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A polyhedron with equilateral triangular faces is called a
deltahedron. There are infinitely many of them but only 8 convex
ones. The following three are regular:
tetrahedron, octahedron and icosahedron.
There are 5 more convex polyhedra which are convex but not
regular (because they have differing number of faces meeting at
their vertices) These are Johnson deltahedra (see link).