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Solve the Golden b puzzle!

At a JMM conference, mathematician Keith Cobenhaver called our attention to Robert Ammann’s Golden b tiling. It is unlike the tilings we see around us every day. Most tilings are periodic (see wallpaper groups). In a periodic tiling, you can find two points where the tiling looks identical. In fact, if there are two, there will be infinitely many such points.

Mathematicians discovered aperiodic tilings in the 1960s. Aperiodic tilings must be able to tessellate (i.e. fill in) the entire plane, and use only a fixed number of tile types. Most importantly, any tessellation with the tiles must be non-repeating in this sense: pick any two points in the tessellation, and the tiling must look different at those points.

Robert Ammann was an early researcher in aperiodic tilings. Remarkably, he was not a professional mathematician—he was a postal worker with a passion for discovering new tilings. One of the tilings he discovered is based on a single tile shape, in two different sizes. The ratio of the area of the larger tile to the smaller is the golden ratio, \(\phi \approx 1.618\). Note that the tile is shaped like a chunky letter b, hence the affectionate moniker “Golden b.”

If we take the shortest side of the smaller tile to be one unit long, then we get the other sides by successively multiplying by \(\sqrt{\phi}\). And we create the larger tile by scaling all side lengths by \(\sqrt{\phi}\).

The elliptical markings on the tiles enforce a matching rule. So, for example, you might combine the two tiles above to create this:

And behold! It is another, larger, Golden b. This suggests you can combine it with the larger tile to create a still larger Golden b.

As you build a tessellation with the Golden b, the ratio of the number of large to small tiles converges to—you guessed it, the golden mean.

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