle nature of the quantum world means that sometimes things appear quantum and sometimes they don't and this appears to be an actual property of matterenergy and not just an issue with uncertainty.
Waveparticle duality of particle physics has nothing to do with whether space is quantized or not, as both particles and waves are above the resolution of plank length.
Which changes nothing about spacetime not being quantized.

News flash: Spacetime doesn't exist. None of this wild abstract stuff that humans have pulled out of their ass to feel smarter than the Creator of the Universe is real.

News flash: Spacetime doesn't exist. None of this wild abstract stuff that humans have pulled out of their ass to feel smarter than the Creator of the Universe is real.
You don't think space or time exists? Tell us more... In another thread preferably.

News flash: Spacetime doesn't exist. None of this wild abstract stuff that humans have pulled out of their ass to feel smarter than the Creator of the Universe is real.
You don't think space or time exists? Tell us more... In another thread preferably.
It seems perfectly fair to discuss in this thread, considering very little of this conversation has to do with Polaris.
I believe time is an abstract concept used to keep track of our existence, nothing more. Space, in regards to 3D plane based XYZ coordinates obviously exist, and the various ways to display and chart points in space do have uses, but the higher levels of mathematics have little to no practical use, or relation to reality.

Usually it deals with energy intensity, but we can also use it to see that the space to fill is increasing at an exponential rate away from the source.
Of course I am very familiar with the inverse square law!
And no, the area to fill is increasing at a square rate and the volume to fill is increasing at a cube rate.
Neither of those comes to anything like an exponential rate.
x^{2} or x^{3} don't increase at anything like the rate of 2^{x} when x gets large.
And I fail to see where quantising can come into anything on this scale.
Yes, I suppose if you choose to disbelieve that anything can be outside you little world 25,000 miles is a huge distance.
Mind you the distance to the moon was measured long before Rowbotham (or NASA) came along and then measured by radar in the 1940's and again by laser in the late 1960's and 1970's.
Guess what, the measurements all agree to within the accuracies expected in each case. Even Hipparchus in the 2nd century BC used parallax to estimate the distance at about 400,000 km, about 7% away from the current figures.
So unless you can come up with something that slows radio waves and light down by a factor of 100 or so,
I'll stick to things being a bit bigger than you tiny universe!

Yes, I suppose if you choose to disbelieve that anything can be outside you little world 25,000 miles is a huge distance.
Mind you the distance to the moon was measured long before Rowbotham (or NASA) came along and then measured by radar in the 1940's and again by laser in the late 1960's and 1970's.
Guess what, the measurements all agree to within the accuracies expected in each case. Even Hipparchus in the 2nd century BC used parallax to estimate the distance at about 400,000 km, about 7% away from the current figures.
Hippachus assumed the earth was round in his calculations.
http://www.michaelbeeson.com/interests/GreatMoments/Hipparchus.pdf
(http://www.astro.cornell.edu/academics/courses/astro201/images/hipparchus.gif)

Hippachus assumed the earth was round in his calculations.
Yes, he did. Interesting thing about those eclipse observations though: even on a flat earth the only way for the moon to completely obscure the sun's disc as viewed from Syene while obscuring only 80% of the sun's disc as viewed from Alexandria is if the moon is closer to the earth than the sun. This is what enables the viewer in Alexandria to see around the edge of the moon. And since we know the moon only just barely obscures the sun, it follows that if it is closer than the sun it must also be smaller than the sun. This contradicts your Wiki, someone should redo the math there.
Just in case somebody wants to dispute the observations of the ancients (and I have to be fair, we have no way of knowing how rigorous or accurate those observations were) we have an opportunity coming up in 18 months to do some realworld observation of our own. On August 21, 2017 we have a total eclipse predicted for the United States. I imagine that many of the board participant live in the US, and we could pool resource and conduct simultaneous observations at various locations. The projected ground track of the eclipse could not be more ideal for this kind of experiment.
http://www.eclipsewise.com/solar/SEprime/20012100/SE2017Aug21Tprime.html (http://www.eclipsewise.com/solar/SEprime/20012100/SE2017Aug21Tprime.html)

And since we know the moon only just barely obscures the sun, it follows that if it is closer than the sun it must also be smaller than the sun. This contradicts your Wiki, someone should redo the math there.
What does the Wiki say about how high and large the moon is?

