4D prismic zonotopes & the duocylinder

[quickfur]

I've just discovered something really fascinating: the existence of a class of 4D zonotopes that consist of identical regular 3D prisms.

Prismic zonotopes

Consider the tetracube with its 8 cells in cell-first perspective projection. This is the "cube-within-a-cube" projection. Let's call the outer cube A, and the inner cube C. Call the upper frustum D and the lower frustum B. The cubes A, B, C, D, form a cycle in the ZW plane. Now consider the 4 side frustums. Let's call these cubes P, Q, R, and S. These form another cycle, in the XY plane. These two cycles are orthogonal.

Now, let's substitute the cubes with, for example, triangular prisms. The ZW cycle can be reduced to a 3-member cycle of triangular prisms, attached to each other by their triangular faces. The XY cycle can likewise be reduced to a 3-member cycle, also attached to each other by their triangular faces. The square faces of the prisms in these two cycles exactly match each other. The result is a closed 4D zonotope made of 6 triangular prisms. These prisms form two 3-member cycles that are interlocked with each other.

We can also do the substitution with other types of prisms, for example, with pentagonal prisms. Then we expand the ZW cycle to include a 5th member. The cycle of 5 square faces of adjacent prisms precisely form a gap that can be filled in by other pentagonal prisms, which will form a 5-member XY cycle. In other words, now we have a prismic zonotope made of 10 pentagonal prisms.

This process actually works with all n-gonal prisms, as long as all edges in the prisms are equal. The result will be a zonotope of 2n prisms. This is fascinating, because all facets of these zonotopes are identical prisms. (The tetracube is just a special case, when square prisms (i.e. cubes) are used.)

Double-torus

Now, what is interesting is when we take the limit of these prisms as n approaches infinity. The limit of the ZW cycle is a circular torus formed by rotating a circle in the XY plane around the ZW plane. The limit of the XY cycle is also a circular torus, formed by rotating a circle in the ZW plane around the XY plane. In other words, this is a "double-torus"-like 4D volume bounded by two torii. (Does anybody know what this object is called, if it does have a name? If not, may I suggest to name this object the 4D double-torus? :-) ) What is fascinating is that, if my intuition is right, this object can roll in two orthogonal planes... which means it can cover the same area as a spherical cylinder. (Alkaline should add this to his collection of rotatopes ;-))

[Update (2005-06-13): I just found out that this shape is in fact the same as the duocylinder.]

Mixed prismic polytopes

The n-gonal prismic zonotopes aren't the only possibilities. We can also mix two different kinds of prisms: for example, we can use cubes in the ZW cycle, but add a 5th cube to it. This produces a cycle of 5 square faces on the sides of the cycle, which can be closed by 4 pentagonal prisms. This will produce a polytope made of 5 cubes and 4 pentagonal prisms.

In general, we can take n m-gonal prisms and m n-gonal prisms and construct analogous polytopes in this way. For example, we can build a polytope from 7 triangular prisms and 3 heptagonal prisms. If n=4, we get the same polytope we would get by extruding an m-gonal prism into 4D. Similarly for m=4. Of course, things are a lot more interesting when neither n and m equal 4. The cool thing about these polytopes is that the gonality of the prisms in one cycle is equal to the number of prisms in the other cycle, and vice versa, so they are completely determined by the two numbers n and m.

Is there a consistent naming convention for these things? (Wendy?) These aren't really prisms in the 4D sense; they are just built from 3D prisms. I was thinking maybe I can name them the "prismic bicycles" ("bicycle" as in bi- + -cycle, for the ZW and XY cycles, not the vehicle :-)) I like the sound of a "5,6-prismic bicycle", or a "7,7-prismic bicycle". :-) In general, the name would be "n,m-prismic bicycle" where n and m are integers greater than 2.

(And I have this inkling that in 6D, we could have tri-cycles... although I'm not sure what the 5D equivalent of prisms would be.)

[wendy]

This is of course, the prism-product of two polygons.

[quickfur]

This is of course, the prism-product of two polygons.

Interesting. I looked up prism product in the polygloss, but didn't really understand how exactly one forms a prism product. Care to elaborate?

