Apollonian Gasket

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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

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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. In Ja-

An Apollonian gasket is built up through successive “generations.” For instance, in generation 1 (top left), each of the red circles is inscribed in one of the four triangular pores formed by the black circles. The final gasket shown here, whimsically named “bugeye” by Katherine Sanden, an under- graduate student of Peter Sarnak at Princeton University, has circles with bends –1 (for the largest circle that encloses the rest), 2, 2 and 3. The list of bends that appears in a given gasket (here, 2, 3, 6, 11, etc.) form a number sequence whose properties Sarnak would like to explain—but, he says, “the necessary mathematics has not been invented yet.” (Im- age courtesy of Katherine Sanden.)

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pan, during the Edo period, a delight- the screensaver. If the first two circles Four circles to the kissing come ful tradition arose of posting beauti- have bends 1 and 1, then the circle be- The smaller are the benter. ful mathematics problems on tablets tween them will have bend 4, because The bend is just the inverse of – – – that were hung in Buddhist or Shinto √1 +√1=√4. The next circle will have The distance from the center. – – – temples, perhaps as an offering to the bend 9, because √1 +√4=√9. Needless Though their intrigue left gods. One of these “Japanese temple to say, the pattern continues forever. Euclid dumb, problems,” or sangaku, is to find the (This also explains what the numbers There’s now no need for rule radius of a circle that just touches two in the first figure mean. Each circle is of thumb. circles and a line, which are themselves labeled with its own bend.) Since zero bend’s a dead mutually tangent. This is a restricted Apollonian circles experienced per- straight line, version of the Apollonian problem, haps their most glorious rediscovery And concave bends have where one circle has infinite radius (or in 1936, when the Nobel laureate (in minus sign, zero bend). The anonymous author chemistry, not mathematics) Frederick The sum of the squares of all – – – shows that, in this case, √a +√b=√c, Soddy became mesmerized by their four bends a sort of demented version of the Py- charm. He published in Nature a poetic Is half the square of their sum. thagorean theorem. This formula, by version of Descartes’ theorem, which the way, explains the pattern I saw in he called “The Kiss Precise”: Soddy went on to state a version for three-dimensional spheres (which he was also not the first to discover) in the final stanza of his poem. Ever since Soddy’s prosodic effort, it has become something of a tradition to publish any extension of his theorem in poetic form as well. The following year, Thorold Gosset published an n- dimensional version, also in Nature. In 2002, when Wilks, Mallows and Jeff Lagarias published a long article in the American Mathematical Monthly, they ended it with a continuation of Soddy’s poem entitled “The Complex Kiss Precise”: Yet more is true: if all four discs Are sited in the complex plane, Then centers over radii Obey the self-same rule again. (The authors note that the poem is to be pronounced in the Queen’s English.)

A Little Bit of Gasketry To this point I have only written about the very beginning of the gasket- making process—how to inscribe one circle among three given circles. How- ever, the most interesting phenomena show up when you look at the gasket as a whole. The first thing to notice is the foam- like structure that remains after you cut out all of the discs in the gasket. Clearly the disks themselves take up an area that approaches 100 percent of the area within the outer disk, and so the area of the foam (known as the “residual set”) must be zero. On the other hand, the foam also has infinite length. Thus, in fact, it was one of the first known examples of a fractal—a In 1643 René Descartes gave a simple formula relating the radii of any four mutually tangent cir- curve of dimension between 1 and 2. δ cles. More than 350 years later, Allan Wilks and Colin Mallows noticed that the same formula re- Even today its dimension (denoted ) lates the coordinates of the centers of the circles (expressed as complex numbers). Here Descartes’s is not known exactly; the best-proven formula is used to find the radius and center of the fourth circle in the “bugeye” packing. estimate is 1.30568.

