From Colorings to Weavings

Abstract

We present a methodology for constructing weavings from 2-colorings of the plane. In particular, we consider tilings T of the plane by triangles and their corresponding triangle groups G. We derive 2-colorings of T using the index 2 subgroups of G.

Equivalent weavings

“…In [2], we introduced a concept of black and white patches of a colored tiling T H . Intuitively, a (black) B-patch of a colored tiling is composed of edge-adjacent black tiles of the same color all having a unique point in common.…”

“…The vertex sets V B and the V W are the set of centroids of the B-patches and W -patches of T H respectively. In Figure 2 Given an index 2 subgroup H = h 1 , h 2 , ..., h l of G, we can determine a constructible fundamental region H = ∪ g with respect to the given the generating set of H. This is elaborated in [2]. We call the points where the edges of the nets N B and N W intersect, the weaving points of the overlapping net and denote the set of all weaving points in O H by I O H .…”

“…If ω H (I O H ) = {⊕, }, then W H is a proper weaving. This method is outlined in Figure 3 and discussed in detail in [2]. Since ω H 2 (I ) = {⊕}, W H 2 is not a proper weaving.…”

“…When this scenario occurs, consider an index 2 subgroup N i of H, identify the motif N i of N i in T (6, 4, 2) and assign values to the weaving points in N i . This method of overcoming a No Weaving Scenario is discussed in detail in [2].…”

“…We begin with a summary of the method for constructing weavings of overlapping nets from 2-colorings of the plane and a definition of weavings as presented by Miro, Garciano and Zambrano in [2]. We then present a definition of combinatorial and weaving equivalence and give some examples of equivalent weavings from colorings of the plane.…”