And since we know the moon only just barely obscures the sun, it follows that if it is closer than the sun it must also be smaller than the sun. This contradicts your Wiki, someone should redo the math there.
What does the Wiki say about how high and large the moon is?
i am sure I can answer that!
The moon is a rotating sphere. It has a diameter of 32 miles and is located approximately 3000 miles above the surface of the earth.
When one observes the phases of the moon he sees the moon's day and night, a shadow from the sun illuminating half of the spherical moon at any one time.
The lunar phases vary cyclically according to the changing geometry of the Moon and Sun, which are constantly wobbling up and down and exchange altitudes as they rotate around the North Pole.
When the moon and sun are at the same altitude one half of the lunar surface is illuminated and pointing towards the sun, This is called the First Quarter Moon. When the observer looks up he will see a shadow cutting the moon in half. The boundary between the illuminated and unilluminated hemispheres is called the terminator.
When the moon is below the sun's altitude the moon is dark and a New Moon occurs.
When the moon is above the altitude of the sun the moon is fully lit and a Full Moon occurs.
Mind you if you look at the geometry it does strike me as being a bit weird, but then from my distorted perspective light travels in straight lines! Silly me!
Also "a shadow from the sun" sounds a bit strange  a shadow from a light source?

And since we know the moon only just barely obscures the sun, it follows that if it is closer than the sun it must also be smaller than the sun. This contradicts your Wiki, someone should redo the math there.
What does the Wiki say about how high and large the moon is?
i am sure I can answer that!
The moon is a rotating sphere. It has a diameter of 32 miles and is located approximately 3000 miles above the surface of the earth.
I see the word approximately in there.
When one observes the phases of the moon he sees the moon's day and night, a shadow from the sun illuminating half of the spherical moon at any one time.
The lunar phases vary cyclically according to the changing geometry of the Moon and Sun, which are constantly wobbling up and down and exchange altitudes as they rotate around the North Pole.
When the moon and sun are at the same altitude one half of the lunar surface is illuminated and pointing towards the sun, This is called the First Quarter Moon. When the observer looks up he will see a shadow cutting the moon in half. The boundary between the illuminated and unilluminated hemispheres is called the terminator.
When the moon is below the sun's altitude the moon is dark and a New Moon occurs.
When the moon is above the altitude of the sun the moon is fully lit and a Full Moon occurs.
Mind you if you look at the geometry it does strike me as being a bit weird, but then from my distorted perspective light travels in straight lines! Silly me!
Also "a shadow from the sun" sounds a bit strange  a shadow from a light source?
What sounds strange about that? The sun doesn't make shadows?

Also "a shadow from the sun" sounds a bit strange  a shadow from a light source?
What sounds strange about that? The sun doesn't make shadows?
The sun only makes shadows because an object in interposed between the sun and the shadowed object (here the moon).
What third object is casting a shadow on the moon to cause the phases?

Also "a shadow from the sun" sounds a bit strange  a shadow from a light source?
What sounds strange about that? The sun doesn't make shadows?
The sun only makes shadows because an object in interposed between the sun and the shadowed object (here the moon).
What third object is casting a shadow on the moon to cause the phases?
When one observes the phases of the moon he sees the moon's day and night, a shadow from the sun illuminating half of the spherical moon at any one time.

Tom, it doesn't help for you to merely quote the Wiki when the question being asked is " What does this line from the wiki mean?" Especially if the line you quote was actually already quoted by the other guy, which means he saw it already and it didn't help the first time he read it. It probably won't help the second time either, especially removed from the surrounding context.
So I will tell you what I nderstand Rabinoz's question to be. Maybe I'm wrong, in which case I apologize, but I would like an answer myself, so nothing lost. Here it is: How can a shadow illuminate anything? That's not what shadows do, illuminate. They block illumination.
Here's the line again, with my emphasis addd to bring attention to the part we're asking about: "When one observes the phases of the moon he sees the moon's day and night, a shadow from the sun illuminating half of the spherical moon at any one time." Clearly states that a shadow is illuminating the moon, we/I just wonder what that means. I think it might just be a typo, but I would lke to know for sure.

