# What is the name of this hexagon/pentagon polyhedron?

What is the name of this convex polyhedron? $$(V,E,F)=(14,36,24)$$.
The top and bottom vertices are degree-$$6$$, spanning hexagons, which are zigzag connected in the band between the two hexagons. The faces are approximate isosceles triangles in this physical model.

• I don't know whether it has a name but there's an interesting generalization. You can build something like this by adding pyramids to the top and bottom faces of an antiprism. All the faces can be isosceles. – Ethan Bolker May 1 at 12:06
• @EthanBolker: So perhaps a capped antiprism. – Joseph O'Rourke May 1 at 12:10
• I think we can go with Dennis, maybe? – Asaf Karagila May 1 at 12:57
• @AsafKaragila: According to @ zwim, apparently "Frank" is more apt. – Joseph O'Rourke May 1 at 13:10
• Pentahex polyhedera were extensively studied by Buckminster Fuller in relation to the design of geodesic domes. See his book, "Critical Path". When I was an engineering student at UC Berkeley in the early 1960's, he had a weekly show on local radio station KPFA. – richard1941 May 7 at 18:52

I believe it is a gyroelongated hexagonal bipyramid.

https://en.wikipedia.org/wiki/Gyroelongated_bipyramid

The example shown in the Wikipedia article uses equilateral triangles, which results in coplanar faces around the degree-6 vertices, but as you mention, the model uses isosceles triangles.

I entered $$14$$ vertices and $$24$$ faces, and it seems to be a C14 Frank Kasper polyhedron (seems to be named from chemist researcher...)

The degrees of vertices $$5$$ and $$6$$ correspond.

http://rcsr.anu.edu.au/polyhedra/fkf

If I am right, the four extra faces make it a tetraicosahedron. But this is a generic term.

As already was said, all faces could be made the same isoceles triangle.

But alternatively the equatorial segment could be made a uniform hexagonal antiprism too. Then the emphasis would be more towards that antiprism than to those attached pyramids. Therefore you also could speak of a "bi-apiculated hexagonal antiprism".

--- rk