his study in symmetry, an origami tessellation, was folded by Chris Palmer, a San Francisco native who now lives in Maryland. When lit from behind, different parts of the design are lighter or darker depending on the number of layers of paper the light has to pass through. When lit from the front, the internal structure becomes invisible, and the model looks completely different. Palmer takes his inspiration from the mosaic designs of Arab mosques, such as the famous Alhambra in Granada, Spain. In these patterns, called tilings or tessellations, a small number of shapes are repeated over and over to fill up a planar region. Most tessellations are not only mirror-symmetric, but have a variety of rotational symmetries. Turn this one around by ninety degrees, and it looks just the same as before. Palmer has developed a method for converting any tiling with regular polygons—triangles, squares, hexagons, and the like—into an origami pattern. In theory, the pattern can be repeated forever. In real life, an origamist is limited only by the size of his paper and his patience. As you can see, Palmer is very patient! Click here for a larger image of Palmer's tessellation.
 Volume 23, Number 2 Exploring Origami - Page 3 of 5 Flower Tower In tessellations, the same folding motif is repeated over and over at the same size. In his "flower tower," Palmer repeats a design in bigger and bigger scales. Many things in nature grow in the same way, spiraling out from the center in larger and larger layers—think of a seashell, an artichoke, or a sunflower. So perhaps it's no accident that this abstract design looks almost as if it were alive.

fter you've seen a few Shafer models, you learn to expect the unexpected. There's the Swiss army knife that opens and closes (and, Shafer promises, really can cut . . . a cobweb). The flyswatter that turns over to reveal a squashed man on the other side. The envelope that opens to reveal a pop-up dragon.

One of Shafer's signature pieces is the "flasher." The flasher hat below folds up instantly to a cylinder no bigger than a roll of toilet paper. In fact, some of Shafer's flashers fold up by themselves if you let go of them.

According to Shafer, he designed the first flasher with Christopher Palmer, after Palmer saw a rose by a legendary Japanese origamist, Toshikazu Kawasaki. The two realized they could use Kawasaki's system of pleats to make a model that could grow or shrink. Shafer estimates that he has designed about thirty-five different flashers, and he even owes his college degree to them. As a student at the University of California at Santa Cruz, he majored in mathematics because "it was the most origami-like of the majors," and he presented a senior seminar on flashers. Even his professors were baffled, Shafer recalls. "I taught them how to fold a flasher," Shafer says. "I was practically having to fold it for them, because they had the instructions in front of them but they couldn't follow them. That was quite a turnaround, you know—you have all those geniuses, and they can't even fold the basic flasher!"

Click here for a larger image of Shafer's Flasher Hat.

 Watch as Jeremy Shafer displays a large example of a flasher. See how it expands dramatically, but also shrinks down to a very compact size

 One of Jeremy Shafer's flashers, transformed into a large and colorful flasher hat.
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