Conan Chadbourne

Intricate Elaboration of Order
archival inkjet print
2015

The icosahedral group (A₅) 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 A₅ by two generators of order 5 (i.e. elements which fix a face and a vertex): <a,b | a⁵=b⁵=(ab)²=(a^-1b)³=1>. The Cayley diagram for this presentation of A₅ 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 A₅.