The cube twists all over. And the dodecahedron squashes fairly flat. You may have to make cross-braces to make these rigid. The cube is made up of squares, which are equilateral quadrilaterals. An equilateral triangle cannot be distorted.
All equilateral triangles have the same shape.
Models of regular solids make nice Christmas tree ornaments. These would have regular polygons as faces. But the vertices may vary. We can simply make a chain of octahedra, end to end, of any length we want. So we can make infinitely many different polyhedra this way, each of which has equilateral triangles as faces.