# Infinite plane...flat or curved...horizon or not... **URL:** \ **Category:** Factual Questions\ **Created:** [October 11, 2011, 1:38pm UTC](https://boards.straightdope.com/t/infinite-plane-flat-or-curved-horizon-or-not/599290 "2011-10-11T13:38:52Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![john93940](https://avatars.discourse-cdn.com/v4/letter/j/e36b37/32.png) [@john93940](https://boards.straightdope.com/u/john93940)\ **Post date:** [October 11, 2011, 1:38pm UTC](https://boards.straightdope.com/t/infinite-plane-flat-or-curved-horizon-or-not/599290/1 "2011-10-11T13:38:52Z")
Assume an observer standing on an infinite seemingly flat plane. Assume the plane is illuminated so that any part can be seen no matter how far away. Assume unobstructed vision and therefore infinite vision (anything can be seen if it’s big enough and bright enough and nothing obstructs line of sight). Will the observer see an horizon? or will he seem to be inside a bright bowl? Fact: as long as the observer’s eyes are above the plane then all of it can be seen. Even though your ‘line of sight’ = an angle higher than the surface’s, by definition if you’re above the surface all of it can be seen. Therefore, there will be no horizon and one can look straight up and see the same surface being stood upon. This wacko conclusion = the absurdity (in our frame of reference) of the concept of ‘infinite’. ---
**Author:** ![Keeve](https://avatars.discourse-cdn.com/v4/letter/k/f07891/32.png) [@Keeve](https://boards.straightdope.com/u/Keeve)\ **Post date:** [October 11, 2011, 2:08pm UTC](https://boards.straightdope.com/t/infinite-plane-flat-or-curved-horizon-or-not/599290/2 "2011-10-11T14:08:54Z")
This forum is called “General Questions”. Do you have a question? ---
**Author:** ![coremelt](https://avatars.discourse-cdn.com/v4/letter/c/4491bb/32.png) [@coremelt](https://boards.straightdope.com/u/coremelt)\ **Post date:** [October 11, 2011, 2:09pm UTC](https://boards.straightdope.com/t/infinite-plane-flat-or-curved-horizon-or-not/599290/3 "2011-10-11T14:09:10Z")
> [@john93940](#): > > Therefore, there will be no horizon and one can look straight up and see the same surface being stood upon. This wacko conclusion = the absurdity (in our frame of reference) of the concept of ‘infinite’. No, No you won’t be able to. There will be a horizon just fine, perspective effects don’t rely on the curvature of the earth. ---
**Author:** ![Mangetout](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mangetout/32/19_2.png) [@Mangetout](https://boards.straightdope.com/u/Mangetout)\ **Post date:** [October 11, 2011, 2:13pm UTC](https://boards.straightdope.com/t/infinite-plane-flat-or-curved-horizon-or-not/599290/4 "2011-10-11T14:13:18Z")
> [@john93940](#): > > Even though your ‘line of sight’ = an angle higher than the surface’s, by definition if you’re above the surface all of it can be seen. Therefore, there will be no horizon and one can look straight up and see the same surface being stood upon. This wacko conclusion = the absurdity (in our frame of reference) of the concept of ‘infinite’. No - just because something extends away from you infinitely doesn’t mean it should appear to curve back around above your head. There is a horizon at infinity - the further away a point on the plane, the closer it approaches the horizon. That’s just geometry (well, and a bit of calculus) ---
**Author:** ![Half\_Man\_Half\_Wit](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/half_man_half_wit/32/21766_2.png) [@Half\_Man\_Half\_Wit](https://boards.straightdope.com/u/Half_Man_Half_Wit)\ **Post date:** [October 11, 2011, 2:15pm UTC](https://boards.straightdope.com/t/infinite-plane-flat-or-curved-horizon-or-not/599290/5 "2011-10-11T14:15:08Z")
In particular, imagine a line of sight emanating from your eyes; you can see whatever it intersects. If you look straight ahead, the line will be parallel to the plane, never intersecting it; a line any infinitesimal angle looking down will ultimately intersect the plane. So, you see a horizon at your eye level. ---
