This paper makes a new observation about arbitrary dimensional Wang Tilings,
demonstrating that any d -dimensional tile set that can tile periodically along d − 1 axes must be able to tile periodically along all axes.
This work also summarizes work on Wang Tiles up to the present day, including
definitions for various aspects of Wang Tilings such as periodicity and the validity of a tiling. Additionally, we extend the familiar 2D definitions for Wang Tiles and associated properties into arbitrary dimensional spaces. While there has been previous discussion of arbitrary dimensional Wang Tiles in other works, it has been largely informal. This work formalizes and proves several key assertions that previous work has referenced as folklore, including the fact that periodicity of a tiling is captured by a lattice.

Author Bio

Ian Tassin is an undergraduate student at Oregon State University. His undergraduate research, with the mentorship of Dr. Huck Bennett, focused on Wang Tiles. He plans to pursue a Ph.D. in Artificial Intelligence after graduation. In particular he is interested in genetic algorithms and use of Reinforcement Learning (RL) in game-like environments. Also inspired by his love of games Ian created and authored a series of tabletop roleplaying games called Apotheosis in his free time during his undergraduate career.