Re: Polaris proves the earth is round.
« Reply #40 on: February 15, 2016, 07:16:30 PM »
I'm completely lost as to why this became an abstract mathematics lecture. Does Polaris prove the earth is round or not?

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Offline Tom Bishop

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Re: Polaris proves the earth is round.
« Reply #41 on: February 15, 2016, 11:44:33 PM »
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 money-driven 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.
« Last Edit: February 15, 2016, 11:53:41 PM by Tom Bishop »
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Offline Tom Bishop

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Re: Polaris proves the earth is round.
« Reply #42 on: February 16, 2016, 12:23:56 AM »
Quote from: Rounder
well, even a jagged not-perfect 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.
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Re: Polaris proves the earth is round.
« Reply #43 on: February 16, 2016, 01:35:37 AM »
"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.
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Re: Polaris proves the earth is round.
« Reply #44 on: February 16, 2016, 04:15:36 AM »
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:


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.


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.
« Last Edit: February 16, 2016, 04:30:40 AM by garygreen »
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Re: Polaris proves the earth is round.
« Reply #45 on: February 16, 2016, 12:15:39 PM »
. . . . . . . . . . . . . . . .
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 de-railer.
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!):

Re: Polaris proves the earth is round.
« Reply #46 on: February 16, 2016, 09:09:52 PM »
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.

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Offline Tom Bishop

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Re: Polaris proves the earth is round.
« Reply #47 on: February 16, 2016, 09:19:46 PM »
"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.

Quote
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.
« Last Edit: February 16, 2016, 09:53:02 PM by Tom Bishop »
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Offline Tom Bishop

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Re: Polaris proves the earth is round.
« Reply #48 on: February 16, 2016, 09:26:24 PM »
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

Quote
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.



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 × 10-35 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.

Quote from: garygreen
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.


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.
« Last Edit: February 18, 2016, 04:15:14 AM by Tom Bishop »
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Re: Polaris proves the earth is round.
« Reply #49 on: February 16, 2016, 09:42:55 PM »
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.

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Offline Tom Bishop

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Re: Polaris proves the earth is round.
« Reply #50 on: February 16, 2016, 10:29:45 PM »
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.
"The biggest problem in astronomy is that when we look at something in the sky, we don’t know how far away it is" — Pauline Barmby, Ph.D., Professor of Astronomy

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Offline rabinoz

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Re: Polaris proves the earth is round.
« Reply #51 on: February 16, 2016, 10:44:04 PM »
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.

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Re: Polaris proves the earth is round.
« Reply #52 on: February 17, 2016, 12:29:25 AM »
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.


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Offline Tom Bishop

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Re: Polaris proves the earth is round.
« Reply #53 on: February 17, 2016, 01:24:11 AM »
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.

Quote
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.
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Re: Polaris proves the earth is round.
« Reply #54 on: February 17, 2016, 02:01:28 AM »
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.

Quote
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!??

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Offline Tom Bishop

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Re: Polaris proves the earth is round.
« Reply #55 on: February 17, 2016, 06:27:15 AM »
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.
"The biggest problem in astronomy is that when we look at something in the sky, we don’t know how far away it is" — Pauline Barmby, Ph.D., Professor of Astronomy

Re: Polaris proves the earth is round.
« Reply #56 on: February 17, 2016, 06:49:20 AM »
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?
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Offline Unsure101

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Re: Polaris proves the earth is round.
« Reply #57 on: February 17, 2016, 09:42:43 AM »
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!

Re: Polaris proves the earth is round.
« Reply #58 on: February 17, 2016, 04:24:18 PM »
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.

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Offline Panzerfaust

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Re: Polaris proves the earth is round.
« Reply #59 on: February 17, 2016, 05:39:06 PM »
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 back-calculated 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.