The n-Enclosing Voderberg Tile's Cavity Neighborhoods
Discipline: Mathematics and Statistics
Subcategory: Mathematics and Statistics
Gabriel Lopez - California State University San Bernardino
Co-Author(s): Jadie Adams, Westminster College, Salt Lake City, Utah; Nhi Tran, University of Washington, Bothell, Bothell, WA
H. Voderberg constructed a tile with the property that a pair of these tiles could enclose one or two other copies of itself. This tile’s design can be extended to a general form which has the property that any number of copies can be enclosed within just two tiles. This property is known as the n-enclosing property, where n is the number of tiles enclosed. Grunbaum and Shephard pose the question: Does there exists a tile with the n-enclosing for each n >= 3 that admits a monohedral tiling? We settle this question by describing how any general form of the Voderberg tile can be used to construct a periodic, as well as a nonperiodic, monohedral tiling. Furthermore, we use one such tiling to disprove another conjecture posed by Grunbaum and Shephard: the neighborhood of any given tile in a monohedral tiling will be equivalent to the patch generated by that tile. We demonstrate that double spiral tilings generated by Voderberg tiles with the m-enclosing property (where m >= 3) provide counterexamples. Such tilings contain tiles with cavity neighborhoods (neighborhoods that are not simply connected), so the patches they generate are not equivalent to their neighborhoods.
Not SubmittedFunder Acknowledgement(s): We would like to thank the NSF for funding our REU program, grant number 1460699, along with the University of Washington, Bothell, for their hospitality.
Faculty Advisor: Casey Mann, cemann@uw.edu
Role: I aided my teammates in constructing the relevant tilings, generating the graphics used, and making our presentations.