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Difficult to put into words, but let's say you have a tetragonal bipyramid. In any cases where you can draw 3 axis through opposite vertices and all three intersect at one point at right angles, how many topologically distinct forms are there?

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what do you mean by topologically distinct? – Willie Wong Jul 6 '11 at 19:27

If you paste together two tetrahera along a common face one gets a "bipyramid." The vertex-edge graph of this polyhedron will have 5 vertices and 6 faces. If you draw this graph in the plane it will have only triangle (3-gons) faces and all other plane triangulations with 5 vertices are combinatorially equivalent to this graph. However, there are many "different" metrical (that involve lengths and angles) convex 3-dimensional solids which will realize this graph. This web page may be of use:

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