Tom, it doesn't help for you to merely quote the Wiki when the question being asked is " What does this line from the wiki mean?" Especially if the line you quote was actually already quoted by the other guy, which means he saw it already and it didn't help the first time he read it. It probably won't help the second time either, especially removed from the surrounding context.
So I will tell you what I nderstand Rabinoz's question to be. Maybe I'm wrong, in which case I apologize, but I would like an answer myself, so nothing lost. Here it is: How can a shadow illuminate anything? That's not what shadows do, illuminate. They block illumination.
Here's the line again, with my emphasis addd to bring attention to the part we're asking about: "When one observes the phases of the moon he sees the moon's day and night, a shadow from the sun illuminating half of the spherical moon at any one time." Clearly states that a shadow is illuminating the moon, we/I just wonder what that means. I think it might just be a typo, but I would lke to know for sure.
Don't worry about Tom, he never actually answers a question, his replies are just intended to obfuscate!
(Gee, wonder where I dredged up a word like that from.)
You should have seen his answers when I showed that if the earth was flat, then π = 2, he almost had me believing it was 4!
Then that the circumference of a 10 cm disk was 2.5x10^{+33} Planck lengths!
You'll get used to it eventually, but don't ever expect any useful information

Tom, it doesn't help for you to merely quote the Wiki when the question being asked is " What does this line from the wiki mean?" Especially if the line you quote was actually already quoted by the other guy, which means he saw it already and it didn't help the first time he read it. It probably won't help the second time either, especially removed from the surrounding context.
So I will tell you what I nderstand Rabinoz's question to be. Maybe I'm wrong, in which case I apologize, but I would like an answer myself, so nothing lost. Here it is: How can a shadow illuminate anything? That's not what shadows do, illuminate. They block illumination.
Here's the line again, with my emphasis addd to bring attention to the part we're asking about: "When one observes the phases of the moon he sees the moon's day and night, a shadow from the sun illuminating half of the spherical moon at any one time." Clearly states that a shadow is illuminating the moon, we/I just wonder what that means. I think it might just be a typo, but I would lke to know for sure.
The wording is correct. Here is an alternative version:
a shadow [caused by] the sun illuminating half of the spherical moon
There is not much difference between the word "from". From can have several meanings. "Caused by" is the clear meaning here. Hardly anyone would interpret it as a shadow illuminating anything in this context. It clearly says the sun is the one doing the illuminating, not that the shadow is illuminating, as the words "sun" and "illuminating" are right next to each other and the words "shadow" and "illuminating" are not.

Tom, it doesn't help for you to merely quote the Wiki when the question being asked is " What does this line from the wiki mean?" Especially if the line you quote was actually already quoted by the other guy, which means he saw it already and it didn't help the first time he read it. It probably won't help the second time either, especially removed from the surrounding context.
So I will tell you what I nderstand Rabinoz's question to be. Maybe I'm wrong, in which case I apologize, but I would like an answer myself, so nothing lost. Here it is: How can a shadow illuminate anything? That's not what shadows do, illuminate. They block illumination.
Here's the line again, with my emphasis addd to bring attention to the part we're asking about: "When one observes the phases of the moon he sees the moon's day and night, a shadow from the sun illuminating half of the spherical moon at any one time." Clearly states that a shadow is illuminating the moon, we/I just wonder what that means. I think it might just be a typo, but I would lke to know for sure.
The wording is correct. Here is an alternative version:
a shadow [caused by] the sun illuminating half of the spherical moon
There is not much difference between the word "from". From can have several meanings. "Caused by" is the clear meaning here. Only a challenged person would interpret it as a shadow illuminating anything in this context. It clearly says the sun is the one doing the illuminating, not that the shadow is illuminating, as the words "sun" and "illuminating" are right next to each other and the words "shadow" and "illuminating" are not.
I repeat my previous post!
The sun only makes shadows because an object in interposed between the sun and the shadowed object (here the moon).
What third object is casting a shadow on the moon to cause the phases?
The sun cannot of itself cast or cause a shadow on the moon!
I ask again What third object (apart from the SUN and MOON) is casting a shadow on the moon to cause the phases?

The wording is correct. Here is an alternative version:
a shadow [caused by] the sun illuminating half of the spherical moon
I see! Thank you, that is perfectly clear to me now, and I would suggest this wording be adopted to replace the text found in the wiki (and I would change it to "...illuminating only half..." as well) It appears we are in agreement that the dark side of the moon is in shadow not because of a third object, but because it faces away from the light source.

The wording is correct. Here is an alternative version:
a shadow [caused by] the sun illuminating half of the spherical moon
I see! Thank you, that is perfectly clear to me now, and I would suggest this wording be adopted to replace the text found in the wiki (and I would change it to "...illuminating only half..." as well) It appears we are in agreement that the dark side of the moon is in shadow not because of a third object, but because it faces away from the light source.
In the FE model, is the moon rotating at a similar speed as the sun?
If the sun is the only light source illuminating the moon, the further south from the north pole you go, you should be able to see multiple phases in a single night as the moon passes overhead.