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The concept of fractional dimension was popularized by Benoît Mandelbrot in his enormously influential book The Fractal Geometry of Nature. Although the meaning of dimension 1.30568 is somewhat opaque, this number is re- lated to other properties of the foam that have direct physical meaning. For instance, if you pick any cutoff radius r, how many bubbles in the foam have radius larger than r? The answer, denoted N(r), is roughly proportional to Physicists study random Apollonian packings as a model for foams or powders. In these simula- δ tions, new bubbles or grains nucleate in a random place and grow, either with rotation or without, r . Or if you pick the n largest bubbles, until they encounter another bubble or grain. Different geometries for the bubbles or grains, and what is the remaining pore space be- different growth rules, lead to different values for the dimension of the residual set—a way of tween those bubbles? The answer is measuring the efficiency of the packing. (Image courtesy of Stefan Hutzler and Gary Delaney.) roughly proportional to n1–2/δ. Physicists are very familiar with this an existing bubble, and then stop. Or first generation. The other has bend sort of rule, which is called a power law. a tree in a forest may grow until its d'=38, as predicted by the formula, As I read the literature on Apollonian canopy touches another tree, and then 2+38=2(2+3+15). More importantly, packings, an interesting cultural differ- stop. In this case, the new circles do even if we did not know d', we would ence emerged between physicists and not touch three circles at a time, but still be guaranteed that it was an inte- mathematicians. In the physics litera- only one. Computer simulations show ger, because a, b, c and d are. ture, a fractional dimension δ is de facto that these “random Apollonian pack- Hidden behind this “baby Descartes equivalent to a power law rδ. However, ings” still behave like a fractal, but equation” is an important fact about mathematicians look at things through with a different dimension. The em- Apollonian gaskets: They have a very a sharper lens, and they realize that pirically observed dimension is 1.56. high degree of symmetry. Circles a, b and there can be additional, slowly increas- (This means the residual set is larger, c actually form a sort of curved mirror ing or slowly decreasing terms. For and the packing is less efficient, than that reflects circle d to circle d' and vice instance, N(r) could be proportional in a deterministic Apollonian gasket.) versa. Thus the whole gasket is like a ka- to rδlog(r) or rδ/log(r). For physicists, More recently, Stefan Hutzler of Trin- leidoscopic image of the first four circles, who study foams empirically (or semi- ity College Dublin, along with Gary reflected again and again through an empirically, via computer simulation), Delaney and Tomaso Aste of the Uni- infinite collection of curved mirrors. the logarithm terms are absolutely un- versity of Canberra, studied the effect Kontorovich and Oh exploited this detectable. The discrepancy they intro- of bubbles with different shapes in a symmetry in an extraordinary and duce will always be swamped by the random Apollonian packing. They amusing way to prove their estimate of noise in any simulation. But for mathe- found, for example, that squares be- the function N(r). Remember that N(r) maticians, who deal in logical rigor, the come much more efficient packers than simply counts how many circles in the logarithm terms are where most of the circles if they are allowed to rotate as gasket have radius larger than r. Kon- action is. In 2008, mathematicians Alex they grow, but surprisingly, triangles torovich and Oh modified the function Kontorovich and Hee Oh of Brown become only slightly more efficient. N(r) by introducing an extra variable of University showed that there are in As far as I know, all of these results are position—roughly equivalent to put- fact no logarithm terms in N(r). The begging for a theoretical explanation. number of circles of radius greater than For mathematicians, however, the r obeys a strict power law, N(r)∼Crδ, classical, deterministic Apollonian gas- where C is a constant that depends on ket still offers more than enough chal- the first three circles of the packing. For lenging problems. Perhaps the most the “bugeye” packing illustrated in the astounding fact about the Apollonian second figure, C is about 0.201. (The gasket is that if the first four circles tilde (∼) means that this is not an equa- have integer bends, then every other cir- tion but an estimate that becomes more cle in the packing does too. If you are and more accurate as the radius r de- given the first three circles of an Apol- creases to 0.) For mathematicians, this lonian gasket, the bend of the fourth is was a major advance. For physicists, found (as explained above) by solving the likely reaction would be, “Didn’t a quadratic equation. However, every we know that already?” subsequent bend can be found by solving a linear equation: Random Packing For many physical problems, the classi- d+d'=2(a+b+c) cal definition of the Apollonian gasket For instance, in the “bugeye” gas- A favorite example of Sarnak’s is the “coins” is too restrictive, and a random model ket, the three circles with bends a=2, gasket, so called because three of the four may be more appropriate. A bubble b=3, and c=15 are mutually tangent generating circles are in proportion to the siz- may start growing in a randomly cho- to two other circles. One of them, with es of a quarter, nickel and dime, respectively. sen location and expand until it hits bend d =2, is already given in the (Image courtesy of Alex Kontorovich.)

