Additions to

Zome Geometry

George W. Hart   and  Henri Picciotto

The Expanded 120-Cell

Here is a rather large 4D polytope project for brave Zomers with plenty of time and parts.  Visualize the 120-cell, then just separate adjacent pairs of dodecahedra with a pentagonal prism.  As three dodecahedra surround each edge of the 120-cell, the expanded 120-cell will have three pentagonal prisms surrounding a triangular prism in the corresponding places.  The bases of the triangular prisms come together in groups of four, making regular tetrahedra in the places corresponding to the 120-cell's vertices.

Every vertex of this expanded 120-cell is identical---the meeting place of one dodecahedron, three pentagonal prisms, three triangular prisms, and a tetrahedron.  So it is a uniform polytope.  There are two types of edges, those that are dodecahedron edges and those that are tetrahedron edges.  Both types of prisms contain both types of edges.  Every pentagon is the junction of a dodecahedron and a pentagonal prism; every square is the junction of a pentagonal prism and a triangular prism; every triangle is the junction of a triangular prism and a tetrahedron.  The cells immediately surrounding each dodecahedron combine to form a rhombicosidodecahedron, so the structure can also be seen as 120 intersecting rhombicosidodecahedra.  Another way to derive it is by expanding the 600-cell.

The mathematical structure was first discovered by Alicia Boole Stott, and described in her 1910 paper on "semiregular polytopes".  This model was constructed and photographed by Mira Bernstein and Vin de Silva with the help of a crew of eight students at Stanford.  They counted that it requires 1260 balls, 960 b2, 1200 y2, 720 r2, and 720 r1.  I don't know what they were eating, but it took them three evenings to complete.