Two LEDs are soldered together in anti-parallel and facing each other, by bending one lead on each by 180º. This unit forms one edge of the polyhedron.
The free leads each form part of a vertex. By building the appropriate number of edge segments and soldering them together, a 3-dimensional polyhedron is created.
The microcontroller is suspended in the center of the shape by attaching each vertex of the polyhedron to one IO pin on the μC using a fine strand of wire.
The following schematic shows the arrangement of the LEDs and the IO pin assignments for the LED tetrahedron.
I have only built a tetrahedron (the smallest platonic solid) so far. If you want to try a larger one, here is an overview of what you need:
The idea of building this came from JSK during 31C3 in Hamburg, and I set out to build it on the very same day. One night of soldering and programming later, and the tetrahedron was up and blinking :-)