Duocylinder: euclidean torus: True shape of Asteroids screen

Discussion of shapes with curves and holes in various dimensions.

Duocylinder: euclidean torus: True shape of Asteroids screen

Postby Eric B » Thu Aug 18, 2005 2:12 pm

I was looking at the definition of the duocylinder at the eusebia site, and trying to figure out exactly what this was. I knew about the cubinder (extruded cylinder: surface=2 flat dimensions, 1 curved) and the spherinder (extruded sphere: 2 curved dimensions, 1 flat), but wondered how a duocylinder was formed from a cylinder.
Studying it again, this time I noted the progression from the duoprisms, to the prismic cylinders, finally to the duocylinder. It basically involves taking the values of m and n to infinity (Note, Wendy, this implies that a circle is "the limit of... [the sides of the polygons] taken to infinity").
Well finally coming to understand it now, I realized that this was the "Euclidean [true] 2-torus" discussed by Luminet, Starkman & Weeks in the Scientific American article "Is Space Finite?"

I, like many others always wondered what the global shape of the Asteroid screen was. You would think, at first, it would be a sphere. But then you realize that the top and bottom and left and right sides are joined at lines, not points. Then, Michio Kaku's Hyperspace said that it was a torus or donut shape, formed by curling a square into a cylinder, and then joining the cylinder's circlular ends to each other. He suggests doing that with a piece of paper. I had thought of this, but realized, that it could not be exact, because if you try to do that with paper, you end up scrunching one side, and ripping the other.
So then, I come across the Sciam aticle. It points out that the "true" torus, or Asteroids screen shape could not sit in 3-space at all! "Doughnuts may do so, because they have been bent into a spherical geometry around the outside [i.e. the stretching], and a hyperbolic geometry around the hole [i.e. the crunching] Without this curvature, doughnuts could not be viewed from the outside". The article does not explain how the true torus is formed, then; but I figured out, it was from taking the two ends of the cylinder, and curling and joining them in four dimensional space. The original square you started out with could be curled so easily because it was "flat". But once it becomes a cylinder, it no longer is flat --in 3 space that is. But it is still "flat" in 4-space! A lower dimensional analogy woulf be a belt or ribbon. If you take a circle in 3-space, and step down the dimensions, the circle becomes two points bounding a "hollow" line segment. The curved 2D surface conmecting the circles is then represented by two lines joining the opposite pairs of points. Basically, a belt or ribbon. Try to imagine joining the ends of the ribbon by keeping it flat on the 2D surface. Once again, you scrunch the side that becomes the inside, while stretching what becomes the outside. Now instead, simply lift the two ends up through 3-space and join them. Much easier!

This, basically, is what the duocylinder is in the higher dimension. The curling of the square represents m, and the curling through the higher space represents n. As circles, both would be the limit of polygons at infinity. In a simply extruded polytope, for instance, all lines generate squares, which sets the value of n at 4)

Is all of this right?
Last edited by Eric B on Sun Aug 21, 2005 1:58 am, edited 1 time in total.
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Postby houserichichi » Thu Aug 18, 2005 2:59 pm

I'll leave the duocylinder argument to the geometers (of which I am not), but the Asteroids screen is, indeed, the surface of a torus. Instead of working with paper (which was only meant to be an analogy), work with the famous sheet of rubber that topologists are used to. Take a rectangle of rubber, identify the top and bottom, then (if you're getting technical about crunching ends, stick a slinky inside to keep its shape...but we'll work with perfect movement here instead) identify the left side with the right side. Now take a pencil (or some other writing tool) and draw lines.

No matter what lines you draw on the torus, if you were to unfold it the same way you made it, you'd see that the lines follow the exact same path that the little triangular spaceship makes in the Asteroids game. Of course we could always come up with a more rigorous proof than that, but it definitely serves our purpose.

Topologists were never very fond of working with paper. :wink:
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Postby pat » Thu Aug 18, 2005 7:32 pm

The book The Shape of Space by Jeffery Weeks is an excellent book that goes into much detail about how a torus need not be bent. The cover picture shows a globe inside a very small, flat, 3-D torus. I highly recommend the book.

From what I understand, the more recent editions come with a CD-ROM with software that lets you look around in various flat spaces. Oops... no, that's a separate, shorter (133pp vs. 328pp) book...

And, houserichi, I think Eric B's point about the paper is that most people's image of a torus is the donut shape. This would imply some sort of bentness and dilation on the Asteroids screen. However, the asteroids screen clearly is meant to have zero curvature.
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Postby wendy » Fri Aug 19, 2005 12:48 am

A torus is one of those 3d solids.

Topologists use it for a particular kind of surface, formed by identifying opposite sides of a square.

The hedrix that forms the margin on a duo-cylinder, or the cartesian product of circle-surfaces, is free from linear variations, but is still distorted.

In a true topological torus, one can draw a straight line through any pair of points. This is not the case for the 3d or 4d representations of the torus. In fact, the only straight lines in the 4d torus are the diagonals, and their parallels.
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Re: Duocylinder: euclidean torus: True shape of Asteroids sc

Postby quickfur » Mon Sep 19, 2005 2:45 am

Eric B wrote:I was looking at the definition of the duocylinder at the eusebia site, and trying to figure out exactly what this was. I knew about the cubinder (extruded cylinder: surface=2 flat dimensions, 1 curved) and the spherinder (extruded sphere: 2 curved dimensions, 1 flat), but wondered how a duocylinder was formed from a cylinder.
Studying it again, this time I noted the progression from the duoprisms, to the prismic cylinders, finally to the duocylinder. It basically involves taking the values of m and n to infinity (Note, Wendy, this implies that a circle is "the limit of... [the sides of the polygons] taken to infinity").
Well finally coming to understand it now, I realized that this was the "Euclidean [true] 2-torus" discussed by Luminet, Starkman & Weeks in the Scientific American article "Is Space Finite?"

