My work is motivated by a fascination with the occurrence of mathematical and scientific imagery in traditional art forms, and the frequently mystical or cosmological significance that can be attributed to such imagery. Mathematical themes both subtle and overt appear in a broad range of traditional art, from Medieval illuminated manuscripts to Buddhist mandalas, intricate tilings in Islamic architecture to restrained temple geometry paintings in Japan, complex patterns in African textiles to geometric ornament in archaic Greek ceramics. Often this imagery is deeply connected with how these cultures interpret and relate to the cosmos, in much the same way that modern scientific diagrams express a scientific worldview.
Here, another isohedral tiling of the genus 3 Klein quartic is rendered, and projected as a 14-gon in the Poincaré disk. The edge identifications of the 14-gon which make this into a surface of genus 3 are drawn in explicitly, as interlaced loops connecting corresponding edges.
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): . 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.
The Klein Quartic is a genus 3 surface whose orientation-preserving automorphism group is isomorphic to PSL(2,7), the projective special linear group of 2x2 matrices over the field of 7 elements. This is the second smallest non-Abelian simple group, and appears in many interesting contexts. In this image, the Klein Quartic is given a hyperbolic metric and projected into the Poincaré disk as a hyperbolic 14-gon with edges identified to give a surface of genus 3. An isohedral tiling of the surface is shown such that each of the 168 tiles can be put into one-to-one correspondence with the elements of PSL(2,7), with an arbitrary choice of an "identity" tile. The tiles are colored according to the conjugacy class of the corresponding element.