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 mean is to consider tilings, our objects of interest, as being points of some ‘moduli space of tilings’. So if we associate to our tilings ‘spaces of associated tilings’, then some sort of classification of these spaces, or computations of topological invariants of them, may aid us in classifying our tilings and computing invariants for tilings.
Here’s a little more detail on this space. We consider two tilings as being ‘close’ if they agree to a large radius about the origin ‘up to a small perturbation’ (note that a tiling is considered as a decomposition of Euclidean space, so a tiling and its translate are generally not ‘equal’ here). So one defines a tiling metric on tilings (although I also like to use uniformities for this). Then, given a tiling, one may consider all of its translates (or rigid motions): this gives a collection of tilings which may be considered as a metric space with the above metric. One takes the completion of this space to obtain the tiling space of our tiling (sometimes called the translational hull when we take translates of tilings or rigid hull for rigid motions of tilings). Although we’ve taken a completion, (at least in ‘nice’ cases) we may still view the points of this space as themselves tilings. They may be considered as those tilings which are ‘locally isomorphic’ to our original tiling, that is, tilings which share the same finite patches to it (up to translation/rigid motion). It’s nice to try and justify this given what I said about how the tiling metric is defined.
For so called finite local complexity, repetitive, aperiodic tilings (such as the Penrose kite and dart), one may think of the tiling space as an infinite bunch of Euclidean spaces which are screwed up together into a compact space. It will be a fibre bundle with base space a $d$-torus (of the same dimension $d$ as the original tiling) and fibre the Cantor set. Its topology is complicated: it will be connected but not path connected or locally connected.
It turns out that ‘classical’ topological invariants are pretty useless for these spaces. In degree zero the singular (co)homology and homotopy groups are uncountably generated (which just says that this space has uncountably many path components) and in higher degrees are trivial (this is related to the fact that there are no identifications on these ‘Euclidean leaves’). However, it turns out that Čech cohomology and K-theory seem to give quite rich information, and in a range of examples are actually computable.
The Čech cohomology is isomorphic to the ‘pattern-equivariant cohomology’, originally pioneered by Kellendonk and Putnam, e.g. here:
• Johannes Kellendonk, Pattern equivariant functions and cohomology.
One considers cellular cochains on the tiling which ‘respect’ the structure of the tiling, that is, cochains for which there exists some radius such that whenever two cells see the same patch of tiles about them to this radius, up to translation, then they are assigned the same coefficient. With the usual cellular coboundary map this gives you a cochain complex which has isomorphic cohomology groups to the Čech cohomology of the translational hull.
The pattern-equivariant (or PE, for short) homology groups defined in my paper are a simple adaptation of the PE cohomology groups but where one takes the boundary maps instead of the coboundary maps. Pattern-equivariant chains aren’t examples of cellular chains in the usual sense: since they are non-compactly supported, these chains are sometimes known as Borel–Moore chains. For non-compact manifolds one has a Poincaré duality isomorphism between the cohomology and Borel–Moore homology of the manifold. The same is true here for the PE (co)homology: for the translational hull of a tiling we have an isomorphism between the PE groups (with degree shift based on the dimension of the tiling) and hence between the PE homology and Čech cohomology of the tiling space.
Now (finally!) to the picture. The red lines illustrate a PE 1-chain on the tiling. If you imagine drawing in the tile boundaries, the chain trails the bottoms of the dart tiles of the Penrose kite and dart tiling. Of course, there should be orientations there but I’ve removed them to decrease clutter. So, for example, we could choose for the 1-chain about that central star to be going anticlockwise so that the chain always points to the right when the dart points upwards.
By being PE (with respect to translations) these chains should be such that there exists some radius for which the chain is equal at any two cells where the patches agree there to this radius, up to a translation. This is clearly true here, in fact, more is true, we could take any rigid motion. The translational and rigid hull have natural actions by rotations on them (remember that the points of each may be considered as tilings). It turns out, by another Poincaré duality like result, that the PE homology with “respect to rigid motions” in degree one will correspond (for a 2d tiling) to the degree one Čech cohomology of the quotient of the translational/rigid hull by rotations. The degree one Čech group for this quotient space in the case of the Penrose kite and dart is isomorphic to the integers and this red chain may be thought of as a generator for this group.
So, what are the green lines? Well, the Penrose kite and dart is given by a tiling substitution which gives it a hierarchical structure. We may find a tiling of ‘supertiles’, inflated kite and dart tiles, which sits on top of our original tiling and decomposes to it under the substitution rule. Trailing the ‘super-dart’ tiles gives the green lines. This is also a PE 1-chain as described above. The picture nicely shows how it is homologous, although with the opposite orientation, to the original red chain, through the boundary of the blue PE 2-chain defined by the dart tiles. Opposed to just being a nice observation, this is actually involved in a method for the computation of this group for substitution tilings, in passing from one ‘approximant homology’ to the next (so in this case we get isomorphisms between the ‘approximant homologies’).
Something that I quite like about this picture is that, although its main job is just to illustrate this PE 1-chain, it actually gives you a clearer picture of some of the properties of the Penrose tiling itself. In a very specific sense, passing to the picture defined by this 1-chain loses no information: there is a rule to reconstruct a Penrose tiling from the pattern of red lines. In tiling theory, we would say that this picture (extended infinitely, of course) and the original tiling are ‘mutually locally derivable’ (MLD). But I think that the hierarchical nature of the Penrose is actually more clear in this picture than the original tiling. You can see how different motifs of the tiling are grouped together, and you could probably even guess how to extend this picture to a larger patch, taking another five copies of this entire motif and placing them about this one (and perhaps deleting a few line segments here and there). There is a choice in how this is done for ‘each stage up the hierarchy’; this is related to the fact that there are uncountably many Penrose kite and dart tilings up to translation, despite all of them being ‘locally isomorphic’. This hierarchical nature forces aperiodicity: any non-zero translation preserving the tiling would need to preserve this hierarchy, but this is impossible since it would then need to preserve motifs of arbitrarily large size.
I apologise for the discussion of the background to this picture outweighing the discussion of the picture itself. The truth is that we almost certainly still don’t know everything that there is to know about what these invariants tell us about tilings. There are already enough interesting connections, though, to be sure that these sorts of techniques are a good approach; for example, the K-theory of the tiling space contains information about gaps in the spectrum of the Hamiltonian of a particle moving in a solid modeled by the tiling. Part of the motivation for defining these PE homology groups was to give another angle of attack on what these topological invariants describe. At least personally, I can relate to these geometric chains a lot more easily than to rather abstract Čech cochains on a rather abstract space of tilings. The hope is that they will inspire more interpretations of these groups.
For more details, read his paper:
• James J. Walton, Pattern-equivariant homology.