Here are models of the five Platonic solids made from the four smallest symmetrical tetrahedra (A, B, Icosa, and Dodeca Quanta) that will build them all:
tetrahedron - 24 A Quanta
hexahedron (cube) - 48 A Quanta + 24 B Quanta
octahedron - 48 A Quanta + 48 B Quanta
icosahedron - 120 Icosa Quanta
dodecahedron - 120 Dodeca Quanta
The rhombic dodecahedron can also be built with 96 A Quanta + 48 B Quanta, or two cubes divided into quarters turned inside out.
The A and B Quanta have the same volume. I have not determined whether the Icosa Quanta and Dodeca Quanta share that characteristic.
The scale used in these patterns is based upon the one right triangle which is common between all the Quanta, a triangle of 90º, 54º 44', and 35º 16'.
colored acetate and PVC
Update: Now with video.
Prepared these models for the Synergetics Symposium at RI School of Design over a year ago. These models all have magnets at the center of each face. Next edition, if I ever get the inclination to build it, should be wire framework models with ball magnets at each vertex, which I think is more realistic.
If you are interested in these kind of geometrics, you should be aware of the Synergetics Collaborative http://www.synergeticscollaborative.org/
Dancing the Cube in Jamaica
Six Right Triangles