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enough information from the spectral decomposition. If there were too many, then previously known methods, such as the ones Wiles used, would already answer all your questions. Because Apollonian gaskets fall right in the middle, they generate a host of unsolved number-theoretic problems. For example, which numbers actually appear as bends in a given gasket? These numbers must satisfy certain “congruence restrictions.” For example, in the bugeye gasket, the only legal bends have a remainder of 2, 3, 6 or 11 when divided by 12. So far, it seems that every number that satisfies this congruence restriction does indeed appear in the figure somewhere. (The reader may find it amusing to hunt for 2, 3, 6, 11, 14, 15, 18, 23, etc.) “Computation indicates that every number occurs, but we can’t prove that even 1 percent of them actually occur!” says Ron Graham of the University of Cali- fornia at San Diego. For other Apollo- nian gaskets, such as the “coins” gasket in the fifth figure, there are some absentees—numbers that obey the congruence restrictions but don’t appear in the gasket. Sarnak believes, however, that the number of absentees is always finite, and beyond a certain point any number Many variations on the Apollonian gasket construction are possible. In this beautiful ex- that obeys the congruence restrictions ample, each pore is occupied by three inscribed circles rather than by one. Light blue arcs rep- does appear somewhere in the gasket. At resent five “curved mirrors.” Reflections in these curved mirrors—known technically as circle this point, though, he is far from proving inversions—create a kaleidoscopic effect. Every circle in the gasket is generated by repeated this conjecture—the necessary math just inversions of the first six circles through these curved mirrors. (Image courtesy of Jos Leys.) doesn’t exist yet. And even if all the problems con- ting a lightbulb at a point x and asking function was all that Kontorovich and cerning the classic Apollonian gaskets how many circles illuminated by that Oh needed to figure out what happens are solved, there are still gaskets ga- lightbulb have radius larger than r. The to N(r) as r approaches 0. lore for mathematicians to work on. count will fluctuate, depending on ex- In this way, a simple problem in ge- As mentioned before, they could study actly where the bulb is placed. But it ometry connects up with some of the random Apollonian gaskets. Another fluctuates in a very predictable way. For most fundamental concepts of mod- modification is the gasket shown in the instance, the count is unchanged if you ern mathematics. Functions that have last figure, where each pore is filled by move the bulb to the location of any of a kaleidoscopic set of symmetries are three circles instead of one. Mallows its kaleidoscopic reflections. rare and wonderful. Kontorovich calls and Gerhard Guettler have shown that This property makes the “lightbulb them “the Holy Grail of number theo- such gaskets behave similarly to the counting function” a very special kind ry.” Such functions were, for instance, original Apollonian gaskets—if the of function, one which is invariant under used by Andrew Wiles in his proof of first six bends are integers, then all the the same symmetries as the Apollonian Fermat’s Last Theorem. An interesting rest of the bends are as well. Ambi- gasket itself. It can be broken down into new kaleidoscope is enough to keep tious readers might want to work out a spectrum of similarly symmetric func- mathematicians happy for years. the “Descartes formula” and the “baby tions, just as a sound wave can be decom- Descartes formula” for these configura- posed into a fundamental frequency and Gaskets Galore tions, and investigate whether there are a series of overtones. From this spectrum, Kontorovich learned about the Apol- congruence restrictions on the bends. you can in theory find out everything lonian kaleidoscope from his mentor, Perhaps you, too, will be inspired to you want to know about the lightbulb Peter Sarnak of Princeton University, write a poem or paint a tablet in honor of counting function, including its value at who learned about it from Lagarias, Apollonius’ ingenious legagy. “For me, any particular location of the lightbulb. who learned about it from Wilks and what’s attractive about Apollonian gas- For a musical instrument, the funda- Mallows. For Sarnak, the Apollonian kets is that even my 14-year-old daugh- mental frequency or lowest overtone gasket is wonderful because it has nei- ter finds them interesting,” says Sarnak. is the most important one. Similarly, ther too few nor too many mirrors. If “It’s truly a god-given problem—or per- it turned out that the first symmetric there were too few, you would not get haps a Greek-given problem.”

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