# What shape is this?

im doing a question that involves a shape with 8 faces, 10 vertices and 16 edges. Can anyone enlighten me as to what this shape is called? Many Thanks

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It's not clear to me that the shape is unique given only this information. – Qiaochu Yuan Mar 15 '11 at 14:32
@Qiaochu: I believe it is unique. It takes a lot cases to prove this though. There are only a couple of possible degree sequences, and only a couple of possible choices for the sequences of faces, and eventually we eliminate all but the degree sequence 3,3,3,3,3,3,3,3,5,5 and the possibility that all the faces are squares. I wonder if there is a slicker way? – Eric Naslund Mar 15 '11 at 14:51
I am going to leave my previous comment, but Dan Moores answer definitely shows it is completely wrong. – Eric Naslund Mar 15 '11 at 16:47

Additionally, any polyhedron represented at this link could also be the shape you're looking for:

http://www.uwgb.edu/dutchs/symmetry/poly8f2.htm

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This seems quite definitive. – Grumpy Parsnip Mar 15 '11 at 16:17

The tetragonal trapezohedron seems to fit your bill.

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This paper, that enumerates and shows plane 3-connected graphs for the convex polyhedra with 4 to 8 faces, may be of interest: Federico, P. J. Polyhedra with 4 to 8 faces. Geometriae Dedicata 3 (1974/75), 469–481.

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It is called an octahedron.

Any polyhedron with eight faces is an octahedron. Unfortunately there are 76 distinct graphs corresponding to polyhedra with the amount of faces and vertices you require, so it is not possible to assign your shape a more specific name.

Please note that even after you restricted your search to a specific graph, all the possible spatial embeddings of such a graph into a polyhedron would result in different names (consider the cube and the truncated square pyramid for instance).

Please also note that Steven Dutch's list - at the time I am writing this post - is not completely accurate, since it contains 77 distinct graphs.

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