A American Scientist Previous Page|||| Contents Zoom in Zoom out Front Cover | Search Issue | Next Page BEF MaGS Computing Science A Tisket, a Tasket, an Apollonian Gasket Dana Mackenzie n the spring of 2007 I had the it onto T-shirts. And in a book about I good fortune to spend a semester Fractals made fractals with the lovely title Indra’s at the Mathematical Sciences Research Pearls, mathematician David Wright Institute in Berkeley, an institution of compared the gasket to Dr. Seuss’s The higher learning that takes “higher” to of circles do Cat in the Hat: a whole new extreme. Perched precari- The cat takes off his hat to reveal ously on a ridge far above the Univer- funny things to Little Cat A, who then removes sity of California at Berkeley campus, his hat and releases Little Cat B, the building offers postcard-perfect mathematicians who then uncovers Little Cat C, vistas of the San Francisco Bay, 1,200 and so on. Now imagine there are feet below. That’s on the west side. not one but three cats inside each Rather sensibly, the institute assigned cat’s hat. That gives a good im- me an office on the east side, with a sequence was 1-squared, 2-squared, pression of the explosive prolifer- view of nothing much but my com- 3-squared, and so on. ation of these tiny ideal triangles. puter screen. Otherwise I might not Before I became a full-time writer, have gotten any work done. I used to be a mathematician. Seeing However, there was one flaw in the those numbers awakened the math geek Getting the Bends plan: Someone installed a screen-saver in me. What did they mean? And what Even the first step of drawing an Apol- program on the computer. Of course, it did they have to do with the fractal on lonian gasket is far from straightfor- had to be mathematical. The program the screen? Quickly, before the screen- ward. Given three circles, how do you drew an endless assortment of fractals saver image vanished into the ether, I draw a fourth circle that is exactly tan- of varying shapes and ingenuity. Every sketched it on my notepad, making a gent to all three? couple minutes the screen would go resolution to find out someday. Apparently the first mathematician blank and refresh itself with a complete- As it turned out, the picture on the to seriously consider this question was ly different fractal. I have to confess that screen was a special case of a more gen- Apollonius of Perga, a Greek geom- I spent a few idle minutes watching the eral construction. Start with three circles eter who lived in the third century b.c. fractals instead of writing. of any size, with each one touching the He has been somewhat overshadowed One day, a new design popped up other two. Draw a new circle that fits by his predecessor Euclid, in part be- on the screen (see the first figure). It was snugly into the space between them, cause most of his books have been lost. different from all the other fractals. It and another around the outside enclos- However, Apollonius’s surviving book was made up of simple shapes—cir- ing all the circles. Now you have four Conic Sections was the first to system- cles, in fact—and unlike all the other roughly triangular spaces between the atically study ellipses, hyperbolas and screen-savers, it had numbers! My circles. In each of those spaces, draw a parabolas—curves that have remained attention was immediately drawn new circle that just touches each side. central to mathematics ever since. to the sequence of numbers running This creates 12 triangular pores; insert One of Apollonius’s lost manu- along the bottom edge: 1, 4, 9, 16 … a new circle into each one of them, just scripts was called Tangencies. Accord- They were the perfect squares! The touching each side. Keep on going for- ing to later commentators, Apollonius ever, or at least until the circles become apparently solved the problem of too small to see. The resulting foam-like drawing circles that are simultane- Dana Mackenzie received his doctorate in math- structure is called an Apollonian gasket ously tangent to three lines, or two ematics from Princeton University in 1983 and (see the second figure). lines and a circle, or two circles and a taught at Duke University and Kenyon College. Something about