Sixth Platonic solid # 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.

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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