Marek14 wrote:Hm, being a Polyhedron dude, perhaps you could tell me what are the simplest analogues of Stewart toroids for the 5 4D toratopes?
wendy wrote:You might notice that the tiger is the swirled torus. You can take a swirl dodecahedron, and run a torus-like hole up the middle of it. Whereas the tiger has two holes, the hollowed out swirl dodecahedron has twelve, and the swirl-icosahedron-torus has twenty. You can have hundreds of toric holes.
It gets worse. What do you do when you take something like the octagonny, and replace all the faces and the octahedrons, with empty space. Kind of like a spherated edge frame with the triangles filled in?
The 'pancake toruses' are frightening enough, and all of the torotopes are pancake toruses. These work like this. In 2d, there are no connected figures with holes connected to the outside.
If you now make a pancake, which is in effect, a thin prism of 2d, you can only make it circular, but you can poke circular holes in it. The number of circular holes is in 3D, the 'genus' or kind of thing. All solids with three holes are topologically equal to a pancake with three holes in it. You can make the pancake into a hollow sphere by shrinking the outline of the pancake into a circle. So a hollow sphere with N holes let into it, gives a pancake with N-1 holes.
In four dimensions, both the pancake shape and the holes let into it, can themselves have holes. So you are not restricted to a sphere-shaped pancake: it can be a torus-shape, or something with two holes let in. Inside the pancake, you can drop not just spherical holes, but even torus-shaped holes, and so forth. And these can also be knotted (like a trifoil knot), or even linked (like a chain of links). The tiger gives a torus-shaped pancake, with a single tube cut around inside the tyre. Philip would appreciate it if you said that the tiger-shape is what you get between the tire and the tube of a bicycle wheel.
Polyhedron Dude wrote:Marek14 wrote:Hm, being a Polyhedron dude, perhaps you could tell me what are the simplest analogues of Stewart toroids for the 5 4D toratopes?
I haven't explored 4-D Stewart toroids before, this could make another interesting group of polychora to investigate like the CRFs.
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