## What exactly is a gyrochoron?

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

### What exactly is a gyrochoron?

I've seen them in Bowers' dice page, and I've played with them in 4D toys, but I haven't found any precise description of how they're built.

I think I have an idea, though. Since gyrochora exist with any amount (n ≥ 5) of cells, I figured that their symmetry group must be (in general) cyclic. So I then thought that perhaps, an (m,n)-gyrochoron could be the dual of the convex hull of the action of a point under a double rotation of angles 2π/m and 2π/n. This would also explain why gyrochora don't exist for m or n equal to 1 (all points would lie on a two-dimensional plane).

Am I correct?
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Dionian

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Joined: Sat Apr 11, 2020 5:44 am
Location: Monterrey, Mexico

### Re: What exactly is a gyrochoron?

Near as I understand, n,m-gyrochora are constructed like this.

Take the vertices of an n,n-duoprism and label them (0,0) to (n-1,n-1), such that any set of vertices with the same value in one of the coordinates forms a n-gonal face of the duoprism. take a subset of vertices starting at (0,0) and then moving along in such away so that the first coordinate increases by 1 and the second one increases by m (if this sends you to a number n or higher, loop back arounnd to 0 from n-1). The convex hull of the selected vertices constructs the n,m-"step prism". THe gyrochora are then the duals of these polychora.

Of course, I still have trouble actually imagining what these things actually look like in 4D. I usually understand polytopes easier by looking at their elements (for example, what cells they have) and this construction doesn't really give any explicit information on that.
Trionian

Posts: 119
Joined: Sat Mar 18, 2017 1:42 pm

### Re: What exactly is a gyrochoron?

I believe both constructions are equivalent, though underlining the relationship to the duoprism certainly helps underline the structure of these polychora.

I think that rather than imagining how these things look, it might be better to imagine how they're constructed. I visualize them as a sort of discrete version of the duocylinder, in the same way a regular polygon is a discrete version of a circle.
I'm building a library of polyhedra and polychora: https://drive.google.com/drive/u/0/folders/1nQZ-QVVBfgYSck4pkZ7he0djF82T9MVy
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Dionian

Posts: 28
Joined: Sat Apr 11, 2020 5:44 am
Location: Monterrey, Mexico

### Re: What exactly is a gyrochoron?

Yeah that makes sense. Still goes against how I usually think about polytopes though.

Apparently analogous things exist in all even dimensions - for example similar constructions based on trioprisms in 6D and so on.