Stewart Toroids

[wendy]

Tom Ruen has me thinking. He's pretty good at that.

Anyway, we can make a polytope of uniform edges and regular polygons, by using a couple of extra wythoff operators.

In essence, they're flat lace towers, with the core removed, and various tunnels drilled through. We could use a special node marker for these nodes.

For example, there is xx3ox4xx&xt. The difference between top and bottom is (0, 1, 0), which is a unit edge. You can make three toroid polytopes, by first popping out the centre, and then popping out one of the three symmetries, eg xxj3ox4xx&xt which pops out the square cupola ox4xx, or the 12 cubes by xx3oxj4xx&xt, or the 8 triangular cupola, by xx3ox4xxj&xt

In four dimensions, things come interesting,

For the simplex group, the only unit edge is x3o3o3x. You can then use this as a difference eg between ox3xx3oo3ox&xt or rectified|| runcitruncate Pentachoron. The outer bit has four face kinds, the inner has two, so CO || T, 3P /- , 6P / - , tT || O using Conway's runes and Klitzing's atop operator. Obviously we have to have tunnels to the core, so we have to remove 1 or 4 (or both). We also have to preserve continuity of faces, so there must be a margin between the faces on the outside.

So numbering these 1 to 4, we can have 12, 123, 23, 24, 234, and 34. We can then convert our ox3xx3oo3ox&xt into these by placing an extra j at the named nodes, eg oxj3xxj3oo3ox&xt.

You need to watch if you plan to list all of these things, that the surface does not 'break apart'. This means that you elect to keep some faces that have no common margins. It's ok if the faces share margins with some third face, the example 234 here, there is no margin between the 3P x3o2o and the tT, but they are both held together by sharing margins with the 6P.

Note that other symmetries offer more depth. For example, one gets o3x3o4o, o3o3o4x, and xi3o3x4o (xi = -1), as strutt-producing vectors between the layers. So if you start off with x3o3o4o as the core, you can add all of these vectors in any number and order, and then still poke the j at the end. Calling the vectors A, B, C, you can then have ABC or BAC or CBA in every permutation, and all 3*2 permutations of two vectors + 3 sundry vectors. That gives, then 6+6+3=15 different stewart toroids.

x3o4o3x has four vectors x3o4o3o, xi3x4o3o, and the reverses.

The group [3,3,5] does not have a lot of possibilities, but there is the wonderful 11-stratum tower that runs from o3x3o5o to x3o3x5x, with the two shortcuts. This opens up a large range of possible inner / outer shells, but when you are digging the tunnels, you might run into a vertex here and there, although i don't think the initial node j3.3.5 will cause problems.

Much fun

Wendy