III. Quadrilateral Prototiles
All tilings listed here are edge to edge. At this time, I'm aware of two distinct families of such tilings with quadrilateral prototiles.
The tilings are classified as described in the Introduction. The figures available for each tiling are listed as links.
A. This family is divided into two branches, each of which contains three distinct tilings. The two branches differ in how the successive generations of tiles fit together.
A1. In this branch, each prototile is completely convex.
- (p4, r6, g4, s.577, m2)
- (p4, r8, g5, s.414, m2)
- (p4, r12, g7, s.268, m2)
A2. In this branch, the two short edges form a concave "side" for each prototile.
- (p4, r6, g5, s.577, m2)
- (p4, r8, g6, s.414, m2)
- (p4, r12, g8, s.268, m2)
B. There are an infinite number of basic tilings in this family. The first three are shown here. The prototiles for this family can be derived by taking segments of regular polygons containing three adjacent edges of the particular polygon. Polygons with number of sides 6, 10, 14, 18 ... all yield edge-to-edge tilings.
- (p4, r2, g3, s.5, m2)
- (p4, r2, g3, s.382, m2)
- (p4, r2, g3, s.357, m2)
The following paper, downloadable as a pdf file, describes this last family of tilings as well as the family of tilings described in the chapter on triangular prototiles.
Self-similar Tilings Based on Prototiles Constructed from Segments of Regular Polygons,
presented at the Bridges Conference (July 28-30, 2000, Southwestern College, Winfield, Kansas). (File size - 660K.)
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