# 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. May 1, 2019 at 12:06
• @EthanBolker: So perhaps a capped antiprism. May 1, 2019 at 12:10
• I think we can go with Dennis, maybe? May 1, 2019 at 12:57
• @AsafKaragila: According to @ zwim, apparently "Frank" is more apt. May 1, 2019 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. May 7, 2019 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.

• That's quite a mouthful! Actually, kinda beautiful. Thanks! May 1, 2019 at 23:01
• Great video on the subject: youtube.com/watch?v=FqCZhSIXcv4 May 2, 2019 at 9:34

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