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I’ve seen a lot of confused discussion of perspective on this forum as well as on the Wiki. I think some of this confusion arises because “perspective” has at least two distinct meanings in the context of scientific discussions:
(1) A branch of geometry that considers where straight lines drawn between a specified point P and a set of other points in three dimensions will intersect with a plane placed between P and the other points. Perspective in this sense was pioneered by ancient Greeks such as Euclid and was further developed during the Renaissance to assist artists, on the assumption that light travels in straight lines, in producing paintings and drawings with a more realistic 3-D look. To my knowledge no one has found any logical flaws in the proofs of the theorems of perspective in this sense, so it still stands as an established field of geometry. Let’s call this kind of perspective “perspectiveG” for purposes of this discussion.
(2) A branch of optics, which in turn is a branch of physics, that studies what happens to light when it passes through air. This is sometimes called “aerial perspective,” so let’s call it “perspectiveA.”
Let’s further distinguish both of these senses of “perspective” from the study of how the human eye perceives light and how the brain interprets the messages it receives from the eye. That is vision science, which overlaps with neuroscience, psychology, ophthalmology, and other sciences.
Bearing these distinctions in mind, I want to look at some comments from Tom Bishop, not to pick on Tom (whose courtesy and apparent sincerity I appreciate) but because he seems to post frequently on perspective. I hope I am not distorting his meaning by taking his quotes out of context:
Tom Bishop:
I'm asking for some sort of evidence that perspective works the way the Ancient Greek math says it works. Will two parallel lines really recede forever into the distance and never appear to touch? That seems extraordinary.
Why should we believe that just because an ancient greek philosopher said that a perfect world would be that way?
To begin with, ancient Greek math, in this case Euclid, defined parallel straight lines as “straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.” You can’t refute a definition, and you can’t refute math without showing a logical flaw in the reasoning. What Tom seems to be talking about here is how distant objects will actually appear to the human eye and brain. The science of optics covers what happens to light as it travels from the objects to the eye, and vision science studies how our brains interpret the light received by the eye (for instance, whether we perceive one object or two when they are separated by 1 second of arc), so this is a question of optics and vision science.
Tom Bishop:
How do we "know" what happens to perspective tens or hundreds of miles away? Who studied that?
Perspective hasn't been tested at any large distance at all. At what distances has it been tested? Who studied it? Please name names and cite studies rather than claiming that it has been proven.
Here Tom is apparently talking about perspectiveA, not perspectiveG – he’s asking how objects actually appear to us when viewed through hundreds of miles of air. But we actually know, beyond reasonable doubt, a great deal about this. The science of optics is well developed and extremely successful at explaining the behavior of light, including what happens to it as it passes through air containing moisture and dust and with temperature and pressure gradients. We can’t directly test what happens to light when it passes through 6000 miles of air because no one has been able to find 6000 miles of the earth’s atmosphere in a straight line. In any case, modern optics provides absolutely no grounds for supposing that light performs the acrobatics posited by FET to explain how the moon displays the same phase simultaneously to all viewers, or how the sun and moon appear to set, or how they maintain the same angular diameter, all while remaining above a flat earth.
Of course, you can argue that mainstream optics is wrong about this – but then you need to come up with better theories that do a better job of explanation and prediction than the currently accepted ones. This is just one instance of what we may call The Big Problem with Flat Earth Theory, namely the fact that FET tosses out large chunks of mainstream science without offering any mutually consistent alternative theories that do at least as good a job of explanation and prediction as the mainstream theories, which include Round Earth Theory. Since what we want from scientific theories is to explain and predict what we observe, we are left with no rational reason to switch to the FE side.
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Excellent writeup. Now we can use this thread to debate perspective instead of constantly derailing other threads.
I would like to expand on your response to the first quote by Tom Bishop:
Tom Bishop:
I'm asking for some sort of evidence that perspective works the way the Ancient Greek math says it works. Will two parallel lines really recede forever into the distance and never appear to touch? That seems extraordinary.
Why should we believe that just because an ancient greek philosopher said that a perfect world would be that way?
Parallel lines never touch, by definition. If they do touch, then they aren't parallel.
However, things get a little complicated when we project lines that are parallel in 3D space onto a 2D surface ("perspectiveG"). Look at the following picture:
(http://www.math.utah.edu/~treiberg/Perspect/ParVP.GIF)
- The lines that are parallel in 3D space, are no longer necessarily parallel when projected onto a 2D surface. This is what we mean when we say they "appear" to touch. They "appear" to touch at the vanishing point (V' in the picture). Just because they "appear" to touch in the 2D projection, does not mean that they touch in 3D space.
- The vanishing point only exists in the 2D projection, not in 3D space. If you follow the parallel lines in 3D space, you will NEVER arrive at the vanishing point. This is what we mean when we say "the vanishing point is at an infinite distance away"
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- The lines that are parallel in 3D space, are no longer necessarily parallel when projected onto a 2D surface. This is what we mean when we say they "appear" to touch. They "appear" to touch at the vanishing point (V' in the picture). Just because they "appear" to touch in the 2D projection, does not mean that they touch in 3D space.
- The vanishing point only exists in the 2D projection, not in 3D space. If you follow the parallel lines in 3D space, you will NEVER arrive at the vanishing point. This is what we mean when we say "the vanishing point is at an infinite distance away"
Yes. The term "vanishing point" is another source of confusion. Primarily, it refers to a point on the picture plane (e.g., a window pane) where the projection of parallel lines (e.g., railroad tracks on a flat surface) would intersect if the projection lines were extended. It could also refer to a point in your visual field to which the parallel lines (railroad tracks) appear to converge. For example, if you lie on your back in a room, you will see with the aid of a broom handle or yardstick that all the vertical lines in the room appear to converge toward the spot on the ceiling directly above your head, and toward the spot on the floor directly beneath you.
VP does not mean a point where parallel lines or their projections onto a plane actually meet, because they don't. Nor does it mean a point at which objects vanish (become imperceptible to human vision) as they move away from the viewer. I don't know if there is a term for such a point, but it's not what VP means.
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- The lines that are parallel in 3D space, are no longer necessarily parallel when projected onto a 2D surface. This is what we mean when we say they "appear" to touch. They "appear" to touch at the vanishing point (V' in the picture). Just because they "appear" to touch in the 2D projection, does not mean that they touch in 3D space.
- The vanishing point only exists in the 2D projection, not in 3D space. If you follow the parallel lines in 3D space, you will NEVER arrive at the vanishing point. This is what we mean when we say "the vanishing point is at an infinite distance away"
Yes. The term "vanishing point" is another source of confusion. Primarily, it refers to a point on the picture plane (e.g., a window pane) where the projection of parallel lines (e.g., railroad tracks on a flat surface) would intersect if the projection lines were extended. It could also refer to a point in your visual field to which the parallel lines (railroad tracks) appear to converge. For example, if you lie on your back in a room, you will see with the aid of a broom handle or yardstick that all the vertical lines in the room appear to converge toward the spot on the ceiling directly above your head, and toward the spot on the floor directly beneath you.
VP does not mean a point where parallel lines or their projections onto a plane actually meet, because they don't. Nor does it mean a point at which objects vanish (become imperceptible to human vision) as they move away from the viewer. I don't know if there is a term for such a point, but it's not what VP means.
What bugs me is that the FEers claim the "Laws of Perspective" prove this and that (you can look it all up),
but when you try to find references all you get are "Rules of Perspective" relating to drawing and painting, and most seems to date from the 1400s, not Greek times.
Also they seem to insist that the Vanishing Point is on the visible horizon, which is demonstrably untrue!
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The math of the ancient Greeks says that two parallel lines should never touch. But, as admitted, they visibly do touch. How is that a proof that the Greeks were correct in their world model? That is direct evidence that they were wrong about their world model.
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The math of the ancient Greeks says that two parallel lines should never touch. But, as admitted, they visibly do touch. How is that a proof that the Greeks were correct in their world model? That is direct evidence that they were wrong about their world model.
The "math of the ancient Greeks" says that parallel lines SHOULD appear to touch on a 2D projection. And they do. What's the problem?
Please try to understand the distinction:
They DON'T actually touch in reality, by definition.
They DO touch in a 2D projection.
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The math of the ancient Greeks says that two parallel lines should never touch. But, as admitted, they visibly do touch. How is that a proof that the Greeks were correct in their world model? That is direct evidence that they were wrong about their world model.
The lines do not actually touch.
THEY ONLY APPEAR TO TOUCH!!!
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The math of the ancient Greeks says that two parallel lines should never touch. But, as admitted, they visibly do touch. How is that a proof that the Greeks were correct in their world model? That is direct evidence that they were wrong about their world model.
Can't you ever see the difference between touch and appear to touch?
Saying "they visibly do touch" is quite a meaningless statement. It would be far more accurate to say "they appear to touch".
Then you claim "That is direct evidence that they were wrong about their world model." Rubbish! Those Greeks never denied that parallel lines appeared to touch. All they said was that they did not touch.
Imagine the lines in question are railway tracks. They would appear to touch in about 3 miles (at a guess), but quite importantly they clearly do not touch, or that TGV flying past us at 200 mph is going to be in BIG BIG BIG TROUBLE in a bit under one minute!
The rays of light in this photo certainly appear to originate within that cloud,
(https://upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Crepuscular_rays_at_Sunset_near_Waterberg_Plateau_edit.jpg/800px-Crepuscular_rays_at_Sunset_near_Waterberg_Plateau_edit.jpg)
yet that is clearly not possible.
The light rays originate from the sun (either 3,000 or 93,000,000 miles away) and what we see is simply perspective.
The actual light rays in either case are near enough to parallel, yet appear to come from a nearby very small source.
Actually touching in that cloud and appearing to touch in that cloud are very different things.
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The math of the ancient Greeks says that two parallel lines should never touch. But, as admitted, they visibly do touch. How is that a proof that the Greeks were correct in their world model? That is direct evidence that they were wrong about their world model.
A perfect illustration of the fallacy you've fallen into appeared in another thread, I present it here for those who don't read every discussion:
This one shows the Suez Canal going to the horizon. I would rather have had one taken with a normal lens and not a wide angle like this - any appearance of curvature is the camera!
