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#1 2015-07-08 23:34:33

phanthanhtom
Member
Registered: 2012-06-22
Posts: 290

Convex polyhedra with only certain types of regular polygon faces?

All of this started after I read the answer for the following question:

"Given are fifty square cardboard pieces and fifty equilateral triangular cardboard pieces, all of side length 1 cm. Construct a set of different convex polyhedra with these pieces as faces. Two polyhedra with the same numbers of vertices, edges, square faces and triangular faces are not considered different. Use as many of the 100 pieces as possible".

The answer included: the regular tetrahedron, the cube, triangular prism, J1, J7, J8, J14, J16, the cuboctahedron and the small rhombicuboctahedron.

I was curious how many polyhedra could be constructed with only square and equilateral triangular faces, and so eventually, with some research, came to the following questions:

1) How many convex polyhedra have only faces that are squares or equilateral triangles as faces? (+list)

2) How many convex polyhedra have only faces that are regular polygons not triangles? (+list)

3) (Bonus) Does the set of convex polyhedra with only regular polygon faces include only Platonic, Archimedean and Johnson solids, plus regular prisms and antiprisms?

Some additional clarification:

1) Polyhedra that only has squares or only had equilateral triangles are also included (e.g. cube, regular tetrahedron)

2) I will be discounting regular prisms (which are infinitely many in number), except the cube, which, well, is special. Anything that includes only regular polygon faces with at least 4 edges.

Thanks for your help.

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