**Author:** ![Keeve](https://avatars.discourse-cdn.com/v4/letter/k/f07891/32.png) [@Keeve](https://boards.straightdope.com/u/Keeve)\ **Post date:** [October 11, 2011, 2:15pm UTC](https://boards.straightdope.com/t/infinite-plane-flat-or-curved-horizon-or-not/599290/6 "2011-10-11T14:15:49Z")
> [@john93940](#): > > Will the observer see an horizon? or will he seem to be inside a bright bowl? Sorry, these are your questions. I missed them before. No, he will not see a horizon. The reason is because - as you explained - there ISN’T any horizon. It doesn’t exist. If he looks down, he’ll see his feet. If he picks his head up a bit, he’ll see a few yards in front of him. If he picks his head up a little more, he’ll see what is ten miles away. If he looks up just a little more, so that his eyes are looking ALMOST horizontally, but just a tiny bit BELOW horizontal, he will see things eleventy-twelve gazillion miles away. But if he looks ABOVE horizontal, he will see nothing. Well, he will see some buildings, if they are tall enough. But no land whatsoever. Nothing curls up above his head. I don’t know where you get the idea that it will look like a bowl. ---
**Author:** ![Saint\_Cad](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/saint_cad/32/18907_2.png) [@Saint\_Cad](https://boards.straightdope.com/u/Saint_Cad)\ **Post date:** [October 11, 2011, 2:18pm UTC](https://boards.straightdope.com/t/infinite-plane-flat-or-curved-horizon-or-not/599290/7 "2011-10-11T14:18:07Z")
> [@Half\_Man\_Half\_Wit](#): > > In particular, imagine a line of sight emanating from your eyes; you can see whatever it intersects. If you look straight ahead, the line will be parallel to the plane, never intersecting it; a line any infinitesimal angle looking down will ultimately intersect the plane. So, you see a horizon at your eye level. So since looking down one will see the ground, would it appear that the observer is in a bowl with the lip at eye level? ---
**Author:** ![Keeve](https://avatars.discourse-cdn.com/v4/letter/k/f07891/32.png) [@Keeve](https://boards.straightdope.com/u/Keeve)\ **Post date:** [October 11, 2011, 2:24pm UTC](https://boards.straightdope.com/t/infinite-plane-flat-or-curved-horizon-or-not/599290/8 "2011-10-11T14:24:41Z")
> [@Saint\_Cad](#): > > So since looking down one will see the ground, would it appear that the observer is in a bowl with the lip at eye level? OMG, you’re right. But curiously, I perceive that illusion only when looking at a two-dimensional video of such scenes, never when I’m actually in them. This is despite the fact that if I stand still and move only my eyes, I can attest that the curvature seen in such videos is accurate. But I NEVER perceive or feel such curves in the real world unless I’m looking for it. So which is real, and which is the illusion? The curvature or the flatness? ---
**Author:** ![Colophon](https://avatars.discourse-cdn.com/v4/letter/c/f05b48/32.png) [@Colophon](https://boards.straightdope.com/u/Colophon)\ **Post date:** [October 11, 2011, 2:31pm UTC](https://boards.straightdope.com/t/infinite-plane-flat-or-curved-horizon-or-not/599290/9 "2011-10-11T14:31:22Z")
> [@Saint\_Cad](#): > > So since looking down one will see the ground, would it appear that the observer is in a bowl with the lip at eye level? No. The ground will appear flat, just as it is. The distance to the horizon (infinite, in this case) is so huge that the distance between foot level and eye level is totally irrelevant. You would just see flat ground stretching from your feet to a horizontal horizon exactly bisecting your visible world. ---
**Author:** ![Keeve](https://avatars.discourse-cdn.com/v4/letter/k/f07891/32.png) [@Keeve](https://boards.straightdope.com/u/Keeve)\ **Post date:** [October 11, 2011, 2:35pm UTC](https://boards.straightdope.com/t/infinite-plane-flat-or-curved-horizon-or-not/599290/10 "2011-10-11T14:35:13Z")
> [@Colophon](#): > > No. The ground will appear flat, just as it is. The distance to the horizon (infinite, in this case) is so huge that the distance between foot level and eye level is totally irrelevant. You would just see flat ground stretching from your feet to a horizontal horizon exactly bisecting your visible world. I’m not so sure. The points are infinity DO appear to be at eye level. That’s perspective in action. I suspect that the eyes do in fact perceive a slightly curved bowl, but the brain corrects this to be the flat surface which experience has taught us about. ---
