## A curious coincidence

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

### A curious coincidence

Since you people are so quiet, I thought I should stir the pot a little.

Here's a challenge:

1) Name a uniform polyhedron with 8 faces, 18 edges, and 12 vertices.

2) Can you name a different polyhedron with 8 faces, 18 edges, and 12 vertices? Interesting coincidence, isn't it?

If you found the above too trivial, how about:

3) How many 4D uniform polychora can you name that contains the above two polyhedra as cells?

quickfur
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### Re: A curious coincidence

quickfur wrote:1) Name a uniform polyhedron with 8 faces, 18 edges, and 12 vertices.

a hexagonal prism?

Hugh
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### Re: A curious coincidence

quickfur wrote:2) Can you name a different polyhedron with 8 faces, 18 edges, and 12 vertices? Interesting coincidence, isn't it?

a truncated tetrahedron?

Hugh
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### Re: A curious coincidence

Excellent! Now can you answer question (3)?
quickfur
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### Re: A curious coincidence

Hexagonal duoprism
Secret
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### Re: A curious coincidence

Well, a 6,6-duoprism has hexagonal prisms for cells, but it doesn't have truncated tetrahedra. Which polychora have both?
quickfur
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### Re: A curious coincidence

I see four of them - there are runcitruncated simplex, runcitruncated 16-cell, runcitruncated 600-cell and a prism over truncated tetrahedron.
Mrrl
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### Re: A curious coincidence

Cool!! You caught a 4th one that I didn't catch: the truncated tetrahedron prism. The others are right, too. Congrats! Молодец!
quickfur
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