#### Presentation Title

Algorithm to Compute the Subdivision Matrix for Tilings and Its Application to Girih Tiles

#### Faculty Mentor

Prof. Mohamad A. Alwash

#### Start Date

23-11-2019 9:15 AM

#### End Date

23-11-2019 9:30 AM

#### Location

Markstein 208

#### Session

oral 1

#### Type of Presentation

Oral Talk

#### Subject Area

physical_mathematical_sciences

#### Abstract

Ancient Islamic architecture provides us with abundant examples of beautiful designs that feature highly complex geometric patterns that require a very high level knowledge of geometry from their architects and artisans. To produce with these designs, the ancient masters combined shapes taken from a set of five Girih tiles. Amazingly, many of their designs featured complex non-periodic patterns that were not studied in the West until the 1970's. Since then, researchers had shown that there are many other sets of tiles that can produce non-periodic tilings.

Our work describes a procedure that can be used to look for possible tiling compositions and their subdivision matrices given a set of tiles. To accomplish this, our procedure first gathers the areas, diameters, and edge lengths of all tiles in the set. Then, a scaling factor based on a combination of these lengths is chosen. The procedure then looks for the right number of tiles that satisfy several area constraints. Using this procedure on the Girih tiles, we found many more feasible compositions for subdividing the Girih tiles beyond the one and only reported in the literature. All these compositions can be converted to each other by breaking up the decagon tile into one bowtie and three hexagons and vice versa. Among the compositions is one without any decagon. This shows that only two tiles from the Girih set of five are needed to form a tiling.

Finally, we studied the effect of using different compositions on the non-periodicity property of Girih tiles. Non-periodicity is shown by first calculating the eigenvalues of the subdivision matrix and then showing that some of these eigenvalues are irrational numbers. We repeated this procedure using different subdivision matrices that were obtained from the alternative compositions we found earlier and found that the irrational eigenvalues remain the same.

Algorithm to Compute the Subdivision Matrix for Tilings and Its Application to Girih Tiles

Markstein 208

Ancient Islamic architecture provides us with abundant examples of beautiful designs that feature highly complex geometric patterns that require a very high level knowledge of geometry from their architects and artisans. To produce with these designs, the ancient masters combined shapes taken from a set of five Girih tiles. Amazingly, many of their designs featured complex non-periodic patterns that were not studied in the West until the 1970's. Since then, researchers had shown that there are many other sets of tiles that can produce non-periodic tilings.

Our work describes a procedure that can be used to look for possible tiling compositions and their subdivision matrices given a set of tiles. To accomplish this, our procedure first gathers the areas, diameters, and edge lengths of all tiles in the set. Then, a scaling factor based on a combination of these lengths is chosen. The procedure then looks for the right number of tiles that satisfy several area constraints. Using this procedure on the Girih tiles, we found many more feasible compositions for subdividing the Girih tiles beyond the one and only reported in the literature. All these compositions can be converted to each other by breaking up the decagon tile into one bowtie and three hexagons and vice versa. Among the compositions is one without any decagon. This shows that only two tiles from the Girih set of five are needed to form a tiling.

Finally, we studied the effect of using different compositions on the non-periodicity property of Girih tiles. Non-periodicity is shown by first calculating the eigenvalues of the subdivision matrix and then showing that some of these eigenvalues are irrational numbers. We repeated this procedure using different subdivision matrices that were obtained from the alternative compositions we found earlier and found that the irrational eigenvalues remain the same.