(http://i1075.photobucket.com/albums/w433/RabDownunder/From%20-%20Timelapse%20of%20Adrian%20Maeligrsk%20sailing%20down%20the%20expanded%20Suez%20Canal_zpsuug8zfkt.jpg)
From - Timelapse of Adrian Mærsk sailing down the expanded Suez Canal
There are undoubtedly better examples, but...the "vanishing point" is an aid to drawing, not a "physical point". Small objects seem to vanish in quite a short distance. If the resolution of the human eye is about 1' of arc (as I believe the Wiki says) the vanishing distances would be about...130 miles for the 200' width of New Suez canal (but with the quality of that photo and lack of contrast, I doubt it would be that far).
Look at the containers on the ship. They "visibly" shrink as the distance from the camera increases, they are "visibly" converging on a vanishing point, and if they went far enough out they would "visibly" touch. But we all know that in reality, they do not ACTUALLY shrink, they don't ACTUALLY converge, they don't ACTUALLY touch, and the sides that we say are parallel, truly ARE parallel, all the way out. This illustrates the difference between a two-dimensional REPRESENTATION of the world, and the three dimensional REALITY of the world.
Unwillingness to admit (or failure to understand) the difference is the primary failing of the Zetetic "what it LOOKS LIKE is what is IS" approach to inquiry.
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If we see that they appear touch, and the Greek's continuous universe math that they should not appear to touch, that is evidence suggesting that they are wrong in their theories of perspective. It is certainly not evidence that they are correct.
I am not arguing that the lines physically touch, only that they appear to, which goes hand-in-hand with the belief that the sun can appear to touch the earth without it physically doing so. The effect of two parallel lines touching is more evidence towards a Flat Earth model where the celestial bodies can touch the earth than it is evidence for a Greek universe where parallel lines should never touch.
Under the "appearances can be deceiving" mantra you are promoting, where things things don't "ACTUALLY touch" you are also arguing against the physics of your own model, agreeing with the Flat Earth position on this matter that how things appear with perspective may not be how they are.
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If we see that they appear touch, and the Greeks continuous universe math that they should not appear to touch, that is evidence suggesting that they are wrong in their theories of perspective. It is certainly not evidence that they are correct.
Well first of all, Greek math states that they DO appear to touch and that they don't physically touch, not the other way around.
I am not arguing that the lines physically touch, only that they appear to, which goes hand-in-hand with the belief that the sun can appear to touch the earth without it physically doing so. The effect of two parallel lines touching is more evidence towards a Flat Earth model where the celestial bodies can touch the earth than it is evidence for a Greek universe where parallel lines should never touch.
Second of all, it doesn't go "hand in hand" because even if this were true and we had enough distance from the sun for this to theoretically happen, it would appear only to touch the surface and not actually reach any vanishing point(vanishing points are a construct only possible in a 2 dimensional world). This means that if it were able to go any farther away(this distance is almost unfathomably far away at this point), it would uniformly disappear above the horizon due to Rayleigh scattering , which it doesn't do.
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If we see that they appear touch, and the Greeks continuous universe math that they should not appear to touch, that is evidence suggesting that they are wrong in their theories of perspective. It is certainly not evidence that they are correct.
I’m not aware that any Greek geometer or astronomer ever said that parallel lines never appear to touch. And even if they did, it wouldn’t discredit their geometry, because “parallel lines never appear to meet” is not a mathematical proposition, it’s a statement about how certain objects appear to the human eye.
I am not arguing that the lines physically touch, only that they appear to, which goes hand-in-hand with the belief that the sun can appear to touch the earth without it physically doing so. The effect of two parallel lines touching is more evidence towards a Flat Earth model where the celestial bodies can touch the earth than it is evidence for a Greek universe where parallel lines should never touch.
I agree that if the sun were far enough away, it could seem to touch the horizon while remaining at an altitude of 3000 miles. But do the math and see how far it would have to be. The required distance is
d = 3000/(tan\theta) miles,
where \theta is the angle at which the sun appears to be above the horizon. Let’s conservatively estimate that if the bottom edge of the sun were 10 minutes of arc (0.167 degrees) above the horizon, it would appear to touch it. Then we have
d = 3000/(tan 0.167) = 3000/0.002915 = 1,029,263 miles,
if it remained 3000 miles above a flat earth. And that’s not taking refraction into account, which would usually increase the apparent angle above the horizon. There’s no way around this unless you posit hitherto unknown optical phenomena for which I know of no corroborating evidence. And of course RET explains the appearance and path of the sun and moon very simply and accurately.
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From the Wiki (http://wiki.tfes.org/Sinking_Ship_Effect):
On the sinking ship, Rowbotham describes a mechanism by which the hull is hidden by the angular limits of perception - the ship will appear to intersect with the vanishing point and become lost to human perception as the hull's increasingly shallow path creates a tangent on which the hull is so close to the surface of the ocean that the two are indistinguishable. The ship's hull gets so close to the surface of the water as it recedes that they appear to merge together. Where bodies get so close together that they appear to merge is called the Vanishing Point.
That is not what “Vanishing Point” generally means in discussions of perspective. It means a point on the drawing board or canvas toward which lines representing actually parallel lines (such as railroad tracks) appear to converge in the drawing, whether or not these lines actually meet in the drawing. By extension it refers to a similar point in one’s field of vision; for example, all actually vertical lines appear to converge toward the zenith (and nadir). However, if the author of the article wants to use “Vanishing Point” in a special sense as a technical term for purposes of this discussion, that’s fine as long as we don’t confuse it with the normal sense of the term. So let’s call a point at which two objects appear to merge from a specified viewpoint a “VPM.”
The Vanishing Point [VPM] is created when the perspective lines are angled less than one minute of a degree. Hence, this effectively places the vanishing point [VPM] a finite distance away from the observer.
Usually it is taught in art schools that the vanishing point [VPM] is an infinite distance away from the observer […]
Nope. This would be equivalent to saying that the human eye can always distinguish two objects, i.e., see them as two separate objects, no matter how far they are from the observer. Maybe there are art teachers who are that ignorant of their own subject, but I don’t know of any.
[drawings omitted]
This finite distance to the vanishing point [VPM] is what allows ships to shrink into horizon and disappear as their hulls intersect with the vanishing point [VPM] from the bottom up. As the boat recedes into the distance its hull is gradually and perceptively appearing closer and closer to the surface of the sea. At a far off point the hull of the ship is so close to the sea's surface that it is impossible for the observer to tell ocean from hull.
While the sails of the ship may still be visible while the hull is perceptively merged, it's only a matter of time before it too shrinks into the vanishing point [VPM] which rests on the surface of the sea and becomes indiscernible from the surface.
We know that this explanation is true because there are reports of half sunken ships restored by looking at them through telescopes. It has been found that the sinking ship effect effect is purely perceptual, that a good telescope with sufficient zoom will change the observer's perspective and bring the ship's hull back in full view.
That is not at all what has been found. It's a common observation to see only the top parts of ships, the lower parts being below the horizon, whether seen by the naked eye or through a telescope. If flat-earthers believe that a sufficiently high-powered telescope will always bring the hull into view, let's see some convincing documentation of this phenomenon.
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The math of the ancient Greeks says that two parallel lines should never touch. But, as admitted, they visibly do touch. How is that a proof that the Greeks were correct in their world model? That is direct evidence that they were wrong about their world model.
The "math of the ancient Greeks" says that parallel lines SHOULD appear to touch on a 2D projection. And they do. What's the problem?
Please try to understand the distinction:
They DON'T actually touch in reality, by definition.
They DO touch in a 2D projection.
Minor correction: the projection of actually parallel lines from 3-D space onto a plane will always be separate lines that never meet. Projection onto a plane is just like looking through a window pane and imagining what you see to be painted on the glass. However, they could be so close together on the plane that the human eye could not tell whether they actually met or not. That's a question of human eye's perceptual capability, not geometry.
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The math of the ancient Greeks says that two parallel lines should never touch. But, as admitted, they visibly do touch. How is that a proof that the Greeks were correct in their world model? That is direct evidence that they were wrong about their world model.
The "math of the ancient Greeks" says that parallel lines SHOULD appear to touch on a 2D projection. And they do. What's the problem?
Please try to understand the distinction:
They DON'T actually touch in reality, by definition.
They DO touch in a 2D projection.
Minor correction: the projection of actually parallel lines from 3-D space onto a plane will always be separate lines that never meet. Projection onto a plane is just like looking through a window pane and imagining what you see to be painted on the glass. However, they could be so close together on the plane that the human eye could not tell whether they actually met or not. That's a question of human eye's perceptual capability, not geometry.
No. We are talking about a perspective projection here. The lines will absolutely always meet on a 2D perspective projection. They will meet at the vanishing point. The vanishing point is a very real and calculable point on the 2D plane. It has nothing to do with the human eyes' perception capability.
Lines (extended to infinity) will always meet at the vanishing point on the 2D perspective projeciton. Parallel line segments (that don't extend to infinity in 3D) will never actually meet on the 2D projection.
The vanishing point is NOT a real point in 3D space. I think this is what confuses people.
Edit: If you parameterize the lines according to their distance away from the observer in 3D space, then you can never reach the vanishing point on the 2D plane by increasing the parameter. Perhaps this is what you mean? If this is what you mean, I would advise against pushing this point, because it seems to be confusing Tom Bishop. Keep it simple as possible.
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The math of the ancient Greeks says that two parallel lines should never touch. But, as admitted, they visibly do touch. How is that a proof that the Greeks were correct in their world model? That is direct evidence that they were wrong about their world model.
The "math of the ancient Greeks" says that parallel lines SHOULD appear to touch on a 2D projection. And they do. What's the problem?
Please try to understand the distinction:
They DON'T actually touch in reality, by definition.
They DO touch in a 2D projection.
Minor correction: the projection of actually parallel lines from 3-D space onto a plane will always be separate lines that never meet. Projection onto a plane is just like looking through a window pane and imagining what you see to be painted on the glass. However, they could be so close together on the plane that the human eye could not tell whether they actually met or not. That's a question of human eye's perceptual capability, not geometry.
No. We are talking about a perspective projection here. The lines will absolutely always meet on a 2D perspective projection. They will meet at the vanishing point. The vanishing point is a very real and calculable point on the 2D plane. It has nothing to do with the human eyes' perception capability.
