{"id":1689,"date":"2023-06-01T00:01:41","date_gmt":"2023-06-01T04:01:41","guid":{"rendered":"https:\/\/mathvoices.ams.org\/featurecolumn\/?p=1689"},"modified":"2024-07-15T12:07:47","modified_gmt":"2024-07-15T16:07:47","slug":"hat-tricks","status":"publish","type":"post","link":"https:\/\/mathvoices.ams.org\/featurecolumn\/2023\/06\/01\/hat-tricks\/","title":{"rendered":"Hat Tricks"},"content":{"rendered":"

Up until recently, all known aperiodic tilings used a minimum of two shapes, and it has long been a major problem to find a tiling that uses just one. Quite recently this problem has been solved, although only with a qualification…<\/em><\/span><\/p>\n

Hat Tricks<\/h1>\n

Bill Casselman
\nUniversity of British Columbia<\/b><\/p>\n

\"paper <\/div>\n<\/p>\n\n
“But however we analyze the difference between the regular and the irregular, we must ultimately be able to account for the basic fact of aesthetic experience, the fact that delight lies somewhere between boredom and confusion.”<\/em> <\/p>\n

E. H. Gombrich<\/a>, in The sense of order<\/b><\/a> (at the top of page 9)<\/p>\n<\/td>\n<\/table>\n

\nBreaking News.<\/em> Just as this column goes to posting, the authors of the original
\nconstruction described below have announced a new aperiodic tiling that does not use
\nreflected tiles.<\/font><\/b> See Kaplan’s new web page<\/a> or the new arxiv submission<\/a> for more information.<\/font>\n<\/p>\n

A tiling<\/b> of the plane by a set of two-dimensional shapes is a partition of the entire plane by congruent copies of the shapes, without overlaps or gaps. One common example is the tiling by regular hexagons: <\/p>\n

\"a <\/div>\n<\/p>\n

Not all shapes can make a tiling. For example, the only regular polygons that tile the plane are triangles, squares, and hexagons. Regular octagons require squares to fill in, making another common pattern. <\/p>\n

\"a <\/div>\n<\/p>\n

These examples are periodic<\/b>—they are invariant under a full discrete set of translations. More interesting from a mathematical point of view are aperiodic<\/b> tilings, which are not invariant under any translations at all. Best known among these are Penrose tilings<\/b>, in which pentagonal shapes play a role: <\/p>\n

\"a <\/div>\n<\/p>\n

Penrose tilings use two shapes, a thick and a thin rhombus. <\/p>\n

\"a \"a <\/div>\n<\/p>\n

Up until recently, all known aperiodic tilings used a minimum of two shapes, and it has long been a major problem to find a tiling that uses just one. Quite recently this problem has been solved, although only with a qualification: <\/p>\n

\"aperiodic <\/div>\n<\/p>\n

The single shape that makes this tiling resembles a hat. It is in some sense a rather elementary shape, since it fits nicely into the standard hexagonal tiling of the plane: <\/p>\n

\"a <\/div>\n<\/p>\n

This construction is due to the four authors David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss. They have posted a long article<\/a> about it on the arXiv<\/tt>, and it will presumably be eventually published in a professional journal. When it first appeared, it got a lot of media attention in places such as the Guardian<\/em> and the New York Times<\/em>. One thing that might have caused this attention was that the tiling was nicknamed einstein<\/em>. This has nothing to do with the famous physicist. The word “ein” is German for “one”, and “Stein” is German for “stone”, incorporated for example into the word “Spielstein”, a gaming token. <\/p>\n

However, as I have said, there is a qualification to the assertion that it uses just a single tile. That is true, but both sides of that tile are used. In the following image, the mirrored tiles are coloured: <\/p>\n

\"hat <\/div>\n<\/p>\n

In fact, the mirrored tiles play an important role in understanding the structure of this tiling. So although it is a remarkable and valuable construction, it still leaves open the problem of finding an aperiodic tiling by a single shape. <\/p>\n

