the view of a flatlander

[alkaline]

Here's something fun to do:

http://www.kcptech.com/dgylconf/proceedings/flatland/flat_triangle1.htm

It's a little java applet that allows you to move around a flatlander and a triangle, and you can see what his linear view looks like.

[Hugh]

The viewpoint shown is not what a 2d flatlander would really see. There is a height given to the line of vision that wouldn't be present. A 1d line (the view along the plane, edge on) has no height so it couldn't be seen.

A 2d flatlander actually sees zero dimensions, 2 less than what it is, even though it can move around in 2 dimensions, theoretically of course.

[thigle]

i think flatlander can see 1d, one d less than where he's at.

[Hugh]

What could be seen of a 1d line?

[thigle]

a segment. a circle seen side-way as well as square seen edge-on both would appear as lines - 1d segment.
i wonder how visual indeterminacy would apply within 2d vision: a segment (2d 'contour') wouldn't necessarily be indeterminate, but if we suppose perspective size-diminishing-over-distance, shortening would happen. so the square seen edge on would be a segment of constant intensity, while circle seen same way would be most intense in the middle, dimininshing in intensity or whatever towards its 'endpoints',

[Hugh]

A segment has no height, only length. It is a portion of a line between 2 points on a line, but you can't see anything of it. If you were to look at a circle and a square a bit apart in a plane edge on, you wouldn't see two lines separated by a space. There would have to be a thickness to the lines to be able to see them, which 1d lines don't have.

[thigle]

segment is 1 spherepoint high. it is a hypothetical sequence of points, which themselves are for me one of funniest members of mathemathical cast. conventionally, they are nothing(0dimensional), yet they are something, the finest grain, as if it is the smallest distinction possible within considered system. so being nothing, still they are the basis on which all else 'stands'.

a straigt line drawn on paper is conventionaly talked about as being 1d although surely it in actuality happens to possess certain thickness, occupies micro-volume of 3space.

in your context, you talk about ideal line, a pure idea, existing in ma.thematical heaven of pythagoreans. how, if at all, do you see ideas ? by physical eyes ? not really.

the vision does not happen as one way process, it is also actively created from the inside: 2 lights, one "outer" of the sun & stars and one "inner" of the spirit - the 'soulight', have to meet for meaningful vision. purely objectively, one cannot explain vision-experience.

so there are 2 different contexts. idealy, line is one dimensional. it cannot be seen then, as it's just succesion of nothings. then even plane, or sequence of planes, ideally forming a volume, cannot be seen for the same reasons - there is nothing to see. and that's how things really are - noone ever saw a point, line, or plane, or volume, purely by physical eyes. the mind seeing ideas is a necessity for perceiving a plane of table, of paper, or of computer screen, plane of earth surface'plane' ... all the particular instances of 'planarity' are seen due to idea of plane in your mind. seeing this idea itself is another matter.

but for the while of refining the techne of our knowing, we speak of representations as if they were actualities. we have to proceed in refining concepts before transcending them, we use them until they fall off.

the applet really overestimates the thicknes of the line in ratio to its lenght, so it looks like a band.

[PWrong]

segment is 1 spherepoint high. it is a hypothetical sequence of points, which themselves are for me one of funniest members of mathemathical cast. conventionally, they are nothing(0dimensional), yet they are something, the finest grain, as if it is the smallest distinction possible within considered system. so being nothing, still they are the basis on which all else 'stands'.

More precisely, the height of the line is epsilon, for some epsilon>0.
The height of the flatlander's eyes is also epsilon.
epsilon/epsilon = 1, in whatever units we use; In this case, the width of the band on the screen.

No more description is neccessary, all it does is confuse everyone. Epsilon/delta proofs make all arguments about infinity obselete, that's why they were invented.

[thigle]

what are these epsilon/delta proofs & argument about infinity being obsolete ? just curious...

[jinydu]

what are these epsilon/delta proofs & argument about infinity being obsolete ? just curious...

Epsilon-Delta proofs provide a rigorous and precise way of dealing with limits in mathematics, without reference to vague concepts like "infinitely large" or "infinitely small".

