What polyhedron has all equilateral triangles as faces?

Answer 1

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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).