Tiling

Group

Presentation

Cayley diagram

Regular Tilings 
Square

ℤ × ℤ

r b
< r,b  rbr^{1}b^{1} >



r b
< r,b  rbr^{1}b >



r b
< r,b  rbrb >



r b
< r,b  r^{4}, b^{4}, rbrb >


two invalid Cayley diagrams deleted here 
Triangular


r g b
< r,g,b  r^{3}, g^{3}, b^{3}, rgb >


Hexagonal


k r
< k,r  k^{6}, r^{2}, (kr)^{3} >



r g b
< r,g,b  r^{2}, g^{2}, b^{2}, (rg)^{3}, (gb)^{3}, (br)^{3} >


Truncated Tilings 
Truncated squares (octagons and squares)

C4 × ℤ × ℤ

k r
< k,r  k^{4}, r^{2}, (kr)^{4} >


Truncated hexagons (dodecagons and triangles)


k r
< k,r  k^{3}, r^{2}, (kr)^{6} >

