Theo Schaad

Statement

Since the last Bridges Conference, I have been on a quest to find a quasiperiodic tiling with local 10-fold rotational symmetry using the substitution method. Two rhombic 10-fold tilings were discovered by Robert Ingalls in 1992 (Decagonal Quasicrystal Tilings, Acta Cryst. A48) using a pentagrid. The grid lines were spaced in a quasiperiodic Fibonacci series. It is known that if the dual grid is quasiperiodic, the tiling is also quasiperiodic. I learned from personal correspondence that Ingalls knew of a substitution method with an inflation factor of $\tau^3$ (where $\tau=\frac{1+\sqrt{5}}{2}$ is the golden ratio), but it is not elaborated in his paper. I created a puzzle to try out different ways to tile 5- and 10-fold patterns.

Artworks

5- & 10-fold Rosettes are two pattern puzzles. Two different rhombic tiles are color coded on one side and decorated on the other. Many different rosettes can be made as coffee-table decorations, or as recreational or educational pursuit. Illustrations are provided as a starting point. Box #1 contains 8 different tiles with 8 color codes. Box #2 contains 4 different tiles with 4 color codes. The substitution rules are complicated, but worthy for a mathematical conference. If the decorated tiles of Box #1 (or #2) are laid out along the colors of the Box #2 (or #1) decorations, new inflated tiles are created with an inflation factor of $\tau^3$, which are the substitution schemes of Robert Ingalls.