Also, I'm beginning to wonder if there is any correlation between the "double-torus" and alkaline's duocylinder (which I still have a hard time visualizing... intersecting 3-manifolds mentally is not exactly trivial...). I'm beginning to wonder if they are in fact the same object. If they are, then the n,n-prismic bicycle construction ought to be a nice approximation for polygon-based animation programs. That would be a nice thing to interactively rotate in a Java applet or something (pat? :-)).

[quickfur]

[...]Also, I'm beginning to wonder if there is any correlation between the "double-torus" and alkaline's duocylinder (which I still have a hard time visualizing... intersecting 3-manifolds mentally is not exactly trivial...). I'm beginning to wonder if they are in fact the same object. If they are, then the n,n-prismic bicycle construction ought to be a nice approximation for polygon-based animation programs. That would be a nice thing to interactively rotate in a Java applet or something (pat? :-)).

Actually, I just did a quick calculation, and it confirmed my suspicions: the "double-torus" is the duocylinder!!

The ZW cycle is the union of all disks (filled circles) of a fixed radius parallel to the XY plane where their Z and W coordinates satisfy z² + w² = r². The XY cycle is likewise the union of all disks of the same radius parallel to the ZW plane where their X and Y coordinates satisfy x² + y² = r². This gives us a 4D closed 3-manifold made of two pieces:

1. The piece corresponding with the ZW cycle, described by the equations:
x² + y² <= r²;
z² + w² = r².

2. And the piece corresponding with the XY cycle:
z² + w² <= r²;
x² + y² = r².

Note: the inequalities are necessary to form a closed 3-surface; otherwise you only get a 2D surface that constitutes the boundary of the bounding 3-volumes (sorta like only the edges of a cube without the intra-face points. I'm not sure why alkaline omitted these inequalities in the rotatopes page (oversight, I guess?). In any case, it's clear that they describe the same shape.

This is very good news... since the duocylinder is the limiting shape of the n,n-prismic bicycles as n approaches infinity, this means we can simply pick a sufficiently large n to serve as a polychoric approximation of the duocylinder. (Directly rendering 4D quadratics is very painful.) We could use the 8,8-prismic bicycle for simple line-based animations, for example. For polygon-based rendered animations, we could use a better approximation, such as a 12,12-prismic bicycle.

[wendy]

The prism product can be presented as a cartesian product.

That is, if you have 4d, you can divide these into wx, yz or w, xyz.

Into each space you can draw whatever that space supports, eg

wx => pentagon

yz => hexagon

The points that have coordinates wx in the pentagon, and yz in the hexagon, fall into the figure.

One can derive what the surtope consist of this figure is too:


pentagon = 1 pentagon + 5 edges + 5 vertices
hexagon = 1 hexagon + 6 edges + 6 vertices

Product = 1 pentagon.hexagon + 5 edge.hexagon + 5 vertex.hexagon
+ 6 edge.pentagon + 30 edge.edge + 30 edge.vertex
+ 6 vertex.pentagon + 30 vertex.edge + 30 vertex.vertex

We have then a vertex*polytope = polytope prism of zero height = polytope. Also, line * polytope makes for a (normal) prism, for example, line * line = rectangle.

This reduces the above to

1 pentagon.hexagon + 5 hexagonal prisms + 6 pentagonal prisms + 5 hexagons + 6 pentagons + 30 squares + 60 edges + 30 vertices.

The 30 squares form a sheet of 5*6, when unfolded.

Of the torus-figure, this is a bi-circular prism, or 'duocylinder'. You can also have things like a 'hexagon-circle prism'. Circles, spheres etc are just shapes that participate in any product. You can have a bi-circular tegum, or a bi-circular pyramid, etc.

W

[quickfur]

The prism product can be presented as a cartesian product.

That is, if you have 4d, you can divide these into wx, yz or w, xyz.
[...]

Ahh I see.

Of the torus-figure, this is a bi-circular prism, or 'duocylinder'. You can also have things like a 'hexagon-circle prism'. Circles, spheres etc are just shapes that participate in any product. You can have a bi-circular tegum, or a bi-circular pyramid, etc.

Yep, I just realized that the cubinder is just a square-circle prism. One could think of it as the limit of a 4,n-prismic bicycle as n approaches infinity. There is no reason to be limited to 4, so we can have triangle-circular cylinders, pentagonal-circular cylinders, and other n-gonal-circular cylinders. Neat stuff.

