Is a solid from 32 congruent regular triangles a Platonic solid? Imagine a ball (a globe) divided by three circles, rectangular to each over. If one smooth the surface between the circles one get 8 triangles and this is the platonic solid octahedron. Going back to the globe it seems possible to divide each of the eight curved surfaces into four congruent triangles, again using circles for this. If one smooth the surfaces again, we get 8 * 4 triangles and the solid satisfies the definition of Platonic solids.

More precisely: If $v$ is one of the vertices you get when dividing the sphere into 8 equal parts, $v$ will be adjacent to exactly 4 other vertices (its "degree" is 4). Dividing the triangular regions into 4 smaller triangular regions will not change the degrees of these vertices.

When you add the 4 smaller triangles, you are adding 12 new vertices. Each of these vertices is adjacent to two of the original vertices, and 4 new vertices, so they have degree 6.

Not all the vertices have the same degree, so it's not a platonic solid. $~\square$