**Author:** ![Colophon](https://avatars.discourse-cdn.com/v4/letter/c/f05b48/32.png) [@Colophon](https://boards.straightdope.com/u/Colophon)\ **Post date:** [October 11, 2011, 2:38pm UTC](https://boards.straightdope.com/t/infinite-plane-flat-or-curved-horizon-or-not/599290/11 "2011-10-11T14:38:27Z")
> [@Keeve](#): > > I’m not so sure. The points are infinity DO appear to be at eye level. That’s perspective in action. > > I suspect that the eyes do in fact perceive a slightly curved bowl, but the brain corrects this to be the flat surface which experience has taught us about. Yes, the points at infinity do appear to be at eye level, but eye level is only, what, just over 5 feet above the ground. A “bowl” an infinite distance in diameter and only 5 feet deep is flat to within an infinitesimal ratio. ---
**Author:** ![Mangetout](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/mangetout/32/19_2.png) [@Mangetout](https://boards.straightdope.com/u/Mangetout)\ **Post date:** [October 11, 2011, 4:25pm UTC](https://boards.straightdope.com/t/infinite-plane-flat-or-curved-horizon-or-not/599290/12 "2011-10-11T16:25:45Z")
> [@Keeve](#): > > Sorry, these are your questions. I missed them before. > > No, he will not see a horizon. The reason is because - as you explained - there ISN’T any horizon. It doesn’t exist. If the sky is blue and the ground is brown, what will our observer see? At some point, visually, one has to give way to the other. That’s going to look like a horizon. ---
**Author:** ![john93940](https://avatars.discourse-cdn.com/v4/letter/j/e36b37/32.png) [@john93940](https://boards.straightdope.com/u/john93940)\ **Post date:** [October 11, 2011, 4:49pm UTC](https://boards.straightdope.com/t/infinite-plane-flat-or-curved-horizon-or-not/599290/13 "2011-10-11T16:49:55Z")
> [@john93940](#): > > Assume an observer standing on an infinite seemingly flat plane. Assume the plane is illuminated so that any part can be seen no matter how far away. Assume unobstructed vision and therefore infinite vision (anything can be seen if it’s big enough and bright enough and nothing obstructs line of sight). Will the observer see an horizon? or will he seem to be inside a bright bowl? > > Fact: as long as the observer’s eyes are above the plane then all of it can be seen. > > Even though your ‘line of sight’ = an angle higher than the surface’s, by definition if you’re above the surface all of it can be seen. Therefore, there will be no horizon and one can look straight up and see the same surface being stood upon. This wacko conclusion = the absurdity (in our frame of reference) of the concept of ‘infinite’. a little emphasis from the poster: Can it be denied that, by definition, an observer in position above the plane, which stretches in all directions to infinity, can see the plane in its entirety? Furthermore, remember, by definition, all of the plane is observable. So, the wacko conclusion = the observer would see no horizon…he would seem to be inside a globe, a pink globe if the plane were colored pink. ---
**Author:** ![Buck\_Godot](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/buck_godot/32/6573_2.png) [@Buck\_Godot](https://boards.straightdope.com/u/Buck_Godot)\ **Post date:** [October 11, 2011, 4:58pm UTC](https://boards.straightdope.com/t/infinite-plane-flat-or-curved-horizon-or-not/599290/14 "2011-10-11T16:58:12Z")
Simple trigonometry. Draw a triangle from his eye at height H, down to his feet at 0, and then out into along the plane a distance D. The angle Theta sweeping from his feet to the distant point will satisfy the following equation. tan(Theta)=D/H. As D approaches infinity, Theta approaches 90 degrees. Theta will also be 90 degrees, for looking at something of height H no matter how far away (effectively reducing H to 0). The only way for Theta to be greater than 90 degrees (looking slightly up, effectively a negative H) would be looking at something higher than height H. But as these get further away they will also approach an angle of 90 degrees. ---
