With these tiles and rules, we can continue forever in both directions, so we can tile the line. Now, on either side of these B’s, we must put A’s, and so on. Suppose we put an A down first.Īccording to the rules, we must put B’s on either side. Next to a B, on either side you can only place an A.Ĭan we tile the line with these tiles and these rules? Absolutely.Next to an A, on either side you can only place a B.If the tiles and the rules we adopt for placing them allow us to create a string of letters that goes on infinitely in both directions, we can “tile the line.” For example, let’s say we have two tiles, A and B, and two rules for putting them together: Let’s imagine our line-filling tiles to be letters that stick together to form sequences.
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To get a handle on this big discovery, let’s start by thinking about a simpler problem: how to tile a line. It’s this kind of beauty that a retired print technician was seeking when he recently discovered the first “aperiodic monotile”- a single tile that fills up the plane in a non-repeating pattern. But mathematicians also find beauty in irregularity. In math, tilings are often appreciated for their regular patterns. Two-dimensional tilings are admired all around the world, both for their beauty - as seen in the artistry of mosaics in cathedrals and mosques around the world - and for their utility, in walls and floors everywhere. Have you ever admired how the slats of a hardwood floor fit together so cleanly, or how the hexagons underneath your bathroom rug perfectly meet up? These are examples of geometric tilings, arrangements of shapes that fit snugly together while filling up space.
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