If I understand you, you are saying that the line segments in the projection will meet if extended. [Edit: Or you were simply referring to infinite parallel lines, while I was talking about lines with a finite length.] I agree; they will meet at the vanishing point. But line segments that are projections of (actual, finite) parallel lines will not actually meet in the 2D plane, although they can come close enough so that the eye cannot tell the difference. In the same way, vertical lines like telephone poles will meet at the zenith if extended, but will not actually meet there without being extended.
The vanishing point is NOT a real point in 3D space. I think this is what confuses people.
That is certainly one thing that confuses people.
Edit: If you parameterize the lines according to their distance away from the observer in 3D space, then you can never reach the vanishing point on the 2D plane by increasing the parameter. Perhaps this is what you mean? If this is what you mean, I would advise against pushing this point, because it seems to be confusing Tom Bishop. Keep it simple as possible.
I'm not sure what you mean here. In any case, I'm trying to keep things "as simple as possible, but not simpler." ;)
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If I understand you, you are saying that the line segments in the projection will meet if extended. I agree; they will meet at the vanishing point. But line segments that are projections of (actual, finite) parallel lines will not actually meet in the 2D plane, although they can come close enough so that the eye cannot tell the difference. In the same way, vertical lines like telephone poles will meet at the zenith if extended, but will not actually meet there without being extended.
You are exactly right about this. I was being too pedantic. Parallel line segments will never ever meet on the 2D projection. They only appear to meet if they are so far away that the human eye can't distinguish the distance between them.
Edit: blah blah blah... me being stupid... blah
I'm not sure what you mean here. In any case, I'm trying to keep things "as simple as possible, but not simpler." ;)
Feel free to ignore that part...
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Well first of all, Greek math states that they DO appear to touch and that they don't physically touch, not the other way around.
No, they don't. We never learn that in Geometry class. The Ancient Greek geometry math assumes that we live in a continuous universe where resolution is infinite and where perfect circles could exist. The model says that two parallel lines should never touch in such a perfect universe.
However, two parallel lines do seem to touch, perhaps due to several other factors, and therefore, it follows that the model is not an accurate reflection of reality, especially at extreme distances.
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If I understand you, you are saying that the line segments in the projection will meet if extended. I agree; they will meet at the vanishing point. But line segments that are projections of (actual, finite) parallel lines will not actually meet in the 2D plane, although they can come close enough so that the eye cannot tell the difference. In the same way, vertical lines like telephone poles will meet at the zenith if extended, but will not actually meet there without being extended.
You are exactly right about this. I was being too pedantic. Parallel line segments will never ever meet on the 2D projection. They only appear to meet if they are so far away that the human eye can't distinguish the distance between them.
There's no such thing as "too pedantic" in my world. At any rate, our parallel thinking processes seem to have met in the 2D plane of this forum.
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Well first of all, Greek math states that they DO appear to touch and that they don't physically touch, not the other way around.
No, they don't. We never learn that in Geometry class. The greek's geometry math assumes that we live in a continuous universe where resolution is infinite and where perfect circles could exist. The model says that two parallel lines should never touch in such a perfect universe.
For lines of finite length, this is true. This is a pretty good assumption. Why would this not be a good assumption?
However, two parallel lines do seem to touch, perhaps due to several other factors, and therefore, ...
Yes, but they only seem to touch if they are far enough away relative to the distance between them. The specific factor that causes them to seem to touch is the fact that cameras and eyes do NOT have infinite resolution.
...it follows that the model is not an accurate reflection of reality, especially at extreme distances.
If our eyes had infinite resolution, then perhaps that would indeed follow. But they don't, so it doesn't.
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The model the Greeks proposed is clearly wrong when it comes to things that are far away. The simple fact is that the lines touch. There may be varying explanations for why they touch.
It might have something to do with resolution. It might also be more than that. For instance, if we shine a laser beam at the point on railroad tracks where they appear to touch in the distance, the beam will widen and touch both of the tracks at the same time.
From http://wiki.tfes.org/Magnification_of_the_Sun_at_Sunset --
Beam Divergence
This phenomenon of enlarging rays is also seen in lasers. Supposedly "straight" rays of light will spread out when shining over long distances.
(http://wiki.tfes.org/images/0/0e/Beam_divergence.jpg)
From the Wikipedia entry on Beam Divergence (https://en.wikipedia.org/wiki/Beam_divergence) we read:
"The beam divergence of an electromagnetic beam is an angular measure of
the increase in beam diameter or radius with distance from the optical
aperture or antenna aperture from which the electromagnetic beam emerges."
The light is broadcasted towards the small scene in the distance and widens appropriately to cover that area it sees. Under a perfect universe the laser beam should only be able to touch only one of the tracks at a time when it reaches the destination. However, the beam is seen to widen significantly, easily covering both tracks and an area of landscape. It seems to suggest that, if the small laser beam diameter can cover a large area, the squishing of the tracks to a single point is more than a resolution limitation of the eye.
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Well first of all, Greek math states that they DO appear to touch and that they don't physically touch, not the other way around.
No, they don't. We never learn that in Geometry class. The greek's geometry math assumes that we live in a continuous universe where resolution is infinite and where perfect circles could exist. The model says that two parallel lines should never touch in such a perfect universe.
That's because by definition, parallel lines will never intersect on any given plane. If we can show that over any finite distance two parallel lines will not touch, why wouldn't we assume that over longer distances approaching infinity that they would continue to not touch? They only seem to touch because we do not have eyes that can show extremely small distances, i.e infinite resolution.
Also, hey Totes, how you doin?
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The model the Greeks proposed is clearly wrong when it comes to things that are far away. The simple fact is that the lines touch. There may be varying explanations for why they touch.
It might have something to do with resolution. It might also be more than that. For instance, if we shine a laser beam at the point on railroad tracks where they appear to touch in the distance, the beam will widen and touch both of the tracks at the same time.
From http://wiki.tfes.org/Magnification_of_the_Sun_at_Sunset --
Beam Divergence
This phenomenon of enlarging rays is also seen in lasers. Supposedly "straight" rays of light will spread out when shining over long distances.
(http://wiki.tfes.org/images/0/0e/Beam_divergence.jpg)
From the Wikipedia entry on Beam Divergence (https://en.wikipedia.org/wiki/Beam_divergence) we read:
"The beam divergence of an electromagnetic beam is an angular measure of
the increase in beam diameter or radius with distance from the optical
aperture or antenna aperture from which the electromagnetic beam emerges."
The light is broadcasted towards the small scene in the distance and widens appropriately to cover that area it sees. Under a perfect universe the laser beam should only be able to touch only one of the tracks at a time when it reaches the destination. However, the beam is seen to widen significantly, easily covering both tracks and an area of landscape. It seems to suggest that the squishing of the tracks to a single point is more than a resolution limitation of the eye.
Beam divergence occurs because light shines in every direction from a light source. Because it's confined to emit from a one sided light, the "laser beam" will actually be a very narrow cone that spreads out because all of the light in this cone is moving in a straight line. It's not as if the light will start bending midway through transmission. Say instead that I took a single photon and fired it in a direction, never will that photon change direction unless it is reflected or refracted by other outside influence.
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The model the Greeks proposed is clearly wrong when it comes to things that are far away. The simple fact is that the lines touch. There may be varying explanations for why they touch.
It might have something to do with resolution. It might also be more than that. For instance, if we shine a laser beam at the point on railroad tracks where they appear to touch in the distance, the beam will widen and touch both of the tracks at the same time.
Yes, there are multiple reasons why they can appear to touch. Insufficient resolving power of the eye/camera (as has been stated multiple times) is one of them. Decreased clarity due to atmospheric scattering is another. But this is all irrelevant. The basic geometry deals with a perfect world. There is no such thing as perfectly parallel lines, or a perfectly clear atmosphere. But that doesn't mean the entire theory is useless. We can still make VERY accurate approximations with the theory. This is why it is important to be able to estimate how much error is possible in your calculation. How is the divergence of a laser beam going to cause significant errors in how we percieve parallel lines?
Going by your logic, we should never use math for anything, since nothing can be calculated 100% accurately. Is this what you are arguing for?
From http://wiki.tfes.org/Magnification_of_the_Sun_at_Sunset --
Holy Toledo, that page is so full of misinformation... please don't tell me you actually think light sources appear larger the farther you are away from them.
Also, hey Totes, how you doin?
Just chillin' yo
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The model the Greeks proposed is clearly wrong when it comes to things that are far away. The simple fact is that the lines touch. There may be varying explanations for why they touch.
It might have something to do with resolution. It might also be more than that. For instance, if we shine a laser beam at the point on railroad tracks where they appear to touch in the distance, the beam will widen and touch both of the tracks at the same time.
Are you really persisting with this? The simple fact is that the lines do not touch, they only appear to touch!
You claim "There may be varying explanations for why they touch." NO, they DO NOT TOUCH, they appear to touch!
The diverging laser beam means nothing at all. All the laser beam, a spotlight or the sun does is illuminate the railway tracks.
We are NOT looking at the laser beam and how much its beam diverges has absolutely nothing to do with the case.
There are cases when we are looking at the source of the light, such as: - looking directly at the laser pointer (with appropriate protective eyewear)
- looking directly at the sun (again with appropriate protective eyewear)
- looking directly at stars at light.
In all these cases the apparent size of the object (measured as the angle subtended at the eye) is simple the (size of the object)/(distance to the object) so for these cases the apparent sizes would be at a "guess" (taking the laser pointer at 3 miles - where I claimed the railway lines converged):- Guess 1 mm (0.025") diameter - apparent size (0.039")/(3 x 12 x 5280") = 2.1x10-7 rad or 0.043" of arc, far too small to resolve.
- (Globe) Sun's diameter = 900,000 miles, Sun's distance = 93,000,000 miles, so apparent size = 900,000/93,000,000 = 0.01 rad or 0.55° - obviously visible.
- Say Alpha Centauri Estimated diameter = 1,100,000 miles, estimated distance = 4.37 ly or 2.57x1013 miles, so apparent size = 0.009" of arc, far too small to resolve.
I am using Globe figures as the FE supporters seem to have no idea of how big the stars might be.
As I posted before a simple test on whether lines meet is to simply go there and check it out, as with the TGV analogy in my previous post.
Of course this is only practical for achievable distances.
But the bottom line is that your diverging laser beam means absolutely nothing.