You should be able to find more information about them on many math sites.

[RQ]

segment is 1 spherepoint high. it is a hypothetical sequence of points, which themselves are for me one of funniest members of mathemathical cast. conventionally, they are nothing(0dimensional), yet they are something, the finest grain, as if it is the smallest distinction possible within considered system. so being nothing, still they are the basis on which all else 'stands'.

More precisely, the height of the line is epsilon, for some epsilon>0.
The height of the flatlander's eyes is also epsilon.
epsilon/epsilon = 1, in whatever units we use; In this case, the width of the band on the screen.

No more description is neccessary, all it does is confuse everyone. Epsilon/delta proofs make all arguments about infinity obselete, that's why they were invented.

Last time I checked epsilon=0 since 0.999...=1. But the person does bring up an interesting point. With respect to us his view wouldn't exist, but with respect to himself it would. This is like saying we aren't supposed to be seeing the 2D sheet of the world in front of our eyes, because it's infinitely flat and doesn't exist.

[PWrong]

Last time I checked epsilon=0

Actually, in any epsilon-delta proof, the most important statement is
"Let epsilon>0"

So by definition, epsilon can never equal zero, but it can be any real number greater than zero. It doesn't even have to be close to zero, it could be 1000.

The nice thing about this is that division is always defined for epsilon, even when epsilon is very close to zero, like 0.0000000001

0.999...=1

Since you mentioned this, I'll show you how you might say it in terms of epsilons. Let me know if this doesn't make any sense to you.

Consider the sequence 0.9 , 0.99 , 0.999 , 0.9999 , ...
That is,
s_1 = 1 - 1/10
s_2 = 1 - 1/100
s_3 = 1 - 1/1000
s_n = 1 - 1/10^n

We say that "the limit as n approaches infinity is 1"

This means that for every epsilon>0, you can find a number n, so that
|1 - s_n| < epsilon.
By the way, |x| means the absolute value of x. |-3| = 3, |2| = 2

Here's another way to put it. For any epsilon >0,
s_n > 1 - epsilon

For example, if I said that it's not approaching 1, it's approaching exactly
0.9999726, then you could find a term in the sequence to prove me wrong, like the fifth term, 0.99999.

[greg825]

A segment has no height, only length. It is a portion of a line between 2 points on a line, but you can't see anything of it. If you were to look at a circle and a square a bit apart in a plane edge on, you wouldn't see two lines separated by a space. There would have to be a thickness to the lines to be able to see them, which 1d lines don't have.

I agree that we wouldn't be able to see anything of a 2-d world, but your logic implies that nothing from any dimension can ever be seen. Why wouldn't a person in a 4 spacial dimensional world deduce that no one in a 3 spacial dimensional world could see anything because theres no "quaddepth" (or whatever comes after width, height, and depth). Why wouldn't a 5 dimensional person deduce this of the people living in 4 dimensions ad infinitum.

[Hugh]

Hi greg825,

I agree that we wouldn't be able to see anything of a 2-d world, but your logic implies that nothing from any dimension can ever be seen.

I don't believe so. In order for there to be vision, there has to be at least an area (length and width) to look at. 1d lines have no area, only length, so there is nothing to see, in my opinion. Beings, from any dimension, that see with at least an x/y axis "plane of vision" can actually see something of what is around them.

[greg825]

Hi greg825,

I agree that we wouldn't be able to see anything of a 2-d world, but your logic implies that nothing from any dimension can ever be seen.

I don't believe so. In order for there to be vision, there has to be at least an area (length and width) to look at. 1d lines have no area, only length, so there is nothing to see, in my opinion. Beings, from any dimension, that see with at least an x/y axis "plane of vision" can actually see something of what is around them.

Well this is just speculation of course but I'm not certain that a 2-d being couldn't see in 1-d. The assumption that an object can exist at all in just 2 dimensions implies different rules than what we have in this universe. Assuming we live in 3 (spacial) dimensions and considering that we see in 2 dimensions (with paralax used to approximate depth) leads me to think that a being in 2 dimensions could see in 1.