Well, the "true torus" you refer to actually only comprises the ridge of the duocylinder, where its two bounding 3-surfaces meet. It's like the "wireframe" of the duocylinder. To completely close up the 4-volume of the duocylinder, you need the two interlocked torus-shaped 3-volumes described by the inequalities on my page.
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Postby Eric B » Mon Sep 19, 2005 10:19 pm

I was wondering if there was some part of the duocylinder not covered by the torus. I figured that perhaps the torus was only the surface of the duocylinder, or perhaps the duocylinder was the convex hull of the torus. (i.e. "filled in" by those two other toruses you mentioned).
Now; I'll have to try to make sure I understand what you've just described and correct the addition I made to the Wikipedia article on this. :oops:
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Postby quickfur » Mon Sep 19, 2005 11:09 pm

Eric B wrote:I was wondering if there was some part of the duocylinder not covered by the torus. I figured that perhaps the torus was only the surface of the duocylinder, or perhaps the duocylinder was the convex hull of the torus. (i.e. "filled in" by those two other toruses you mentioned).
Now; I'll have to try to make sure I understand what you've just described and correct the addition I made to the Wikipedia article on this. :oops:

When thinking of 4D objects, it is useful to keep in mind that 2D surfaces in 4D cover only a region equivalent to a string. That is to say, it is not sufficient to close a 4D volume, just as the surface of a sphere cannot be covered by a finite piece of string. You need a 3D volume (or more precisely, a 3-manifold) in order to close off a 4D volume.

So the torus you describe really is only the "edge" of the duocylinder, sorta like a wireframe of a cylinder rather than the actual surface. The bounding surface consists of two parts, which are the volumes that fill either side of the torus.

Keep in mind that, even though to our 3D minds the two interlocked toroidal volumes seem to fill up space, they are actually infinitely thin in 4D, and lie completely on the surface of the duocylinder. The torus you describe actually only covers the equivalent of the "rim" of the duocylinder, so to speak. When I said "filling in", I meant it in the sense of stretching a membrane across the rim so that it actually closes off the shape. The interlocked toroidal shapes actually do not include the inside of the duocylinder at all.

Again, the catch in 4D is that you need to think in terms of bounding volumes rather than 2D surfaces, because it takes a 3D volume to wrap around a 4D volume (just as it takes a 2D surface to wrap around a 3D volume). This is the reason supposed "containers" like the Klein bottle actually won't hold any 4D liquid: it only covers a 2D region, so it's like trying to hold water with a piece of string; the water will immediately flow off the Klein bottle. It takes a 3-manifold to hold liquid in 4D, and as far as I can tell, none of them are non-orientable surfaces like Klein bottles. So (un)fortunately you won't have 4D beings pouring water into/out of their Klein bottles and other such strange feats.
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Postby Eric B » Tue Sep 20, 2005 2:37 am

Well, actually, when the 2D surface becomes a torus, it closes off 3D volumes, and those 3D volumes then can close off a 4D volume. (Sort of like making a circle out of a line, and then wrapping that around a sphere).
And when I specified "convex hull"; I was figuring that the object was still "hollow" in 4D. So "conxex hull" is an accurate description, right?

Also. when you say "ridge", that is a 2D surface in 4D, as the d-2 counterpart to a 1D "edge" in our space, right?

[PS. Wikipedia updated!]
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Postby quickfur » Mon Sep 26, 2005 1:56 am

Eric B wrote:Well, actually, when the 2D surface becomes a torus, it closes off 3D volumes, and those 3D volumes then can close off a 4D volume. (Sort of like making a circle out of a line, and then wrapping that around a sphere).

A 2D surface only closes off a 3D volume when it lies completely within a 3-hyperplane (which in this case it doesn't, unfortunately). Just as a circular piece of thread in 3D doesn't necessarily close off a circular area (e.g., it is bent and twisted in the 3D sense, even if it is still circular), a toroidal 2D surface in 4D doesn't uniquely define a closed 3D volume. This is particularly true for the duocylinder ridge, which is circular in two dimensions. There are at least two different 3D volumes that can legally be the 3D closure of the torus shape. In fact, there can be infinitely many (think of how inflating a balloon inside a wireframe can make a bulging cube, for example).

I'm just being nitpicky here, of course... I know what you mean by the 3D regions closed off by the torus. The problem with that statement is that the duocylinder's ridge defines two different 3D volumes to close off, and both of them are the two distinct pieces of the duocylinder's surface. They are mutually perpendicular.

And when I specified "convex hull"; I was figuring that the object was still "hollow" in 4D. So "conxex hull" is an accurate description, right?

Now, the convex hull of the torus might be the duocylinder itself. I'm not 100% sure about this. In any case, my point is that one should not gloss over the fact that the duocylinder's surface is made of two distinct 3D pieces.

Also. when you say "ridge", that is a 2D surface in 4D, as the d-2 counterpart to a 1D "edge" in our space, right?

That's right.

[PS. Wikipedia updated!]

Cool. :-)
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