Now that I have my geometric calculator in a crudely working form, I think I will start creating diagrams of these prismic bicycles / prism products. I just realized upon reflection that the ridges in the duocylinder project onto the surface of a 3D torus in perspective projection. Rotating the 4D viewpoint would produce fascinating transformations of this torus, with various "turning inside-out" behaviours. One of these days I should cobble together an animation of this fascinating object.

[quickfur]

[...]Now that I have my geometric calculator in a crudely working form, I think I will start creating diagrams of these prismic bicycles / prism products. I just realized upon reflection that the ridges in the duocylinder project onto the surface of a 3D torus in perspective projection. Rotating the 4D viewpoint would produce fascinating transformations of this torus, with various "turning inside-out" behaviours. One of these days I should cobble together an animation of this fascinating object.

I seem to be replying to myself a lot recently, hmm... :? But anyway, just thought you guys would be interested in this:



That's the perspective projection of the duocylinder, approximated by a 30,30-prismic bicycle. :-) Note that it is not actually hollow; there are two 30-gonal lids covering the "hole" of the torus but they didn't get rendered 'cos they are planar. The pink part in the center is farther away in the 4th direction. I colored it pink so that it's more easily visible. However, it still looks as if it's "behind" the far side of the torus, which it is not: it lines the "doughnut hole" of the torus. The black lines seem to assert themselves to be in the front, however, and the 4D curvature of the surface unfortunately reinforces this illusion.

I'm gonna do an animation of it sometime... this particular view of the duocylinder looks pretty tame, but some of the other views have that pink part doing bizarre twists and other cool stuff. Stay tuned! :-)

Update: here's another view of the duocylinder, which gives you an idea of some of the weird stuff it does when rotated through 4D:



Again, the "flat" side of this image is covered by a 30-gonal lid, but it isn't apparent because it's flat, so the image appears to be hollow. You can see here how the pink part (the part farthest away in the 4th direction) has rotated to the limb of the image, and is about to emerge. The outer black part has begun to warp itself inwards (rotated away from the viewer in 4D), and eventually will become the new "hole" in the torus (albeit rotated 90 degrees, 'cos the two planes of rotation in the duocylinder are perpendicular).

[wendy]

Maybe one day, you might feed in a bi-circular tegum. This should be interesting.

You make this, from drawing a circle in the wx and yz planes, and then covering the lot with a skin, so that the surface represents the space

sqrt(w^2 + x^2) + sqrt(y^2+z^2) = constant.

You can also make it by setting a cone up in 3d, with the base in yz plane, and height in x, and then rotate the apex in the wx space.

W

[quickfur]

Maybe one day, you might feed in a bi-circular tegum. This should be interesting.

You make this, from drawing a circle in the wx and yz planes, and then covering the lot with a skin, so that the surface represents the space

sqrt(w^2 + x^2) + sqrt(y^2+z^2) = constant.

You can also make it by setting a cone up in 3d, with the base in yz plane, and height in x, and then rotate the apex in the wx space.

I'm willing to give it a try, but I'll need to figure out what's a good polychoric approximation for it. My geometric calculator program doesn't actually do tesselation of arbitrary objects, so I have to explicitly program it to compute vertices and edges, etc.. (This happens to work well with the duocylinder 'cos the m,n-prismic bicycles happen to produce a nice square-based mesh.)

Currently, I'm thinking of approximating the initial cone by an n-gonal-base pyramid, and perhaps split up the pyramid along regular intervals on its apex-base axis so that we get a nicer mesh. But I'm not sure how to join this with the rotated versions of the cone... perhaps add an edge between equivalent vertices between two adjacent cones? The base would have to be excluded, of course, since all the cones share the same circular base. But with this many edges, the result may be less than pleasing... we'll see.

[quickfur]

Alrighty, I've made an animation of the duocylinder turning around the YW plane (the viewpoint is looking into +W). This looks totally cool:



The blue part marks the part of the duocylinder farthest away from the viewer at the beginning. As the cylinder turns in YW, this part eventually exchanges places with the black part and the black part becomes farthest away (becomes the "inside" of torus hole). As it rotates past 180 degrees, the blue part rotates back into the center once again. Truly fascinating.