What I want to do in the rest of this essay is just to give some idea of the peculiar features of the new hat tiling. As opposed to answering questions such as: How was this construction found? Are there variants? Why, exactly, is it aperiodic?<\/em> These are all quite difficult, and at least partially answered in the arxiv<\/tt> paper. But even carrying out my limited intention is not a simple task. In doing this, I shall be basically expanding slightly on the exposition of Smith et al.<\/p>\n

The structure of the tiling<\/h2>\n<\/p>\n

Let’s look more carefully at the tiling to see if we can discern some kind of pattern. One thing that becomes quickly evident is that the neighbourhoods of the mirrored hats are all similar. The most important observation of this kind is that three unmirrored hats in this neighbourhood always form congruent triads: <\/p>\n

\"a <\/div>\n<\/p>\n

I’ll call these groups of four hats H-clusters<\/b>. <\/p>\n

There are now a couple of things that come to mind. One is the frequent occurrence of singleton hats inside triplets of H-clusters. Another is that near corners of the H-clusters are groups of three hats that are invariant under $120^\\circ$ rotation: <\/p>\n

\"a <\/div>\n<\/p>\n

These extend to larger groups of six hats, still invariant under $120^\\circ$ rotation: <\/p>\n

\"a <\/div>\n<\/p>\n

What now remains unmarked is fairly simple. The unmarked tiles are either isolated hats or a couple of hats sandwiched between two H-clusters. The remarkable conclusion is that the hats in the tiling can be grouped into one of four types of clusters, and we therefore arrive at a new tiling by these four types: <\/p>\n

\"tiling <\/div>\n<\/p>\n

Each of these clusters can be associated to a simpler figure, called a metatile<\/b>. They are named, apparently arbitrarily, T<\/b>, H<\/b>, P<\/b>, and F<\/b>. <\/p>\n

\"a \"a <\/div>\n<\/p>\n

\"a \"a <\/div>\n<\/p>\n

Three of these shapes are invariant under some rotation, while the underlying groups of hats are not. So their orientations must be kept track of, here by arrows. <\/p>\n

The tiling by clusters gives rise to a tiling by metatiles! <\/p>\n

\"a <\/div>\n<\/p>\n

This transition from hats to metatiles is the main step to the explicit construction of a tiling of the plane, although it is not at all obvious how this goes.<\/p>\n

Inflation and deflation<\/h2>\n<\/p>\n

The metatiles can themselves be grouped into supertiles<\/b> … <\/p>\n

<\/div>\n<\/p>\n

… which can themselves be clustered into larger supertiles … <\/p>\n

\"a <\/div>\n<\/p>\n

… and so on to infinity. <\/p>\n

The point is that if one is given just one really large supertile one can construct all of the supertiles it contains (its descendants<\/em>), and then repeat this process until one reaches the bottom layer of metatiles, from which one can lay down the hats they are associated to. In this way one can construct arbitrarily large patches of hats in a consistent manner, and some very general reasoning assures you that a covering of the whole plane exists, even though of course you cannot construct it. At first it might seem extraordinary that Smith et al. came up with this scheme, but in fact it is a variation on a well known technique in the construction of aperiodic tilings—for example the Penrose tilings. Stepping up from one layer of hat\/meta\/super tiles to one dominating it is called inflation<\/b>, and stepping down to one it dominates is called deflation<\/b>. The version implemented here has one entirely new and noteworthy feature, though—the tiles in different layers are not similar shapes, but deformed a bit from descendants and parents. <\/p>\n

The parameters of a supertiling<\/h2>\n<\/p>\n

So now we ask: Given a tiling by metatiles, how do you construct the tiling by supertiles that dominates it? More generally, how to step from one tiling by supertiles to the one dominating it? To the one it dominates?<\/em> <\/p>\n

All of the supertilings that occur in this business have the same general appearance—what Smith et al. call a fixed combinatorial configuration. They are characterized by a small number of parameters: lengths $a$, $x$, $y$, a vector $z_{0}$ (along with rotations), and an angle $e$. <\/p>\n

<\/div>\n<\/p>\n