New Class of Polyhdera

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

New Class of Polyhdera

Postby Robert May » Wed Jun 27, 2018 7:21 pm

For n >= 1 and p >= 3 define an (n,p)-polyhedron as a convex polyhedron with exactly n congruent regular p-gonal faces. No restrictions on the other faces.

For n = 1, you can use pyramids with regular p-gonal bases. For n = 2, uniform prisms or uniform antiprisms with regular p-gonal bases.

For p = 3, you can find an example for n = 1, 2, ..., 20 by doing the appropriate truncation at n vertices of a regular dodecahedron. (There are of course other examples.) Antiprisms can be used for all even n > 20. J24 & J26 take care of n = 25, and J47 & J71 take care of n = 35.

For p = 4, use the uniform n-prism for n >= 3.

For p = 5, use the appropriate truncations at n vertices of the regular icosahedron for n = 1, 2, ..., 12.

For p = 6, the truncated icosahedron has n = 20 regular hexagons. Augment with hexagonal pyramids with heights so that the resulting polyhedron remains convex. You can do this to get (n,6)-polyhedra for for n = 1, 2, ..., 19.

For p = 8, the truncated cube has n = 6 regular octagons. Augment with octagonal pyramids with heights so that the resulting polyhedron remains convex. You can do this to get (n,8)-polyhedra for n = 1, 2, 3, 4, 5.

For n = 10, the truncated dodecahedron has n =12 regular decagons. Augment with decagonal pyramids with heights so that the resulting polyhedron remains convex. You can do this to get (n,10)-polyhedra for n = 1, 2, ..., 11.

Question 1: Are there examples for p =3 and odd n > 20 besides n = 25 and n = 35?

Question 2: Are there examples for p = 5 and n > 12?

Question 3: Are there examples for p = 6 and n > 20?

Question 4: Are there examples for p = 8 and n > 6?

Question 5: Are there examples for p = 10 and n > 12?

Question 6: Are there examples for p = 7, 9, 11, 12, 13, ... and n> = 3?
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Re: New Class of Polyhdera

Postby ubersketch » Mon Oct 22, 2018 10:12 pm

I had a similar idea to this, but with vertex figures instead of faces and the faces only had to be semiregular, not regular.
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Re: New Class of Polyhdera

Postby Klitzing » Tue Oct 30, 2018 9:07 am

You could inscribe any (p=3k)-gon (k>1) into the triangles of any convex polyhedron with any amount of regular triangles. And then just consider the convex hull of vertex set of those p-gons, i.e. truncating / beveling / etc. the starting polyhedron accordingly. At least you would end with that amount of p-gons plus some other mostly irregular faces, but those seem not to concern you here.

The same idea would hold for any other p=mk with each m>=3 and any k>1, i.e. inscribing such regular p-gons into the former regular m-gons of some convex starting figure.

--- rk
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