The Way The World Fits Together (Tiling 1)

Jueves 16 de abril, 1pm
Auditorio del IIMAS

Abstract: We are surrounded by an amazing variety of  geometric repeating patterns, on our clothes, floors, buildings -- where there is a surface, people will decorate it. To a mathematician,  though, there are only a few ways patterns are put together. In this presentation we'll learn how the Magic Theorem sorts out the possible symmetry types, using only a simple arithmetic count. We'll apply the Magic Theorem to recognize the mathematical type of repeating patterns -- and then we'll explore new methods of creating patterns using paper-and-scissors topology.

Matching Rules For Rep-tiles (Tiling 2)

Jueves 23 de abril, 4:30pm
Salón de Seminarios Graciela Salicrup
Instituto de Matemáticas

Four copies of this L-shape can be fitted together to form a larger L-shape; four of those can be fitted into a larger L still, and so on ad infinitum, in the end producing a non-periodic, hierarchical tiling of the plane. But this L-shape can form lots of other kinds of patterns too — how can we enforce this hierarchical structure? Today there are a few dozen specific examples (most famously the Penrose tiles and the Hat monotile) and a series of general constructions (Mozes '89, GS '98, Fernique-Ollinger '10), but these are not widely understood. We aim to demystify these methods with a simple to state and easy to prove lemma that applies to each of these constructions  (pushing the hard work into showing the hypotheses of the lemma hold in whatever setting we are considering).