Pursuing punctured polyhedra

The flat faces and straight edges of a cube signify that it belongs to a large family of geometric shapes known as polyhedra. Several years ago, mathematician John H. Conway of Princeton University wondered whether a polyhedron could have a hole passing through every face and remain a polyhedron. He coined the term holyhedron to describe such a form, should it exist.

An infinite lattice of interpenetrating tetrahedrons doesn’t quite meet the mathematical specifications for an object called a holyhedron. Vinson

For a long time, no one could come up with an example, even in principle, that met Conway’s precise specifications for a holyhedron. Conversely, no one could say why it was impossible to construct one. Now, Princeton mathematician Jade P. Vinson has proved that such an object can exist. His surprising solution is slated for publication in Discrete and Computational Geometry.

One way to approach the problem is to consider what happens when a vertex of one polyhedron pierces the face of another. A tetrahedron, for example, has four triangular faces and four vertices. It’s possible to construct an infinite lattice of interpenetrating tetrahedra, where each face of each tetrahedron is pierced by the vertex of another. In effect, each face has a hole where a vertex punctures it.

Conway, however, was looking for a finite structure rather than an infinite array. To solve the problem, Vinson opted to use polyhedra that have more vertices than faces. “The first key idea . . . was to find a simple, repetitive arrangement of polyhedra so that there is a large excess of unused vertices over unpierced faces,” he says. “The second key idea was to ‘trade’ several unused vertices in an inconvenient location for a single new vertex in a better location.”

Vinson’s careful manipulations produced a monstrous holyhedron with 78,585,627 faces. “The current construction is hard to visualize,” he admits. Simple cardboard models give just the roughest idea of how it all fits together.

Conway had offered a reward of $10,000—divided by the number of faces—for finding a holyhedron, so Vinson’s initial effort netted him a minuscule return. Conway suspects that someone may yet find a holyhedron with fewer than 100 faces.

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