We all know that a laser beam has a finite (though very small) divergence angle - but so what, other light sources have much bigger divergent angles.
The sun's light shines over a very large angle (a full sphere), but this has not the slightest effect on the apparent size of objects we see!
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Beam divergence occurs because light shines in every direction from a light source. Because it's confined to emit from a one sided light, the "laser beam" will actually be a very narrow cone that spreads out because all of the light in this cone is moving in a straight line. It's not as if the light will start bending midway through transmission. Say instead that I took a single photon and fired it in a direction, never will that photon change direction unless it is reflected or refracted by other outside influence.
The phenomena of beam divergence is certainly a curiosity, particularly because a laser beam is supposed to be straight due to photons between a series of mirrors and a glass amplifier to produce an extremely bright and straight beam of light. It may be argued that some of the photons are not straight, but then the divergence should have a central hot spot as the beam diverges.
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Yes, there are multiple reasons why they can appear to touch. Insufficient resolving power of the eye/camera (as has been stated multiple times) is one of them. Decreased clarity due to atmospheric scattering is another. But this is all irrelevant. The basic geometry deals with a perfect world. There is no such thing as perfectly parallel lines, or a perfectly clear atmosphere. But that doesn't mean the entire theory is useless.
If their theory doesn't match observations it means the theory is wrong and must be modified or discarded.
Going by your logic, we should never use math for anything, since nothing can be calculated 100% accurately. Is this what you are arguing for?
If the math can't calculate things far away accurately, what reason is there to use that math for things that are far away?
From http://wiki.tfes.org/Magnification_of_the_Sun_at_Sunset --
Holy Toledo, that page is so full of misinformation... please don't tell me you actually think light sources appear larger the farther you are away from them.
Have you never seen headlights in fog?
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The model the Greeks proposed is clearly wrong when it comes to things that are far away. The simple fact is that the lines touch. There may be varying explanations for why they touch.
It might have something to do with resolution. It might also be more than that. For instance, if we shine a laser beam at the point on railroad tracks where they appear to touch in the distance, the beam will widen and touch both of the tracks at the same time.
Are you really persisting with this? The simple fact is that the lines do not touch, they only appear to touch!
You claim "There may be varying explanations for why they touch." NO, they DO NOT TOUCH, they appear to touch!
Maybe the photons coming from the tracks are hitting your eye in a way that the photons are touching (or getting as close as they physically can to each other). It has not been demonstrated that the effect is due to "lack of resolution". The human eye is incredibly sensitive. Tests have been done where the human eye can detect a single photon in a dark room.
A photon is the physical manifestation of that object at distance, after all. This society seeks to ask and explore such vexing questions, not mindlessly scream that "NO, they DO NOT TOUCH".
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An experiment for Mr. Bishop, which is absolutely within the realm of possibility.
Items needed:
1. Yourself and at least one trusted companion
2. Stretch of train track of sufficient length (3 to 5 miles) in a straight line (no sweeps left or right)
3. A means of communication between participants (cell phone, radio, semaphore, etc.)
4. Markers of agreed upon type or method
5. Two measurement devices agreed upon by you and your trusted companion as being accurate
6. Two notebooks
7. At least two pencils, possibly more
This is really quite simple.
To begin, both of you start at the same point in the middle (between the rails) of the track. At this point both of you make note of the width of the track. Have your trusted companion move forward along the track, keeping notes of the width of the track at regular intervals and placing a marker at each point measured, to the point where the rails appear, to you, to touch. Communicate via your chosen method for your trusted companion to stop at this point and make note of the width of the track.
Now, you make your way forward, stopping to measure the width of the track at each point marked by your trusted companion, until you reach the point where you had your trusted companion stop. At this point make a final measurement of the width of the track.
Once you have made your final measurement, compare your own measurements across the entire length you've traveled to see if your measurements have arrived at zero (as they would if the rails touched). Once you have done this, compare your measurements to those of your trusted companion to see if their measurements have arrived at zero (as they would if the rails touched).
I'm only suggesting this because it seems to me that you appear capable of performing such an experiment. Of course my perspective may be skewed.
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Yes, there are multiple reasons why they can appear to touch. Insufficient resolving power of the eye/camera (as has been stated multiple times) is one of them. Decreased clarity due to atmospheric scattering is another. But this is all irrelevant. The basic geometry deals with a perfect world. There is no such thing as perfectly parallel lines, or a perfectly clear atmosphere. But that doesn't mean the entire theory is useless.
If their theory doesn't match observations it means the theory is wrong and must be modified or discarded.
Going by your logic, we should never use math for anything, since nothing can be calculated 100% accurately. Is this what you are arguing for?
If the math can't calculate things far away accurately, what reason is there to use that math for things that are far away?
From http://wiki.tfes.org/Magnification_of_the_Sun_at_Sunset --
Holy Toledo, that page is so full of misinformation... please don't tell me you actually think light sources appear larger the farther you are away from them.
Have you never seen headlights in fog?
You say "If their theory doesn't match observations it means the theory is wrong and must be modified or discarded." But you have never shown that their theory does not match observations!
You earlier said "The model the Greeks proposed is clearly wrong when it comes to things that are far away. The simple fact is that the lines touch" and here YOU are simply mistaken! As I have tried to get across with parallel lines the simple fact is that the lines DO NOT touch, they do APPEAR to TOUCH.
In addition you blame poor Euclid and those other Greeks for everything. But our modern ideas are based on much more than that.
Take a look at: Geometry: A History from Practice to Abstraction (https://nrich.maths.org/6352), where we learn of:- Ancient and Classical Geometries
Aristotle (384-322) BCE, Euclid of Alexandria (325-265) BCE (maybe tacking on Playfair's Axiom), Abul Wafa al-Buzjani (940-998), Omar Khayyam (1048-1131) - yes even him, Nasir al-Din al-Tusi (1201-1274) - Renaissance and Early Modern Developments
The Painters' Perspective (Al-Haytham (965-1039)), Leone Battista Alberti (1404-1472), Piero della Francesca (1412-1492) etc, etc. - Modern Geometries
Girolamo Saccheri (1667-1733), Johan Heinrich Lambert (1728-1777), Adrien-Marie Legendre (1752-1833), Carl Friedrich Gauss (1777-1855),
Nikolai Ivanovich Lobachevski (1792-1856) (finally getting into non-Euclidean stuff), János Bolyai (1802-1860), Eugenio Beltrami (1835-1900), etc, etc.
So, the Greeks Philosophers just laid a foundation.
Again as I said before you can either actually travel to where they appeared to touch, and of course you find that they did not touch, or at least do a "thought experiment". If you try to claim that they DO touch you need a good physical reason to make that claim.
Now, of course I do accept that we live in slightly non-Euclidean space as postulated in Einstein's GR, but the actual curvature in our region is smaller that minute. The sun's huge gravitational field only deflects light by about 1.8 seconds of arc, and half that is due to time dilation.
And again "Have you never seen headlights in fog?" Of course I have seen headlights in a fog, but have you seen the sun setting in a perfectly clear sky and appearing exactly the same size (maybe slightly larger - the Ponzo illusion) as it did at midday?
There is no way "atmospheric magnification" can keep the sun exactly the same size all day. Apart from anything else is happens every day. It can appear fuzzy and larger through cloud, but that is obvious when it happens and can happen in any part of the sky.
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Tom at what distance does math no longer work? 1, 10, 100, 1000, 10000 or further?
Just your estimate where math fails to be able to give good estimates or right answers?
I can use that math to estimate ranges when I sail and had accurate results.
I used it recently to determine how much fuel a rounded tank on my boat could hold and the answer I got was verified to be correct when I put fuel in it.
I used it for celestial navigation and determined my position accurately with noon and star sightings.
Throughout my life I have used it in my career and personal life and consistently had it verified since it gave me the right answers that were verified correct when applied to projects, the amount of material needed, how much liquid something could hold, etc.
So when does math no longer work? I ask because you seem assured you are right and must know at what distances it is no longer accurate. My experience is that it does return the right answers. An experience I highly doubt that the majority of people would say the opposite.
Did you have evidence of being correct or is the only evidence you offer is if the Greeks were right you are wrong about the shape of the Earth?
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The phenomena of beam divergence is certainly a curiosity, particularly because a laser beam is supposed to be straight due to photons between a series of mirrors and a glass amplifier to produce an extremely bright and straight beam of light.
According to whom is a laser beam "straight"? How would mirrors an amplifiers result in parallel photons?
It may be argued that some of the photons are not straight, but then the divergence should have a central hot spot as the beam diverges.
Why? Do you refer to a focal point?
If their theory doesn't match observations it means the theory is wrong and must be modified or discarded.
According to Kant, Euclidean Geometry isn't a theory in the traditional sense. It's based on an a-priori understanding of space, not on observation.
If the math can't calculate things far away accurately, what reason is there to use that math for things that are far away?
Define " accurately".
Maybe the photons coming from the tracks are hitting your eye in a way that the photons are touching (or getting as close as they physically can to each other). It has not been demonstrated that the effect is due to "lack of resolution". The human eye is incredibly sensitive. Tests have been done where the human eye can detect a single photon in a dark room.
A photon is the physical manifestation of that object at distance, after all. This society seeks to ask and explore such vexing questions, not mindlessly scream that "NO, they DO NOT TOUCH".
Even if you want to call the photons emitted a "physical manifestation", the image they produce in you brain isn't.
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Beam divergence occurs because light shines in every direction from a light source. Because it's confined to emit from a one sided light, the "laser beam" will actually be a very narrow cone that spreads out because all of the light in this cone is moving in a straight line. It's not as if the light will start bending midway through transmission. Say instead that I took a single photon and fired it in a direction, never will that photon change direction unless it is reflected or refracted by other outside influence.
The phenomena of beam divergence is certainly a curiosity, particularly because a laser beam is supposed to be straight due to photons between a series of mirrors and a glass amplifier to produce an extremely bright and straight beam of light. It may be argued that some of the photons are not straight, but then the divergence should have a central hot spot as the beam diverges.
It's not a curiosity, it's actually quite the opposite. It just appears that you don't fundamentally understand how light is emitted from a source. Much like sound, light, when produced from a source, moves in all possible directions from its origin point. The casing of a laser pointer isn't exactly dead focal, meaning that the "laser beam" is a uniform and extremely narrow cone. Because it is a narrow cone, it spreads out as it moves farther away from its point of origin. You cannot attribute this to light bending or not moving in a straight line, it's an invalid argument.
