# A Sixth Platonic solid?

[1] Wouldn't gluing a tetrahedron's one triangle to a another tetrahedron's triangle make a platonic solid ? See the picture to see clearly what I mean. Tetrahedron stacked one on each makes an another solid with $6$ faces, $5$ vertices and $9$ edges.

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No: two of the vertices would have three edges while three vertices would have four edges

You are describing a triangular bipyramid

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That is the wrong picture of a triangular bipyramid. OP is interested in a bipyramid whose faces are equilateral; that picture shows a more general bipyramid with irregular faces. upload.wikimedia.org/wikipedia/commons/a/a5/… is rendered from a bad angle, but is the bipyramid that OP wants. –  MJD Feb 10 '13 at 5:01
@MJD - you may be correct, but this one shows the topological point clearly enough –  Henry Feb 10 '13 at 17:54