American Scientist

To the Editors:

In the November–December issue’s Computing Science column, “ The Science of Sticky Spheres ,” Brian Hayes points out that five tetrahedra can almost, but not quite, meet around an edge (see figure below). If you try to make this happen with balls and sticks of equal length, the outer edges are not quite long enough to close the loop—and if you try to do it with five perfect tetrahedra, there is an angular gap between them. Similarly, if you have 20 perfect tetrahedra, they can almost meet around a corner. Their outer faces almost form an icosahedron, but with narrow gaps between them.

I was fascinated by this phenomenon in high school. Having been raised on Martin Gardner’s columns, I realized I was in the same position as a two-dimensional inhabitant of Flatland who tries to make five triangles meet at a corner. All we need is a little curvature to close the gap: In three dimensions, we can make a polyhedron where five triangles meet at each corner, namely an icosahedron.

Similarly, in four dimensions there is a polytope where five tetrahedra meet at every edge, and 20 meet at every corner. This gorgeous creature has 120 corners, 720 edges, 1,200 triangular faces and 600 tetrahedral cells. Along the same lines, four dodecahedra can almost meet at each corner; curving them up in the fourth dimension to make them fit together leads to the dual of the previous polytope, which has 600 corners and 120 dodecahedral cells.

Just as there are five Platonic solids in three dimensions, there are six in four dimensions, and these are two of them. In five or more dimensions, the only Platonic polytopes are the analogs of the cube, the octahedron and the tetrahedron.

Cristopher Moore
Santa Fe Institute, New Mexico