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You earlier said "The model the Greeks proposed is clearly wrong when it comes to things that are far away. The simple fact is that the lines touch" and here YOU are simply mistaken! As I have tried to get across with parallel lines the simple fact is that the lines DO NOT touch, they do APPEAR to TOUCH.
If one looks at the scene, they do touch. It's a factual statement. "Appear" is implied.
According to the mathematical model of the Ancient Greeks, they should never touch.
And again "Have you never seen headlights in fog?" Of course I have seen headlights in a fog, but have you seen the sun setting in a perfectly clear sky and appearing exactly the same size (maybe slightly larger - the Ponzo illusion) as it did at midday?
There is no way "atmospheric magnification" can keep the sun exactly the same size all day. Apart from anything else is happens every day. It can appear fuzzy and larger through cloud, but that is obvious when it happens and can happen in any part of the sky.
Several examples were given in the link showing that the enlargement is proportional to distance, causing the body to seem the same size.
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Tom at what distance does math no longer work? 1, 10, 100, 1000, 10000 or further?
Just your estimate where math fails to be able to give good estimates or right answers?
I can use that math to estimate ranges when I sail and had accurate results.
I used it recently to determine how much fuel a rounded tank on my boat could hold and the answer I got was verified to be correct when I put fuel in it.
I used it for celestial navigation and determined my position accurately with noon and star sightings.
Throughout my life I have used it in my career and personal life and consistently had it verified since it gave me the right answers that were verified correct when applied to projects, the amount of material needed, how much liquid something could hold, etc.
So when does math no longer work? I ask because you seem assured you are right and must know at what distances it is no longer accurate. My experience is that it does return the right answers. An experience I highly doubt that the majority of people would say the opposite.
Did you have evidence of being correct or is the only evidence you offer is if the Greeks were right you are wrong about the shape of the Earth?
It certainly does not work at the vanishing point of railroad tracks, as the math says that they do not touch, when they observably do touch. The observation is evidence that the world model as they described it is wrong.
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It certainly does not work at the vanishing point of railroad tracks, as the math says that they do not touch, when they observably do touch. The observation is evidence that the world model as they described it is wrong.
Only if you cherry pick some observations and ignore others. Euclidean geometry is perfectly in accordance with observation since observation is based on euclidean geometry.
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It certainly does not work at the vanishing point of railroad tracks, as the math says that they do not touch, when they observably do touch. The observation is evidence that the world model as they described it is wrong.
Only if you cherry pick some observations and ignore others. Euclidean geometry is perfectly in accordance with observation since observation is based on euclidean geometry.
Where have we observed perfect circles?
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Tom at what distance does math no longer work? 1, 10, 100, 1000, 10000 or further?
Just your estimate where math fails to be able to give good estimates or right answers?
I can use that math to estimate ranges when I sail and had accurate results.
I used it recently to determine how much fuel a rounded tank on my boat could hold and the answer I got was verified to be correct when I put fuel in it.
I used it for celestial navigation and determined my position accurately with noon and star sightings.
Throughout my life I have used it in my career and personal life and consistently had it verified since it gave me the right answers that were verified correct when applied to projects, the amount of material needed, how much liquid something could hold, etc.
So when does math no longer work? I ask because you seem assured you are right and must know at what distances it is no longer accurate. My experience is that it does return the right answers. An experience I highly doubt that the majority of people would say the opposite.
Did you have evidence of being correct or is the only evidence you offer is if the Greeks were right you are wrong about the shape of the Earth?
It certainly does not work at the vanishing point of railroad tracks, as the math says that they do not touch, when they observably do touch. The observation is evidence that the world model as they described it is wrong.
It certainly does work with the railway tracks. They do not touch, they only appear to touch. How many times do we have to say the same thing?
PROOF:
Imagine the lines in question are railway tracks. They would appear to touch in about 3 miles (at a guess), but quite importantly they clearly do not touch, or that TGV flying past us at 200 mph is going to be in BIG BIG BIG TROUBLE in a bit under one minute!
Go and have a look here if want to see what might happen Tgv crash (http://i1.mirror.co.uk/incoming/article6832735.ece/ALTERNATES/s615b/Rescue-workers-search-the-wreckage-of-a-test-TGV-train-that-derailed-and-crashed-in-a-canal-outside-Eckwersheim-near.jpg), not the same cause.
Yes, I know I posted it before, but sometimes reality takes while to sink in.
Clearly railway tracks DO NOT ACTUALLY MEET, THEY ONLY APPEAR to MEET!
As I posted earlier in many cases we can test whether lines meet, by simply travelling to where they appear to meet.
Of course in some cases we may not be certain that the "lines" are truly parallel. That is no reflection on the geometry.
I'm afraid you must live in a different world to the rest of us, one quite divorced from reality!
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Where have we observed perfect circles?
Nowhere. Which is why I said observation is based on euclidean geometry, not the other way round. We don't observe perfect circles or parallel lines. But euclidean geometry does also account for non-perfect circles and almost-parallel lines, so that observation doesn't conflict with the geometry.
In any case, you have shifted the question. The original point was that converging parallel lines only conflict with euclidean geometry if a.) you fail to account for the fact that the lines aren't actually perfectly parallel and b.) you fail to account for the fact that moving the observer makes the lines appear parallel once more.
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Tom at what distance does math no longer work? 1, 10, 100, 1000, 10000 or further?
Just your estimate where math fails to be able to give good estimates or right answers?
I can use that math to estimate ranges when I sail and had accurate results.
I used it recently to determine how much fuel a rounded tank on my boat could hold and the answer I got was verified to be correct when I put fuel in it.
I used it for celestial navigation and determined my position accurately with noon and star sightings.
Throughout my life I have used it in my career and personal life and consistently had it verified since it gave me the right answers that were verified correct when applied to projects, the amount of material needed, how much liquid something could hold, etc.
So when does math no longer work? I ask because you seem assured you are right and must know at what distances it is no longer accurate. My experience is that it does return the right answers. An experience I highly doubt that the majority of people would say the opposite.
Did you have evidence of being correct or is the only evidence you offer is if the Greeks were right you are wrong about the shape of the Earth?
It certainly does not work at the vanishing point of railroad tracks, as the math says that they do not touch, when they observably do touch. The observation is evidence that the world model as they described it is wrong.
It certainly does work with the railway tracks. They do not touch, they only appear to touch. How many times do we have to say the same thing?
PROOF:
Imagine the lines in question are railway tracks. They would appear to touch in about 3 miles (at a guess), but quite importantly they clearly do not touch, or that TGV flying past us at 200 mph is going to be in BIG BIG BIG TROUBLE in a bit under one minute!
Go and have a look here if want to see what might happen Tgv crash (http://i1.mirror.co.uk/incoming/article6832735.ece/ALTERNATES/s615b/Rescue-workers-search-the-wreckage-of-a-test-TGV-train-that-derailed-and-crashed-in-a-canal-outside-Eckwersheim-near.jpg), not the same cause.
Yes, I know I posted it before, but sometimes reality takes while to sink in.
Clearly railway tracks DO NOT ACTUALLY MEET, THEY ONLY APPEAR to MEET!
As I posted earlier in many cases we can test whether lines meet, by simply travelling to where they appear to meet.
Of course in some cases we may not be certain that the "lines" are truly parallel. That is no reflection on the geometry.
I'm afraid you must live in a different world to the rest of us, one quite divorced from reality!
If one looks at the scene, they do touch. It's a factual statement. "Appear" is implied.
According to the mathematical model of the Ancient Greeks, they should never touch.
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You earlier said "The model the Greeks proposed is clearly wrong when it comes to things that are far away. The simple fact is that the lines touch" and here YOU are simply mistaken! As I have tried to get across with parallel lines the simple fact is that the lines DO NOT touch, they do APPEAR to TOUCH.
If one looks at the scene, they do touch. It's a factual statement. "Appear" is implied.
According to the mathematical model of the Ancient Greeks, they should never touch.
Math is a purely logical construction; it concerns what can be deduced from an initial set of propositions. So the only way to refute a mathematical conclusion (i.e., a statement that proposition P can be deduced logically from the initial premises A, B, C,…) is to show a logical error in the proof.
In the Elements, Euclid defined parallel lines as lines in the same plane that never meet no matter how far they are extended in either direction. So if two lines meet, by definition they are not parallel in Euclid’s sense. Saying “these two parallel lines actually touch, therefore Euclid was wrong,” is like saying “this triangle has four sides, therefore Euclid was wrong about triangles.”
If logically valid math gives you the wrong answer when applied to the real world, the problem is with how you have applied it, not with the math.
Example: Ancient Greek arithmetic says 1 + 1 = 2. So you take one rabbit and put it in an enclosure. Then you put another rabbit of the opposite sex in the enclosure. You come back in a few months, find a dozen rabbits in the enclosure, and declare that the world model of Greek arithmetic is wrong. But the error is actually in how you have applied the math.
Example: Euclid proved that, given the axioms, postulates, and definitions from which he begins, the angles of a triangle on a flat plane sum to two right angles (180 degrees). No one has found an error in his proof in 2,300 years. Now you find a triangle somewhere, measure the three angles, and find that the sum is 181 degrees. Aha, you say, Euclid’s world view is wrong. No, his proof is correct. There could be several reasons why your sum differs from Euclid’s: Your measuring instruments could be inaccurate. Rounding errors could add up to a degree. You lost your glasses and misread your instruments. The sides of the triangle are not perfectly straight. Or, the sides of the triangle are actually great-circle paths on the surface of a spherical planet, whereas Euclid’s proof concerns triangles on a flat plane.
Example: You look at straight railroad tracks extending miles into the distance on a flat plain. You observe that what your brain tells you are rail lines in your field of vision meet at the horizon, and conclude that Euclid’s world view is wrong. No, you’ve just misapplied his reasoning. He never said that parallel lines will never appear to meet in your field of vision, no matter how far away they are. Now you look at the lines through a telescope, and the lines you see don’t meet any more. Hmm.