The flat earth and heliocentric models are opposites in almost every way. Is it really that hard to tell which is correct?
The answer is NO. The earth is a sphere. It's obvious by the apparent positions of the celestial objects in the sky as observed from earth. By simple observation you can determine conclusively that the earth is round and that a flat earth is impossible.
One of the simplest examples illustrating this is Polaris. See why here:
http://debunkingflatearth.blogspot.com/2016/02/debunkingflatearthhowpolarisproves.html (http://debunkingflatearth.blogspot.com/2016/02/debunkingflatearthhowpolarisproves.html)
Your entire post was an unsupported opinion. The link you provided has the following requirements that the information conveyed requires to be valid.
1) The Earth revolves around the Sun.
2) The Sun moves throughout the galaxy.
3) The sphere earth rotates on an axis.
4) The stars also move relative to the galaxy.
Any observer in the moon given these 4 requirements would not be able to produce a photograph with star trails as they are rotating away from the Stars while the planet they are on is moving through the galaxy. The trails presented on these images do not comport with these given requirements. As a natural result one would have to being eliminating the points representing the source of the conflict until the results can be reconciled. This wouldn't leave any points on the list above remaining.
The example you provided is void as the fact that star trail photos have been reproduced numerous times indicates that the star trails form in a singular direction, predictably, and persistently through the ages.

1) The Earth revolves around the Sun.
2) The Sun moves throughout the galaxy.
3) The sphere earth rotates on an axis.
4) The stars also move relative to the galaxy.
Flat Earthers seem to have no concept of time scales and distances.
"The sphere earth rotates on an axis." once in a little under 24 hours.
"The Earth revolves around the Sun" once in a bit over 365 days.
"The Sun moves throughout the galaxy." but, not just the Sun, but the Solar System as a whole and this movement is too slow (in relative terms) for any but astronomers to see.
"The stars also move relative to the galaxy" this is a bit backwards. Most of the stars we see are part of our galaxy, the Milky Way and I assume the Milky Way moves relative to the rest of the universe. But, these movements are completely undetectable to other than astronomers.
Then the distance from the earth to the Sun is almost 1,200 times diameter of the earth.
The distance to the nearest star (Proxima Centauri) is over 250,000 times the distance to the Sun.
The whole point of this is that the only movement that affects star trails on earth is the rotation of the earth and
since the moon rotates on its axis (and around the earth) about once in 27 days
Star trails would be observed on the moon, but would take about 27 days for one rotation.
Any movement around the Sun would have only a minor effect. Any movement relative to the stars is quite negligible.
(I hope my quick calculations are right!)

The flat earth and heliocentric models are opposites in almost every way. Is it really that hard to tell which is correct?
The answer is NO. The earth is a sphere. It's obvious by the apparent positions of the celestial objects in the sky as observed from earth. By simple observation you can determine conclusively that the earth is round and that a flat earth is impossible.
One of the simplest examples illustrating this is Polaris. See why here:
http://debunkingflatearth.blogspot.com/2016/02/debunkingflatearthhowpolarisproves.html (http://debunkingflatearth.blogspot.com/2016/02/debunkingflatearthhowpolarisproves.html)
Your entire post was an unsupported opinion. The link you provided has the following requirements that the information conveyed requires to be valid.
1) The Earth revolves around the Sun.
2) The Sun moves throughout the galaxy.
3) The sphere earth rotates on an axis.
4) The stars also move relative to the galaxy.
Any observer in the moon given these 4 requirements would not be able to produce a photograph with star trails as they are rotating away from the Stars while the planet they are on is moving through the galaxy. The trails presented on these images do not comport with these given requirements. As a natural result one would have to being eliminating the points representing the source of the conflict until the results can be reconciled. This wouldn't leave any points on the list above remaining.
The example you provided is void as the fact that star trail photos have been reproduced numerous times indicates that the star trails form in a singular direction, predictably, and persistently through the ages.
You are mistaken. The information provided requires none of those. What it shows is that the shape of the earth can only be round and the celestial objects can only be very far away, according to the angles at which we observe those celestial objects. It also shows that a flat earth is impossible because it cannot produce those observed angles. This includes the sun which would never set, nor even get close to the horizon. The only way around these facts is to subscribe to the ridiculous argument (already expressed in this thread) that applied geometry and trigonometry somehow, despite all empirical evidence, don't really work in the real world, and that triangles which are very large somehow magically cease to adhere to the physical laws to which other triangles are subject, e.g., they can have irrational properties like 0° angles.