The Penrose kite and dart are a pair of tiles that can be used to create aperiodic tilings of the plane. The kite is a quadrilateral whose four interior angles are 72°, 72°, 72°, and 144°. The dart is a non-convex quadrilateral whose four interior angles are 36°, 72°, 36°, and 216°.
We can fit the kite into the dart to form a rhombus. But if we alternately color the vertices of these tiles black and white in the correct way, and require that vertices of tiles that touch must have matching colors, we can forbid the tiles from forming a rhombus. We can still tile the plane while respecting this rule—but any such tiling must be aperiodic! A tiling is aperiodic if it doesn’t repeat, and it doesn’t contain arbitrarily large periodic patches.
In fact, there are uncountably many ways to tile the plane with kites and darts while respecting the ‘matching rule’ I described… but any finite-sized patch of one such tiling can be found somewhere in any other such tiling!
This image illustrates a ‘pattern-equivariant 1-chain’, a tool used by James J. Walton to study the topology of the kite and dart tiling, and other aperiodic tilings. In email, Walton gave an excellent explanation of how this tool works:
I’ll take the perspective that we are already motivated to study aperiodic tilings (and there are plenty of motivations!). One may then ask the question of what is a good approach to studying them; an approach which may help in some sort of classification, or which will allow computations of certain invariants of tilings (and perhaps even give us clues as to what these invariants should be). There is a time-honoured approach in maths which is to turn a problem into a geometric one. Ok, well, tilings are already rather geometric, but what I me