So what you seem be saying is that if lines in your field of vision actually meet as interpreted by your brain, then the lines out there, miles away, actually do meet. But that doesn’t explain why they don’t meet any more when viewed through a telescope. Nor is it consistent with our everyday observation that what looks like an ellipse turns out to be a circle when viewed from a different angle.
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So Tom thinks the math is wrong since if we look at something like railroad tracks and have enough viewing distance they appear to touch.
The math tells us they do not actually touch and a very simple observation will confirm this. Just watch a train travel down those tracks. Unless the train shrinks as it gets further away.
Tom why not show us using this faulty math where it says two parallel lines actually touch and not appear to touch do to perspective?
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So Tom thinks the math is wrong since if we look at something like railroad tracks and have enough viewing distance they appear to touch.
The math tells us they do not actually touch and a very simple observation will confirm this. Just watch a train travel down those tracks. Unless the train shrinks as it gets further away.
Tom why not show us using this faulty math where it says two parallel lines actually touch and not appear to touch do to perspective?
If one looks at the scene, they do touch. It's a factual statement. "Appear" is implied.
According to the mathematical model of the Ancient Greeks, they should never touch.
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Math is a purely logical construction; it concerns what can be deduced from an initial set of propositions. So the only way to refute a mathematical conclusion (i.e., a statement that proposition P can be deduced logically from the initial premises A, B, C,…) is to show a logical error in the proof.
An error in the proof is that parallel lines seem to converge in contradiction of theory.
In the Elements, Euclid defined parallel lines as lines in the same plane that never meet no matter how far they are extended in either direction. So if two lines meet, by definition they are not parallel in Euclid’s sense. Saying “these two parallel lines actually touch, therefore Euclid was wrong,” is like saying “this triangle has four sides, therefore Euclid was wrong about triangles.”
Elucid was wrong about a lot of things. Look up Zeno's Paradox. The Greek model of the universe is flimsy.
Example: You look at straight railroad tracks extending miles into the distance on a flat plain. You observe that what your brain tells you are rail lines in your field of vision meet at the horizon, and conclude that Euclid’s world view is wrong. No, you’ve just misapplied his reasoning. He never said that parallel lines will never appear to meet in your field of vision, no matter how far away they are. Now you look at the lines through a telescope, and the lines you see don’t meet any more. Hmm.
So what you seem be saying is that if lines in your field of vision actually meet as interpreted by your brain, then the lines out there, miles away, actually do meet. But that doesn’t explain why they don’t meet any more when viewed through a telescope. Nor is it consistent with our everyday observation that what looks like an ellipse turns out to be a circle when viewed from a different angle.
If the Greek models are corrupted by illusions at long distances then we must admit that there are illusions present in the subject matter and that the unsatisfactory Greek models cannot be used as a disproof of what a Flat Earth sun might or might not do.
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So Tom thinks the math is wrong since if we look at something like railroad tracks and have enough viewing distance they appear to touch.
The math tells us they do not actually touch and a very simple observation will confirm this. Just watch a train travel down those tracks. Unless the train shrinks as it gets further away.
Tom why not show us using this faulty math where it says two parallel lines actually touch and not appear to touch do to perspective?
If one looks at the scene, they do touch. It's a factual statement. "Appear" is implied.
According to the mathematical model of the Ancient Greeks, they should never touch.
The Greek model does not say they actually touch. That is why I asked you to show us where it says they actually touch. The important part is appearing to touch and actually touching.
I have not found anything saying two parallel actually touch, only appear to touch as the result of perspective. I can find things saying parallel will never touch and can use what the Greeks came up with those lines do not touch at any distance.
They do not touch:
(http://www.cpalms.org/Uploads/resources/71891/Rubric/49682/graphics/MFAS_ProvingTheSlopeCriterionForParallelLinesOne_Image3.jpg)
They can appear to touch:
(http://www.handprint.com/HP/WCL/IMG/LPR/vpgeom1.gif)
Using the Greek model you claim is wrong it says two parallel lines never touch.
The only argument I can think of you can make is something similar to this:
(https://s-media-cache-ak0.pinimg.com/736x/a1/b3/17/a1b3177f793fcd6ceac3bbc901dee666.jpg)
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The Greek model does not say they actually touch. That is why I asked you to show us where it says they actually touch. The important part is appearing to touch and actually touching.
If one looks at the scene, they do touch. It's a factual statement. "Appear" is implied.
According to the mathematical model of the Ancient Greeks, they should never touch.
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The Greek model does not say they actually touch. That is why I asked you to show us where it says they actually touch. The important part is appearing to touch and actually touching.
If one looks at the scene, they do touch. It's a factual statement. "Appear" is implied.
According to the mathematical model of the Ancient Greeks, they should never touch.
Are you down with your Orwellian double-speak yet?
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Tom at what distance does math no longer work? 1, 10, 100, 1000, 10000 or further?
Just your estimate where math fails to be able to give good estimates or right answers?
I can use that math to estimate ranges when I sail and had accurate results.
I used it recently to determine how much fuel a rounded tank on my boat could hold and the answer I got was verified to be correct when I put fuel in it.
I used it for celestial navigation and determined my position accurately with noon and star sightings.
Throughout my life I have used it in my career and personal life and consistently had it verified since it gave me the right answers that were verified correct when applied to projects, the amount of material needed, how much liquid something could hold, etc.
So when does math no longer work? I ask because you seem assured you are right and must know at what distances it is no longer accurate. My experience is that it does return the right answers. An experience I highly doubt that the majority of people would say the opposite.
Did you have evidence of being correct or is the only evidence you offer is if the Greeks were right you are wrong about the shape of the Earth?
It certainly does not work at the vanishing point of railroad tracks, as the math says that they do not touch, when they observably do touch. The observation is evidence that the world model as they described it is wrong.
It certainly does work with the railway tracks. They do not touch, they only appear to touch. How many times do we have to say the same thing?
PROOF:
Imagine the lines in question are railway tracks. They would appear to touch in about 3 miles (at a guess), but quite importantly they clearly do not touch, or that TGV flying past us at 200 mph is going to be in BIG BIG BIG TROUBLE in a bit under one minute!
Go and have a look here if want to see what might happen Tgv crash (http://i1.mirror.co.uk/incoming/article6832735.ece/ALTERNATES/s615b/Rescue-workers-search-the-wreckage-of-a-test-TGV-train-that-derailed-and-crashed-in-a-canal-outside-Eckwersheim-near.jpg), not the same cause.
Yes, I know I posted it before, but sometimes reality takes while to sink in.
Clearly railway tracks DO NOT ACTUALLY MEET, THEY ONLY APPEAR to MEET!
As I posted earlier in many cases we can test whether lines meet, by simply travelling to where they appear to meet.
Of course in some cases we may not be certain that the "lines" are truly parallel. That is no reflection on the geometry.
I'm afraid you must live in a different world to the rest of us, one quite divorced from reality!
If one looks at the scene, they do touch. It's a factual statement. "Appear" is implied.
According to the mathematical model of the Ancient Greeks, they should never touch.
According to the mathematical model of the Ancient Greeks, they should never touch.
And they never do touch - as I believe I quite convincingly proved with the railway tracks!
YOU claim: "If one looks at the scene, they do touch. It's a factual statement. "Appear" is implied."
And YOU say the "Appear" is implied. But the Greeks did not ever say "they do touch".
In fact what Euclid did say in his Fifth Postulate was: (https://upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Parallel_postulate_en.svg/525px-Parallel_postulate_en.svg.png)
If the sum of the interior angles α and β is less than 180°,
the two straight lines, produced indefinitely, meet on that side.
What Euclid states is that lines which are not parallel do converge. He does not even say that parallel do not converge.
Now I am no expert on Greek geometry (and of course we do not simply "rely on Greek geometry" anyway!), so maybe you can show where they even say that parallel lines "appear to touch" somewhere!
Can't you ever see the difference between what appears to be true and what is actually true?
If the railway tracks actually did meet we would have a horrendous smash. We do not have a smash (hopefully!) so the railway tracks do not meet at their "vanishing point".
Still I guess there is not point discussing this further. For some people waords have well defined meanings, for others:
"When I use a word," Humpty Dumpty said in rather a scornful tone, "it means just what I choose it to mean — neither more nor less."
PS Never, ever change your mind and accept the fact that the earth is really a Globe. We just could not take your logic on our side!
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Tom at what distance does math no longer work? 1, 10, 100, 1000, 10000 or further?
Just your estimate where math fails to be able to give good estimates or right answers?
I can use that math to estimate ranges when I sail and had accurate results.
I used it recently to determine how much fuel a rounded tank on my boat could hold and the answer I got was verified to be correct when I put fuel in it.
I used it for celestial navigation and determined my position accurately with noon and star sightings.
Throughout my life I have used it in my career and personal life and consistently had it verified since it gave me the right answers that were verified correct when applied to projects, the amount of material needed, how much liquid something could hold, etc.
So when does math no longer work? I ask because you seem assured you are right and must know at what distances it is no longer accurate. My experience is that it does return the right answers. An experience I highly doubt that the majority of people would say the opposite.
Did you have evidence of being correct or is the only evidence you offer is if the Greeks were right you are wrong about the shape of the Earth?
It certainly does not work at the vanishing point of railroad tracks, as the math says that they do not touch, when they observably do touch. The observation is evidence that the world model as they described it is wrong.
It certainly does work with the railway tracks. They do not touch, they only appear to touch. How many times do we have to say the same thing?
PROOF:
Imagine the lines in question are railway tracks. They would appear to touch in about 3 miles (at a guess), but quite importantly they clearly do not touch, or that TGV flying past us at 200 mph is going to be in BIG BIG BIG TROUBLE in a bit under one minute!
Go and have a look here if want to see what might happen Tgv crash (http://i1.mirror.co.uk/incoming/article6832735.ece/ALTERNATES/s615b/Rescue-workers-search-the-wreckage-of-a-test-TGV-train-that-derailed-and-crashed-in-a-canal-outside-Eckwersheim-near.jpg), not the same cause.
Yes, I know I posted it before, but sometimes reality takes while to sink in.
Clearly railway tracks DO NOT ACTUALLY MEET, THEY ONLY APPEAR to MEET!
As I posted earlier in many cases we can test whether lines meet, by simply travelling to where they appear to meet.
Of course in some cases we may not be certain that the "lines" are truly parallel. That is no reflection on the geometry.
