Exploring the Penrose Tile

This mathematical poster explores the Penrose tiling, a famous example of an aperiodic tiling constructed from just two tiles: the Penrose Kite and Dart. It introduces the history and mathematical ideas behind aperiodic tilings, before examining the geometric structure and matching conditions that allow the Kite and Dart to cover the plane without ever forming a repeating pattern.

Using inflation and decomposition arguments, the poster demonstrates both that the Penrose tiles can tile the entire plane and that any resulting tiling must be non-periodic. It also explores some interesting real-world connections, including quasicrystals and the unusual use of Penrose tilings in the design of toilet paper.