**Author:** ![newme](https://avatars.discourse-cdn.com/v4/letter/n/e0b2c6/32.png) [@newme](https://boards.straightdope.com/u/newme)\ **Post date:** [October 11, 2011, 4:58pm UTC](https://boards.straightdope.com/t/infinite-plane-flat-or-curved-horizon-or-not/599290/15 "2011-10-11T16:58:32Z")
> [@john93940](#): > > a little emphasis from the poster: Can it be denied that, by definition, an observer in position above the plane, which stretches in all directions to infinity, can see the plane in its entirety? Furthermore, remember, by definition, all of the plane is observable. > > So, the wacko conclusion = the observer would see no horizon…he would seem to be inside a globe, a pink globe if the plane were colored pink. This is not correct. Looking straight at your feet, at 0 degrees, you would see the plane. Looking straight ahead, at 90 degrees, you would see the plane. Looking above 90 degrees, you would not see the plane. ---
**Author:** ![AndrewL](https://avatars.discourse-cdn.com/v4/letter/a/3d9bf3/32.png) [@AndrewL](https://boards.straightdope.com/u/AndrewL)\ **Post date:** [October 11, 2011, 5:20pm UTC](https://boards.straightdope.com/t/infinite-plane-flat-or-curved-horizon-or-not/599290/16 "2011-10-11T17:20:56Z")
Does our hypothetical infinite plane have a gravitational field that bends light? If it does, then wouldn’t light rays coming off the surface of the plane eventually bend downwards such that you would see the plane even if looking upwards? Or does it not work that way? ---
**Author:** ![john93940](https://avatars.discourse-cdn.com/v4/letter/j/e36b37/32.png) [@john93940](https://boards.straightdope.com/u/john93940)\ **Post date:** [October 11, 2011, 5:25pm UTC](https://boards.straightdope.com/t/infinite-plane-flat-or-curved-horizon-or-not/599290/17 "2011-10-11T17:25:46Z")
> [@john93940](#): > > Assume an observer standing on an infinite seemingly flat plane. Assume the plane is illuminated so that any part can be seen no matter how far away. Assume unobstructed vision and therefore infinite vision (anything can be seen if it’s big enough and bright enough and nothing obstructs line of sight). Will the observer see an horizon? or will he seem to be inside a bright bowl? > > Fact: as long as the observer’s eyes are above the plane then all of it can be seen. > > Even though your ‘line of sight’ = an angle higher than the surface’s, by definition if you’re above the surface all of it can be seen. Therefore, there will be no horizon and one can look straight up and see the same surface being stood upon. This wacko conclusion = the absurdity (in our frame of reference) of the concept of ‘infinite’. again from the poster: a flat plane going in all directions to infinity means the plane has NO END. So, how could you stop seeing it if you can see all of it? The point is, the concept of the “infinite” is inconceivable. ---
**Author:** ![Half\_Man\_Half\_Wit](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/half_man_half_wit/32/21766_2.png) [@Half\_Man\_Half\_Wit](https://boards.straightdope.com/u/Half_Man_Half_Wit)\ **Post date:** [October 11, 2011, 5:33pm UTC](https://boards.straightdope.com/t/infinite-plane-flat-or-curved-horizon-or-not/599290/18 "2011-10-11T17:33:29Z")
Think about the plane being covered with circles a fixed distance apart, with you at their common center. Naturally, there’ll be infinitely many of them. The more to the distance you look, the closer together the circles will appear; as your gaze approaches the horizon, the rings will seem ‘infinitely close’ together. That way, you can see all of the infinite plane within the finite area of your visual field it occupies – it’s simple projective geometry. ---