It's possible that us trying to visualize what a 2-d being would see is futile, but from that alone I can't conclude that such a being would see nothing.

[Hugh]

The assumption that an object can exist at all in just 2 dimensions implies different rules than what we have in this universe.

The rules would definitely be different.

It's possible that us trying to visualize what a 2-d being would see is futile, but from that alone I can't conclude that such a being would see nothing.

Fair enough. In our realilty, in our universe, it's at least 3d, so when we think of 2d beings it's more of a thought experiment.

The reason why I got into questioning the whole idea in the first place, was in thinking that if a 2d being has a severely limited viewpoint of what it sees around itself, perhaps we might too if we're in higher than 3d space ourselves.

[Batman3]

Maybe I don't understand what is being talked about here, but if I can respond...

It seems to me that just because we can't see 4d around here there is no quaddepth. In 2d it's like we are walking in a narrow alley filled with boxes and garbage cans. In 3d the 3d light would hit us wih point size in our dimensions but also with a quaddepth in the perpendicular fourth dimension, parallel to the other quaddepths of all the other photons and particles in 3d+quaddepth space. Since everything would be parallel we would be oblivious to the quaddepth.

[Hugh]

Hi Batman3,

It seems to me that just because we can't see 4d around here there is no quaddepth.

Since everything would be parallel we would be oblivious to the quaddepth.

So do you think it's possible that we're actually 4d (or higher), but we're unable to see it because of our limited vision?

[Keiji]

I wouldn't think so, as if we were 4D, then surely our eyes would be able to see in 4D as well. But it's possible.

[Batman3]

If we were 4d our eyes, according to this interpretation would be 4-spheric(i.e. hyperspheric, or (glomic?)). But in 3d our eyes are only spherical and not hyperspherical so they only focus light from 3-space.
On the other hand if we were living in a 3-space bounded on either side by 4d-walls so we only had quaddepth+3-space freedom, our minds might be able to divide or imagine this quaddepth.
That is what I Seem to do sometimes, and it helps imagining 4d.
So yes, I think we may be living in 4d or higher.

[Hugh]

Hi iNVERTED,

if we were 4D, then surely our eyes would be able to see in 4D as well.

Are you sure? A 2d being is limited to a 1d line to look at, which I don't think could even be seen at all. Wouldn't a 4d being have a limit as well? What do you think that might be?

Hi Batman3,

If we were 4d our eyes, according to this interpretation would be 4-spheric(i.e. hyperspheric, or (glomic?)). But in 3d our eyes are only spherical and not hyperspherical so they only focus light from 3-space

If our vision is limited to 3d, but we were actually 4d, we would think we see another person's eye as a 3d ball, but in fact it would only be a 3d slice of a 4d hyperspheric eye.

The actual eye would have extensions in 4d but we couldn't see those if we can only see 3d at any time, in any one direction.

[miseleigh]

To start, let's specify some words so everyone knows what is meant.
'Quaddepth' is termed as trength on the rest of this website, so tetronian space has length, width, depth, and trength. And let's not forget those very useful terms of bionian, trionian, etc.

I think greg825 has the right idea in that, although we are trionian creatures, we see in two dimensions, implying that a bionian would see in one, perhaps approximating two by having two eyes one above the other the way we use ours. This also implies that a tetronian would see in three dimensions, again probably approximating four by (4d)parallax. Therefore I don't think we are four-dimensional creatures, since we only see in two, not three.

If you are not convinced that our eyes see in two dimensions rather than three, try to find out how accurate your depth perception is with one eye closed. We need stereo vision for 3d.

Hugh, your definition of vision intrigues me. How do you propose to see a bionian object that has infinitely small (=0) depth? My understanding of vision is that photons bounce off of objects and are detected by our eyes. For a photon to bounce off of something, it needs to exist in our world, and to do that, it needs volume, and therefore depth...
Besides, I doubt that a bionian would have eyes that work the same way ours do. My guess is he would see a 1D line without any problems, assuming both he and the line he sees exist.

