Here are the step-by-step instructions for building a Zome model of a Pythagopod, which is one of the ways to nest the five Platonic Solids. The nesting order is, from inner to outer, octahedron, tetrahedron, hexahedron, dodecahedron, and icosahedron.
This configuration was described by architect Christopher Glass in a paper titled The Pythagopod in the Nexus Network Journal. That in turn was derived from a version described by Matila Ghyka in the book The Geometry of Art and Life.
For the Zome version, you need the following parts: 68 nodes, 12 B3, 24 HG3, 60 HB3, and 60 HB2 struts.
These instructions builds the "large" version, which uses half-blue and half-green struts. It is also possible to build a version using only full-length struts. For that, start with a 2B2 cube, then all the other half-lengths become full lengths that are one size smaller. For example, the HG3's become G2's, the HB2's become B1's, etc. The parts list for the full-length version is 80 nodes, 84 B2, 24 G2, 60 B1 struts.
|Step 1: Build a B3 cube with 12 B3 struts and 8 nodes.|
|Step 2: Build a 2HG3 tetrahedron inside the cube with 12 HG3 struts and 6 nodes.|
|Step 3: Build a 2HG3 octahedron inside the tetrahedron with 12 HG3 struts.|
|Step 4: Build a HB2 roof over one face of the cube with 10 HB2 struts and 7 nodes.|
|Step 5: Build a 2HB2 dodecahedron around the cube by building a step 4 roof for each remaining face of the cube, using 50 HB2 struts and 35 nodes.|
|Step 6: Build a HB3 pyramid with 5 HB3 struts and 1 node.|
|Step 7: Build a 2HB3 icosahedron that intersects the dodecahedron at the edge midpoints by building a step 6 pyramid for each remaing face of the dodecahedron, using 55 HB3 struts and 11 nodes.
The geometrically astute will notice that this is not a true nesting because the dodecahedron and icosahedron intersect each other, rather than touching at a vertex, edge or face. Note that it is also possible in Zome to build the icosahedron as the innermost solid, inside the octahedron. That nesting of the Platonic solids is called a "cosmogram" by mathematician John Conway.