I'm afraid you must live in a different world to the rest of us, one quite divorced from reality!
If one looks at the scene, they do touch. It's a factual statement. "Appear" is implied.
According to the mathematical model of the Ancient Greeks, they should never touch.
According to the mathematical model of the Ancient Greeks, they should never touch.
And they never do touch - as I believe I quite convincingly proved with the railway tracks!
YOU claim: "If one looks at the scene, they do touch. It's a factual statement. "Appear" is implied."
And YOU say the "Appear" is implied. But the Greeks did not ever say "they do touch".
In fact what Euclid did say in his Fifth Postulate was: (https://upload.wikimedia.org/wikipedia/commons/thumb/e/ed/Parallel_postulate_en.svg/525px-Parallel_postulate_en.svg.png)
If the sum of the interior angles α and β is less than 180°,
the two straight lines, produced indefinitely, meet on that side.
What Euclid states is that lines which are not parallel do converge. He does not even say that parallel do not converge.
Now I am no expert on Greek geometry (and of course we do not simply "rely on Greek geometry" anyway!), so maybe you can show where they even say that parallel lines "appear to touch" somewhere!
Can't you ever see the difference between what appears to be true and what is actually true?
If the railway tracks actually did meet we would have a horrendous smash. We do not have a smash (hopefully!) so the railway tracks do not meet at their "vanishing point".
Still I guess there is not point discussing this further. For some people waords have well defined meanings, for others:
"When I use a word," Humpty Dumpty said in rather a scornful tone, "it means just what I choose it to mean — neither more nor less."
PS Never, ever change your mind and accept the fact that the earth is really a Globe. We just could not take your logic on our side!
Well, I don't believe the sun is actually touching the earth. Obviously I'm saying that they apparently touch when I say that they touch in the parallel line example. Can't you just move on, or do I need to repeat myself another 10 times?
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Well, I don't believe the sun is actually touching the earth. Obviously I'm saying that they apparently touch when I say that they touch in the parallel line example. Can't you just move on, or do I need to repeat myself another 10 times?
Well, neither do I. But, if that is what you were trying to say whyever did you make such silly claims about the ancient Greeks.
What I do claim is that the sun appears to go down behind the horizon (that is much further from us than the horizon).
(http://i1075.photobucket.com/albums/w433/RabDownunder/25%20-%20Sunset%20at%20Barnhill_zpssg7be5xb.jpg)
Sun Setting at Barhhill, WA | | (http://i1075.photobucket.com/albums/w433/RabDownunder/26%20-%20Sunset%20to%20North_zpsytujy348.jpg)
Sun has set, Barnhill, WA |
the sun which is actually some 3,000 miles (your figures not mine!) above the earth and
at distance of roughly from the observer (on the equator at the equinox) 8,800 miles hence
at an elevation of roughly 20° can ever appear behind or even below the visible horizon.
When you marry that with Rowbotham's explanation of the sun staying the same size throughout the day. CAUSE OF SUN APPEARING LARGER WHEN RISING AND SETTING THAN AT NOONDAY.
IT is well known that when a light of any kind shines through a dense medium it appears larger, or rather gives a greater "glare," at a given distance than when it is seen through a lighter medium. This is more remarkable when the medium holds aqueous particles or vapour in solution, as in a damp or foggy atmosphere. Anyone may be satisfied of this by standing within a few yards of an ordinary street lamp, and noticing the size of the flame; on going away to many times the distance, the light or "glare" upon the atmosphere will appear considerably larger.
This is utter rubbish because the sun staying the same says is the rule (rather than the exception) and happens in dry or wet conditions.
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The Greek model does not say they actually touch. That is why I asked you to show us where it says they actually touch. The important part is appearing to touch and actually touching.
If one looks at the scene, they do touch. It's a factual statement. "Appear" is implied.
According to the mathematical model of the Ancient Greeks, they should never touch.
Repetition of something may give a sense of security and fact to you as an individual but, ultimately, the repetition of a lie does not make it true.
Somehow you are, in your argument, defining appear as fact. By the definition you're trying to cram appear into above, every time I stand in front of a mirror I've made a second version of myself since it appears that I'm standing there looking at myself.
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Yes, there are multiple reasons why they can appear to touch. Insufficient resolving power of the eye/camera (as has been stated multiple times) is one of them. Decreased clarity due to atmospheric scattering is another. But this is all irrelevant. The basic geometry deals with a perfect world. There is no such thing as perfectly parallel lines, or a perfectly clear atmosphere. But that doesn't mean the entire theory is useless.
If their theory doesn't match observations it means the theory is wrong and must be modified or discarded.
Not necessarily. It might mean your basic assumptions are wrong. For example, the "ancient greek math" assumes that you have perfect resolving power and no atmospheric effects. Clearly these are bad assumptions. So what should we do? Should we just give up on basic geometry and declare that math is useless? Of course not, that would be silly.
Instead, let's come up with some more theories that describe how these "bad assumptions" affect the result from the original theory.
Theory A: Anything smaller than about one arcminute in diameter (.017 degrees) is not distinguishable by the human eye. (https://en.wikipedia.org/wiki/Naked_eye#Basic_accuracies)
Theory B: Stuff becomes blurry when viewed from really far away through an atmosphere (https://en.wikipedia.org/wiki/Aerial_perspective)
So now lets combine the "ancient greek math" (which describes a perfect world) with Theory A (which takes into account one of those imperfections).
Train Track Example:
You are looking down some train tracks that extend a long way into the distance. They are separated by a distance of 4 feet. Let's calculate how far apart those train tracks should appear 3 miles (15840 feet) away using the "ancient greek math".
angle = tan-1(4/15840) = 0.014 degrees
So, according to the "ancient greek math", the tracks should appear to be separated by 0.014 degrees. However, according to "Theory A", the human eye can't distinguish anything less than 0.017 degrees. Therefore, we should expect the tracks to appear to touch according to the human eye, even though the "ancient greek math" says they technically aren't touching.
The "ancient greek math" doesn't tell the entire story. It doesn't bother to take into account the limitations of the human eye, or cameras, or the atmosphere. But that doesn't mean it is useless. It is a starting point.
If you want to show that the "ancient greek math" is wrong, then you need to show an error in their proof. You have not even attempted that.
However, if you want to show that the "ancient greek math" is not applicable to the real world, then you need to show us a contradiction between the mathematical result and reality that can't be explained by some known phenomenon. So far, you have only repeatedly stated "the math says they should never touch, but they do touch". This contradiction can easily be explained by the limitations of the human eye or camera (as shown in my example above).
Sun Example:
On the other hand, the "ancient greek math" states that the sun should be separated from the horizon by about 20 degrees at a minimum. Assuming the sun is 3000 miles high, and an equatorial diameter of 8000 miles.
angle = tan-1(3000/8000) = 20 degrees
Is there any phenomenon that you know of that can make 20 degrees appear to be 0 degrees? I don't. Most cameras can easily distinguish stuff less than 20 degrees assuming they have more than 3 pixels. We can't blame atmospheric scattering, because the we can still distinguish the actual sun, which is much less than 20 degrees.
Unless you can come up with an explanation for this difference, blaming the "ancient greek math" just seems like a refusal to accept that your model might be wrong.
Sorry for the late reply. Busy busy.
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TL;DR of previous long-winded post:
So far, you have only repeatedly stated "the math says they should never touch, but they do touch". This apparent contradiction can easily be explained by the limitations of the human eye or camera.
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Math is a purely logical construction; it concerns what can be deduced from an initial set of propositions. So the only way to refute a mathematical conclusion (i.e., a statement that proposition P can be deduced logically from the initial premises A, B, C,…) is to show a logical error in the proof.
An error in the proof is that parallel lines seem to converge in contradiction of theory.
I’m having a hard time making sense of your responses, Tom. You seem to be saying that if parallel lines, such as railroad tracks on a flat surface, seem to your eyes (your brain, actually) to meet in the distance, then that refutes Euclid somehow. But so far as I know he never claimed that parallel lines will never appear to meet no matter how distant they are, and even if he had made such a claim, it wouldn’t have been as a part of any geometrical theorem.
As others have pointed out, our eyes have limited resolution capabilities: our vision can’t separate two objects 0.001 seconds of arc apart in our field of view. But they are still two separate objects.
Look, Tom, if you’re trying to make converts to FET, this isn’t the way to do it. You need to write in such a way that RE believers as well as fence-sitters can at least make sense of what you’re saying.
In the Elements, Euclid defined parallel lines as lines in the same plane that never meet no matter how far they are extended in either direction. So if two lines meet, by definition they are not parallel in Euclid’s sense. Saying “these two parallel lines actually touch, therefore Euclid was wrong,” is like saying “this triangle has four sides, therefore Euclid was wrong about triangles.”
Elucid was wrong about a lot of things. Look up Zeno's Paradox. The Greek model of the universe is flimsy.
Zeno’s Paradoxes are interesting philosophical questions, but I don’t see what they have to do with perspective, the topic of my OP.
And I haven’t seen a reply from any FE believer to my demonstration upthread that the sun would have to be at least a million miles away for it to appear to touch the horizon while remaining 3000 miles above the earth.
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The lines don't touch. The sky doesn't meet the ocean. But it appears to. We know it doesn't, it would be like chasing a rainbow trying to find the spot where the sky touches the ocean. It's an optical "illusion," but that doesn't mean it can't be described mathematically. What do you think every 3d engine ever has done to show rigid geometric 3d world in a way that mimics reality?
Euclid geometry obviously isnt used to describe the world as it appears, rather as it is. Even then it can be debated whether or not any of these perfect straight lines and circles even exist in reality.
Math, anyway you chalk it up, is a purely imaginary construct. It doesn't "exist," it just aids us in our description of phenomena and our natural desire for logical order in our universe. Even Einstein said "As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality."
That being said it can never be used to completely describe the whole of our existence.
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The lines don't touch. The sky doesn't meet the ocean. But it appears to.
Yes, but WHY does it appear to touch? Hint: because the angle between them is really really small.
The angle between the sun and the horizon on a flat earth is NOT really really small. 20 degrees is not small at all. There is absolutely no reason why 20 degrees should appear as 0 degrees to us. Does this obvious inconsistency really not bother any flat-earthers?
Here is a to-scale diagram. If you really don't trust trigonometry, you can use a protractor.
