Cayley Diagrams drawn on Plane Tilings

This page gives the Cayley diagrams of a very few infinite groups. Their presentations are also given. Above each presentation is a short list of elements in bold type: the colours used in the list correspond to the colours of the lines in the diagram.

You can click on any of the diagrams to see a larger region of it.

Tiling Group Presentation Cayley diagram

Regular Tilings

Square ℤ × ℤ r b
< r,b | rbr-1b-1 >
  r b
< r,b | rbr-1b >
  r b
< r,b | rbrb >
  r b
< r,b | r4, b4, rbrb >
two invalid Cayley diagrams deleted here
Triangular   r g b
< r,g,b | r3, g3, b3, rgb >
Hexagonal   k r
< k,r | k6, r2, (kr)3 >
  r g b
< r,g,b | r2, g2, b2, (rg)3, (gb)3, (br)3 >

Truncated Tilings

Truncated squares (octagons and squares) C4 × ℤ × ℤ k r
< k,r | k4, r2, (kr)4 >
Truncated hexagons (dodecagons and triangles)   k r
< k,r | k3, r2, (kr)6 >

Some more Cayley diagrams
and other pages on groups
Copyright N.S.Wedd 2007