the Apollonian line, or three circles. The hardest case Since 1996 he has been a freelance writer special- gasket makes ordinary, sensible math- of all was the case where the three izing in math and science, and he has frequently ematicians get a little bit giddy. It circles are tangent. edited articles on mathematical topics for Ameri- inspired a Nobel laureate to write a No one knows, of course, what can Scientist. His published books include The Big Splat, or How Our Moon Came to Be (Wiley, poem and publish it in the journal Na- Apollonius’ solution was, or whether it 2003), and volumes 6 and 7 of What’s Happening ture. An 18th-century Japanese samu- was correct. After many of the writings in the Mathematical Sciences (American Math- rai painted a similar picture on a tablet of the ancient Greeks became available ematical Society, 2007 and 2009). Email: ____scribe@ and hung it in front of a Buddhist tem- again to European scholars of the Re- danamackenzie.com ple. Researchers at AT&T Labs printed naissance, the unsolved “problem of 10 American Scientist, Volume 98 A American Scientist Previous Page|||| Contents Zoom in Zoom out Front Cover | Search Issue | Next Page BEF MaGS A American Scientist Previous Page|||| Contents Zoom in Zoom out Front Cover | Search Issue | Next Page BEF MaGS 81 64 49 36 25 81 49 64 25 49 81 81 49 25 64 49 81 25 36 49 64 81 81 64 49 36 25 81 49 64 25 49 81 81 49 25 64 49 81 25 36 49 64 81 Apollonius” became a great challenge. 84 16 76 76 16 84 84 16 76 76 16 84 52 9 9 52 52 9 9 52 81 81 81 81 In 1643, in a letter to Princess Elizabeth 72 28 28 72 72 28 28 72 57 57 57 57 of Bohemia, the French philosopher 96 4 96 96 4 96 64 64 64 64 and mathematician René Descartes cor- 33 97 97 33 33 97 97 33 rectly stated (but incorrectly proved) a 88 12 88 88 12 88 73 73 73 73 24 24 beautiful formula concerning the radii 40 40 60 60 of four mutually touching circles. If the 84 84 radii are r, s, t and u, then Descartes’s formula looks like this: 84 84 60 60 40 40 24 24 2 73 73 73 73 1⁄r2+ 1⁄s2+ 1⁄t2+ 1⁄u2= 1⁄2 (1⁄r+ 1⁄s+ 1⁄t+ 1⁄u) . 88 12 88 88 12 88 33 97 97 33 33 97 97 33 All of these reciprocals look a little1 64 64 1 64 64 1 96 96 96 96 bit extravagant, so the formula is usu- 57 57 57 57 72 28 28 72 72 28 28 72 ally simplified by writing it in terms 81 81 81 81 52 4 52 52 4 52 76 76 76 76 84 9 9 84 84 9 9 84 of the curvatures or the bends of the 25 16 25 25 16 25 25 16 25 25 16 25 81 64 49 36 81 49 64 49 81 81 49 64 49 81 36 49 64 81 81 64 49 36 81 49 64 49 81 81 49 64 49 81 36 49 64 81 circles. The curvature is simply de- fined as the reciprocal of the radius. Numbers in an Apollonian gasket correspond to the curvatures or “bends” of the circles, with Thus, if the curvatures are denoted by larger bends corresponding to smaller circles. The entire gasket is determined by the first four a, b, c and d, then Descartes’s formula mutually tangent circles; in this case, two circles with bend 1 and two “circles” with bend 0 reads as follows: (and therefore infinite radius). The circles with a bend of zero look, of course, like straight lines. (Image courtesy of Alex Kontorovich.) a2+b2+c2+d2=(a+b+c+d)2/2. As the third figure shows, Des- 1990s by Allan Wilks and Colin Mal- the circle centers as complex num- cartes’s formula greatly simplifies the lows of AT&T Labs, and Wilks used bers. Imaginary and complex num- task of finding the size of the fourth it to write a very efficient computer bers were not widely accepted by circle, assuming the sizes of the first program for plotting Apollonian gas- mathematicians until a century and a three are known. It is much less obvi- kets. One such plot went on his office half after Descartes died. ous that the very same equation can door and eventually got made into the In spite of its relative simplicity, be used to compute the location of the aforementioned T-shirt. Descartes’s formula has never become fourth circle as well, and thus com- Descartes himself could not have widely known, even among mathema- pletely solve the drawing problem. discovered this procedure, because it ticians. Thus, it has been rediscovered This fact was discovered in the late involves treating the coordinates of over and over through the years.