(http://i.imgur.com/u7pKbqD.png)
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The lines don't touch. The sky doesn't meet the ocean. But it appears to.
Yes, but WHY does it appear to touch? Hint: because the angle between them is really really small.
The angle between the sun and the horizon on a flat earth is NOT really really small. 20 degrees is not small at all. There is absolutely no reason why 20 degrees should appear as 0 degrees to us. Does this obvious inconsistency really not bother any flat-earthers?
Here is a to-scale diagram. If you really don't trust trigonometry, you can use a protractor.
(http://i.imgur.com/u7pKbqD.png)
Show us a real world example of how objects at that sort of distance appear and behave.
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The lines don't touch. The sky doesn't meet the ocean. But it appears to.
Yes, but WHY does it appear to touch? Hint: because the angle between them is really really small.
The angle between the sun and the horizon on a flat earth is NOT really really small. 20 degrees is not small at all. There is absolutely no reason why 20 degrees should appear as 0 degrees to us. Does this obvious inconsistency really not bother any flat-earthers?
Here is a to-scale diagram. If you really don't trust trigonometry, you can use a protractor.
(http://i.imgur.com/u7pKbqD.png)
Show us a real world example of how objects at that sort of distance appear and behave.
Show us any reason at all why an object would behave differently at that distance than at any other distance.
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Show us any reason at all why an object would behave differently at that distance than at any other distance.
A laser appears to be straight and constant locally, but at far distances, the beam widens out.
https://en.wikipedia.org/wiki/Beam_divergence
(http://www.olympusmicro.com/primer/techniques/lasers/microscopelasersfigure2.jpg)
Ice bergs may seem to float on the water locally, but from far away they may seem to float in the air.
https://en.wikipedia.org/wiki/Mirage
(http://www.astronomycafe.net/weird/lights/mirage3.jpg)
So yes, we will need some kind of real evidence that things work as you say they work, not a diagram.
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Show us any reason at all why an object would behave differently at that distance than at any other distance.
A laser appears to be straight and constant locally, but at far distances, the beam widens out.
https://en.wikipedia.org/wiki/Beam_divergence
(http://www.olympusmicro.com/primer/techniques/lasers/microscopelasersfigure2.jpg)
Ice bergs may seem to float on the water locally, but from far away they may seem to float in the air.
https://en.wikipedia.org/wiki/Mirage
(http://www.astronomycafe.net/weird/lights/mirage3.jpg)
So yes, we will need some kind of real evidence that things work as you say they work, not a diagram.
Both of those phenomena are well understood, and would not cause the sun to appear below the horizon when it is actually 20 degrees above the horizon.
The iceberg mirage is caused by refraction. It is heavily dependent on air and surface temperature. It varies wildly. Sunsets happen every single day, everywhere, regardless of the weather. So no, this is not a possibility.
The laser beam divergence... I have no idea how it is relevant. Perhaps you can explain to us how it would cause the sun to appear below the horizon?
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The laser beam divergence... I have no idea how it is relevant. Perhaps you can explain to us how it would cause the sun to appear below the horizon?
I'm guessing he's inferring that the light diverges to a point where the sun appears the same size near sunset as it does at midday. Then maybe it's light scattering and divergence that eventually makes the sun invisible through the atmosphere.
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The laser beam divergence... I have no idea how it is relevant. Perhaps you can explain to us how it would cause the sun to appear below the horizon?
I'm guessing he's inferring that the light diverges to a point where the sun appears the same size near sunset as it does at midday. Then maybe it's light scattering and divergence that eventually makes the sun invisible through the atmosphere.
That's what I was guessing too. Unfortunately for him, light divergence/scattering off the atmosphere wouldn't change the apparent location or size of the sun. All it would do is cause the sun to be dimmer and add some glare around it. It's pretty easy to filter out the glare to reveal the exact edge of the sun.
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Both of those phenomena are well understood, and would not cause the sun to appear below the horizon when it is actually 20 degrees above the horizon.
The iceberg mirage is caused by refraction. It is heavily dependent on air and surface temperature. It varies wildly. Sunsets happen every single day, everywhere, regardless of the weather. So no, this is not a possibility.
The laser beam divergence... I have no idea how it is relevant. Perhaps you can explain to us how it would cause the sun to appear below the horizon?
You asked for examples for why we should believe that things might be different when things are viewed by afar. I've satisfied this query. Now how about a real world example to show that things are how they should be under your theory?
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Tom Bishop:
Show us a real world example of how objects at that sort of distance appear and behave.
So yes, we will need some kind of real evidence that things work as you say they work, not a diagram.
This is like saying we don’t know what would happen to a 15 kg cannonball dropped from an aircraft at 20 km above the earth’s surface, because no one has done this exact experiment. Actually, we know to a practical certainty what would happen in those cases, because we have an extraordinarily well-developed science of physics that is extremely successful at making predictions and that can predict what would happen in this case, and even take into account air resistance and the Coriolis effect.
Similarly, we have an extremely successful and well-developed science of optics that explains what happens to light as it passes through air, taking into account moisture, dust, and temperature and pressure gradients. So we can predict how the sun would appear to the observer in the diagram, and on the basis of the science of optics there is no reason to believe that the sun would appear to be on or near the horizon. Nor does optics give us any reason to believe that the sun would maintain the same angular diameter throughout its course. You know, scientists have studied light and its interactions with matter for centuries.
Sure, it’s possible that a very successful theory could be wrong in some respects. But then you have to provide some basis, in theory or experiment, for believing the theory to be wrong, if you want to be taken seriously.
And here’s the thing: we already have a theory that explains, simply, with great accuracy, and consistently with the rest of our scientific knowledge, how the sun and other celestial objects appear and their positions and paths through the sky. I understood this theory and how it explains these phenomena at around the age of 5 or 6, not because I was precocious but because it’s that simple to understand. And FET, in order to fix what isn’t broken, posits new, unsupported and unconfirmed theories of physics that are inconsistent with the rest of our scientific knowledge. What about Occam’s Razor?
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You asked for examples for why we should believe that things might be different when things are viewed by afar. I've satisfied this query. Now how about a real world example to show that things are how they should be under your theory?
Sorry, you are right. I was loose with my wording. Let me try again so you can't wiggle out of giving a straight answer:
Show us any reason at all why an object would behave differently at that distance than at any other distance... which can be used to explain how the sun appears to be behind the horizon when a simple diagram shows it to be 20 degrees away from the horizon.
Roundabout: well stated.
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This is like saying we don’t know what would happen to a 15 kg cannonball dropped from an aircraft at 20 km above the earth’s surface, because no one has done this exact experiment. Actually, we know to a practical certainty what would happen in those cases, because we have an extraordinarily well-developed science of physics that is extremely successful at making predictions and that can predict what would happen in this case, and even take into account air resistance and the Coriolis effect.
The math is very limited, and assumes that local effects hold true endlessly. Unless you have accurately experimented at all scales, it cannot be said that we know how things will look like at all scales based on math alone.
Should we just assume that the earth will get infinitely hotter the deeper we dig, because mines have been seen to get hotter with greater depth?
Simple math says that it should. But to assume that is a fallacy, as we do not have full knowledge of the conditions beyond our immediate reach.
Similarly, we have an extremely successful and well-developed science of optics that explains what happens to light as it passes through air, taking into account moisture, dust, and temperature and pressure gradients. So we can predict how the sun would appear to the observer in the diagram, and on the basis of the science of optics there is no reason to believe that the sun would appear to be on or near the horizon. Nor does optics give us any reason to believe that the sun would maintain the same angular diameter throughout its course. You know, scientists have studied light and its interactions with matter for centuries.
You know, scientists also believe in experimentation before coming to a conclusion.
And here’s the thing: we already have a theory that explains, simply, with great accuracy, and consistently with the rest of our scientific knowledge, how the sun and other celestial objects appear and their positions and paths through the sky.
"Zeus and the other gods did it" is also a theory that explains everything. That's why experimentation is necessary, I am afraid.
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This is like saying we don’t know what would happen to a 15 kg cannonball dropped from an aircraft at 20 km above the earth’s surface, because no one has done this exact experiment. Actually, we know to a practical certainty what would happen in those cases, because we have an extraordinarily well-developed science of physics that is extremely successful at making predictions and that can predict what would happen in this case, and even take into account air resistance and the Coriolis effect.
The math is very limited, and assumes that local effects hold true endlessly. Unless you have accurately experimented at all scales, it cannot be said that we know how things will look like at all scales based on math alone.
Should we just assume that the earth will get infinitely hotter the deeper we dig, because mines have been seen to get hotter with greater depth?
Simple math says that it should. But to assume that is a fallacy, as we do not have full knowledge of the conditions beyond our immediate reach.
If by "simple math" you mean data extrapolation, then yes, it tells us that the earth gets hotter the deeper we dig. There is no reason why it would get INFINITELY hotter, because the earth isn't infinitely deep. I suspect that people have much better ways to determine the temperature of the earth than just linear extrapolation though. Seismic data, computer models.... I'm no geologist, so I can't say much more on this issue.
You know, scientists also believe in experimentation before coming to a conclusion.
Indeed. But any experiment involving space just gets labelled as fake and ignored. Which is why we have to resort to little diagrams, and stuff that everybody can see with their own eyes. If you really want experiments having to do with the sun, this (http://sohowww.nascom.nasa.gov/home.html) may be a good place to start looking.
I noticed you STILL haven't provided a reason why the sun would appear to set below the horizon when it is actually 20 degrees above the horizon. For a Zetetic, you seem to be stubbornly rejecting what is plainly visible with your own eyes: the sun goes below 20 degrees from the horizon. Isn't that the exact opposite of Zeteticism?
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As the story goes, I think that the earth is flat in the same way that parallel lines touch.
In our perceived reality the earth is flat and if we travel south we might appear to see a wall of ice. Parallel lines touch because our perspective allows us to see only that much.
The fact is parallel lines don't touch and the earth is round, but it's not shameful to say that you they appear to be so.
Truth is for most of us it doesn't make any difference if the earth is flat or round. Things are sorted out for us already so we don't have to take earth's shape into account for our whole lives. However, when people will have to do something so big that they would have to take earth's shape into account, they will realize for themselves which theory works best for them. If you believe only what you see, go up there and see for yourself.