The only thing I've found that could possibly be termed as a less than three-dimensional object is a quark. They are considered to have no volume, and therefore no width, depth, etc. However, we have not detected them directly- the quark model simply fits all the data we have, and from that physicists are convinced they are real. Besides, I don't think photons can bounce off of these to allow us to see them anyway- especially since photons are made of (3?) quarks. The quark would bounce off the photon.
A little on quarks for those who are interested, although the volume of quarks does not seem to be mentioned at all in any of these (used a number of sites off Google to find that):
http://scout.wisc.edu/Reports/NSDL/PhysSci/2002/ps-020809-topicindepth.html

And thigle, a pencil works by leaving bits of stuff (graphite) on paper, and a pen by letting bits of stuff (ink) be absorbed between paper fibers. So yes, a pencil line occupies space. That's why you have to sharpen pencils and refill pens occasionally.

I wish I knew more about quantum mechanics. It's funny how closely it ties in with the idea of a spatial fourth dimension and any creatures living in it. Anybody here who knows some QM who could enlighten us on the dimensional properties of quarks? Google is rather limited unless you have hours to search.

[miseleigh]

I would like to apologize, it appears that some of my statements were wrong and my ideas could have been better stated.

Easier way to say it would be to say something that is truly bionian (2D) can not be seen by a trionian (3D) because it would have to exist in the 3D realm of the trionian for them to see it and that would mean it was 3D.

Light can exist in 2D... it would only move in 2 directions instead of 3.

Light is something that isn't restricted by depth, height and width because it is energy and not matter. It takes up no space.

from a neighboring and related forum http://tetraspace.alkaline.org/forum/viewtopic.php?t=424 where I was kindly corrected. A photon's 'size' is generally talked about in terms of wavelengths.

I have also since discovered that photons do not always in fact 'bounce' off of objects; they are also absorbed by electrons, causing the electrons to gain energy which they then re-release in the form of a different photon.

So I apologize for those two paragraphs.
I also apologize for making my first post so long.

[Hugh]

Hi miseleigh,

I doubt that a bionian would have eyes that work the same way ours do. My guess is he would see a 1D line without any problems, assuming both he and the line he sees exist.

The problem is that the line has only one dimension, length. It's not a "very thin line" or now that I think about it an "infinitely thin line" or even one with a thickness of "epsilon".

P Wrong states that the most important statement in any epsilon-delta proof is "Let epsilon>0". This doesn't apply here because the second dimension isn't available here to give any height to the line of vision, it's only available as a perpendicular direction in which to look in.

[houserichichi]

Even more succinctly, miseleigh, photons NEVER bounce off things, not even eachother. Either they are absorbed and re-emitted or they pass right through the thing in their way.

For instance, when a photon strikes a mirror it is actually absorbed by an electron within and spit out in a different direction. You're right to say that a different photon is re-emitted, but since all photons are identical it's kind of a moot point. Also of note is that the photon doesn't necessarily bounce off the 2-dimensional "skin"/boundary of the mirror. Some, in fact, make it inside (since no mirrors are 2-dimensional) and are absorbed/emitted within but appear otherwise.

Also, if two photons were to "collide" in mid air with one another they would pass right through because two photons CAN occupy the same place at the same time (they're gauge bosons...which are allowed to do that).

[moonlord]

So bosons can't bounce off anything? Interesting... Got to learn QM...

[houserichichi]

Well bosons don't bounce, they get absorbed or annihilate. The two photons hitting eachother wasn't exactly true - they annihilate, form particle-antiparticle pairs, which in turn form more photons, etc etc.

Nothing bounces off anything at the quantum level because all "things" have no structure to bounce off of.

[miseleigh]

So are all bosons measured by wavelength? And does anybody consider them 'particles' as such? I agree with moonlord- learning some QM would be good.

But back to a slightly earlier topic, now that I've thought about it a little more, I don't think a bionian would see a one dimensional line. That would be analagous to us looking at a 2d plane. If Fred lives in a bionian world, the object he's looking at would have to be bionian as well, such as a circle or square lying in his plane. So extrapolating from this seems to suggest that Fred can have no interaction with anything in our world and vice versa, and therefore the same for Emily's tetronian space.

So how can we possibly know if either of these exist?