# Frank Gould

Adjunct Professor of Mathematics (Retired)
Math Department, Central Connecticut State University
New Britain, Connecticut, USA

My wife, Louise, has drawn me into her explorations of polyhedra and polytopes over the years. This piece more or less forced itself on me because of a misstep in putting together a model of a classic Petrie-Coxeter skew-regular infinite polyhedron. It was striking in that it appeared to be a portion of a semi-regular (rather than regular) space-filling polyhedron, but with dodecahedral symmetry, which is not possible. Another striking aspect was that the surface appeared to separate space into a union of soccer balls on one side and a union of truncated tetrahedra on the other side. So, what was it? The one section of the polyhedron had its own compelling beauty and seemed worthy of a careful rendering for a math-art display.

Impossible Polyhedron?
12 x 16 x 16 cm
card stock
1917

A portion of a polyhedron with 4 hexagons at each vertex that appears to extend naturally into 3-space, but with dodecahedral symmetry. The gaps shown in pink along the outer boundary of the model betray the fact that the surface cannot be accommodated in 3-space without distortion.

The polyhedron exists inside a polytope in Euclidean 4-space as a finite surface bounded by 120 truncated icosahedra on one side and 600 truncated tetrahedra on the other side. The surface has 2400 hexagonal faces, 7200 edges and 3600 vertices. If one fills in the triangular and pentagonal holes, then one obtains the semi-regular "bi-truncated 120-cell". This object consists of 720 3-cells, 4320 faces, 7200 edges and 3600 vertices.