Three Color “2 : 1 : 1” Designs

Darrah Chavey
Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture (2014)
Pages 341–344 Short Papers


We analyze an extension of symmetric colored patterns where there are three colors, one of which occurs twice as much as the other two, but where we still have full symmetry between the colors: There is at least one symmetry that fixes the larger color while interchanging the two smaller colors, and at least one symmetry that interchanges the larger color with the union of the two smaller colors. Such “2:1:1” designs occur in traditional patterns in central, frieze, and wallpaper designs. Here we classify the 33 possible types of 2:1:1 frieze designs, and the 4 types of 2:1:1 central designs, we provide constructed examples of all of them, and describing the ways in which these designs arise in the arts of various cultures. An associated web page gives photos of many examples.