Intricate Elaboration of Order

archival inkjet print

2015

The icosahedral group (A_{5}) is isomorphic to the group of orientation-preserving symmetries of the Great Dodecahedron, a self-intersecting polyhedron having 12 pentagonal faces, 5 of which meet at each vertex. This genus 4 polyhedron naturally suggests a presentation of A_{5} by two generators of order 5 (i.e. elements which fix a face and a vertex): <a,b | a^{5}=b^{5}=(ab)^{2}=(a^{-1}b)^{3}=1>. The Cayley diagram for this presentation of A_{5} cannot be embedded in a plane, but can be associated with the edge graph of the rhombidodecadodecahedron, a genus 4 uniform star polyhedron with 12 pentagonal faces, 12 pentagrammic faces and 30 square faces. This image uses a projection of that polyhedron to render the Cayley graph for this presentation of A_{5}.