Tiling Hamiltonian Cycles on the 24-Cell

We present a construction for tiling the 24-cell with congruent copies of a single Hamiltonian cycle, using the algebra of quaternions. http://dx.doi.org/10.4169/amer.math.monthly.119.10.872 MSC: Primary 00A08, Secondary 05C45; 51M20 872 c © THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 119 This content downloaded from 195.187.72.155 on Wed, 19 Feb 2014 05:45:28 AM All use subject to JSTOR Terms and Conditions 1. AROUND THE WORLD. In modern terms, a Hamiltonian cycle is a path on a graph, which passes through every vertex exactly once before returning to its starting point. Hamilton’s name is attached to the idea because of his “Icosian Game”, a puzzle that amounted to finding such cycles among the vertices and edges of a dodecahedron [2]. Anyone with a few spare minutes can verify that the graphs of the five Platonic solids all contain Hamiltonian cycles. The octahedron, however, is unique among the five in one interesting respect. The cycle shown in Figure 1 uses exactly half of the edges in the graph, and can be rotated 90 degrees to give a congruent copy of the cycle that covers the other half of the edges. Thus, the edges of the octahedron can be covered by disjoint, geometrically congruent copies of a single Hamiltonian cycle; briefly, the octahedron can be tiled by one of its Hamiltonian cycles. None of the other Platonic solids can be tiled in this way. This is simply a matter of divisibility; for such a tiling to exist, the number of edges must be divisible by the number of vertices. In this note, we show how to construct a tiling Hamiltonian cycle on the edges of the 24-cell, one of the six regular convex polytopes in dimension four [5]. Finding such a cycle is within the realm of brute-force computation [6], so the interest here is in the construction, which is algebraic and can be verified by hand. We begin with an overview of the 24-cell’s structure and the particular features that make our construction possible. Figure 1. Hamiltonian cycle that tiles the edges of the octahedron 2. STRUCTURE OF THE 24-CELL. Combinatorially, the 24-cell consists of 24 vertices, with 8 edges meeting at each vertex for a total of 96 edges. It can be constructed by gluing the triangular faces of 24 octahedra together in pairs. Concretely, we can situate the 24-cell in R4 with 16 vertices at (±1/2,±1/2,±1/2,±1/2) and additional vertices at all 8 permutations of (±1, 0, 0, 0). Vertices are connected by an edge if they are unit distance apart. These are the vertices of a 4-dimensional hypercube together with the vertices of its dual “cross polytope”, and it is peculiar to 4-dimensional geometry that taking the cube and its dual together in this way results in a regular polytope [4]. Besides the simplicity of the coordinates, this particular embedding of the 24-cell has algebraic significance. If we identify the vertex at (a, b, c, d) with the quaternion a + bi+ cj+ dk, then the vertices form a nonabelian group under quaternion multiplication, which we will call G. This 24-element group is known as the binary tetrahedral group [3]. December 2012] NOTES 873 This content downloaded from 195.187.72.155 on Wed, 19 Feb 2014 05:45:28 AM All use subject to JSTOR Terms and Conditions Adjacency of vertices can be characterized algebraically. Define g1 = (1+ i+ j− k)/2, g2 = (1+ i− j+ k)/2, g3 = (1− i+ j+ k)/2,