**Author:** ![dracoi](https://avatars.discourse-cdn.com/v4/letter/d/90db22/32.png) [@dracoi](https://boards.straightdope.com/u/dracoi)\ **Post date:** [October 11, 2011, 5:35pm UTC](https://boards.straightdope.com/t/infinite-plane-flat-or-curved-horizon-or-not/599290/19 "2011-10-11T17:35:04Z")
> [@john93940](#): > > a little emphasis from the poster: Can it be denied that, by definition, an observer in position above the plane, which stretches in all directions to infinity, can see the plane in its entirety? Furthermore, remember, by definition, all of the plane is observable. > > So, the wacko conclusion = the observer would see no horizon…he would seem to be inside a globe, a pink globe if the plane were colored pink. The assertion that you can see the entire plane does not lead to the conclusion that you can see anything above you. Others have answered this in other ways, but I’ll take a whack at it. At the ground nearest to you, a small change in the angle of your eyes produces a small change in how far away the ground you see is. Looking straight down, you’re seeing ground that is 5’ away. Looking at 45 degrees down, the ground is only about 7’ away now. At the ground farther away from you, a small change in the angle of your eyes produces an enormous change in how far away the ground is. The difference between 89.9 degrees and 89.91 degrees is going to be miles and miles. (I could do the math, but I’m lazy). So what you see is going to be nice detailed images nearby and smaller images farther away. We see this on Earth, too, but the rounding means that we don’t see smaller and smaller images forever. On a plane, distances 100 miles, 1000 miles and 10,000 miles are all going to appear very small and very close together on the “horizon” - which is not a true horizon, but a line created by the approach to infinity. Maybe a way to illustrate that is by thinking about a series of dots evenly spaced out into the distance. It would appear something like this: ```auto You . . . . . . . . . . .........__ ``` (where eventually the dots are going to blur together because your eyes lack infinite resolution) ---
**Author:** ![LSLGuy](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/lslguy/32/5813_2.png) [@LSLGuy](https://boards.straightdope.com/u/LSLGuy)\ **Post date:** [October 11, 2011, 5:49pm UTC](https://boards.straightdope.com/t/infinite-plane-flat-or-curved-horizon-or-not/599290/20 "2011-10-11T17:49:15Z")
ETA: Not trying to pile on: \*\*Half Man Half Wit \*\*& \*\*dracoi \*\*weren’t there when I started. > [@john93940](#): > > again from the poster: a flat plane going in all directions to infinity means the plane has NO END. So, how could you stop seeing it if you can see all of it? The point is, the concept of the “infinite” is inconceivable. You started with more or less sensible hypothetical but are now spouting nonsense. I suspect you have the mistaken notion that there is only one infinity of only one size. That’s completey wrong. Right now, here in the real world, look towards your feet. There is an infinity of distance down that way, obstructed only by your shoes, the floor, and a trivial few thousand miles of rock. Now look up towards the ceiling. There is an infinity of distance up that way too, obstructed only by your roof & a couple hundred miles of air. The infinity below does not interfere with your ability to see the infinity above. As others have said: If you are standing on an infinite flat plane you will see it as a surface extending outward in all directions horizontally. And you’ll see a sharp line at eye level where the plane ends and whatever the sky is begins. If the sky is the same color as the plane, then no, you won’t see any difference. But if the plane is (say) white & the sky is (say) blue, there will be a sharp line where the plane ends (or more precsiely, where the plane’s projection onto your eye’s backplane ends), and the sky begins. If you have more to add, skip the poetry & bring on the trig. It’s the key to accurate thought about nature, even contrived artificial nature. [Next page](https://boards.straightdope.com/t/infinite-plane-flat-or-curved-horizon-or-not/599290.md?page=2)