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

No. By definition a Platonic solid has the same number of faces meeting at each vertex. Your solid has three faces meeting at each of two vertices and four faces meeting at each of the other three vertices.

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You can also work with the angles: Along any of the edges where the tetrahedrons are spliced together, there will be a planar angle of $120^\circ$, whereas the planar angles on each tetrahedron are $60^\circ$. Since these angles differ, the solid is not regular.

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