In the heaven of the great god Indra is said to be a vast and shimmering net, finer than a spider’s web, stretching to the outermost reaches of space. Strung at the each intersection of its diaphanous threads is a reflecting pearl. Since the net is infinite in extent, the pearls are infinite in number. In the glistening surface of each pearl are reflected all the other pearls, even those in the furthest corners of the heavens. In each reflection, again are reflected all the infinitely many other pearls, so that by this process, reflections of reflections continue without end.

This is a book about serious mathematics, but one which we hope will be enjoyed by as wide an audience as possible. It is the story of our computer aided explorations of a family of unusually symmetrical shapes, which arise when two spiral motions of a very special kind are allowed to interact. These shapes display intricate ‘fractal’ complexity on every scale from very large to very small. Their visualisation forms part of a century-old dream conceived by the great German geometer Felix Klein. Sometimes the interaction of the two spiral motions is quite regular and harmonious, sometimes it is total disorder and sometimes and this is the most intriguing case - it has layer upon layer of structure teetering on the very brink of chaos.

As we progressed in our explorations, the pictures that our computer programs produced were so striking that we wanted to tell our tale in a manner which could be appreciated beyond the narrow confines of a small circle of specialists. You can get a foretaste of their variety by taking a look at the Road Map on the final page. Mathematicians often use the word ‘beautiful’ in talking about their proofs and ideas, but in this case our judgment has been confirmed by a number of unbiassed and definitely non-mathematical people. The visual beauty of the pictures is a veneer which covers a core of important and elegant mathematical ideas; it has been our aspiration to convey some of this inner aesthetics as well. There is no religion in our book but we were amazed at how our mathematical constructions echoed the ancient Buddhist metaphor of Indra’s net, spontaneously creating reflections within reflections, worlds without end.

Most mathematics is accessible, as it were, only by crawling through a long tunnel in which you laboriously build up your vocabulary and skills as you abstract your understanding of the world. The mathematics behind our pictures, though, turned out not to need too much in the way of preliminaries. So long as you can handle high school algebra with confidence, we hope everything we say is understandable. Indeed given time and patience, you should be able to make programs to create new pictures for yourself. And if not, then browsing through the figures alone should give a sense of our journey. Our dream is that this book will reveal to our readers that mathematics is not alien and remote but just a very human exploration of the patterns of the world, one which thrives on play and surprise and beauty.

What to do with the pictures? Two thoughts surfaced: the first was that they were unpublishable in the standard way. There were no theorems, only very suggestive pictures. They furnished convincing evidence for many conjectures and lures to further exploration, but theorems were the coin of the realm and the conventions of that day dictated that journals only publish theorems. The second thought was equally daunting: here was a piece of real mathematics that we could explain to our non-mathematical friends. This dangerous temptation prevailed, but it turned out to be much, much more difficult than we imagined.

We persevered off and on for a decade. One thing held us back: whenever we got together, it was so much more fun to produce more figures than to write what Dave W. named in his computer TheBook. I have fond memories of traipsing through sub-zero degree gales to the bunker-like supercomputer in Minneapolis to push our calculations still further. The one loyal believer in the project was our ever-faithful and patient editor, David Tranah. However, things finally took off when Caroline was recruited a bit more than a decade ago. It took a while to learn how to write together, not to mention spanning the gulfs between our three warring operating systems. But our publisher, our families and our friends told us in the end that enough was enough.

You know that ‘word problem’ you hated the most in elementary school? The one about ditch diggers. Ben digs a ditch in 4 hours, Ned in 5 and Ted in 6. How long do they take to dig it together? The textbook will tell you 1 hour, 37 minutes and 17 seconds. Baloney! We have uncovered incontrovertable evidence that the right answer is hours. This is a deep principle involving not merely mathematics but sociology, psychology, and economics. We have a remarkable proof of this but even Cambridge University Press’s generous margin allowance is too small to contain it.

Anyway, as a complete lark, I tagged along with David M. while he built a laboratory of computer programs to visualize Kleinian groups. It was a mathematical joy-ride. As it so happened, in the summer of 1980 , there was a great opportunity to share the results of these computer explorations with the world at the historic Bowdoin College conference in which Thurston presented his revolutionary results in three-dimensional topology and hyperbolic geometry. We arranged for a Tektronix terminal to be set up in Maine, and together with an acoustically coupled modem at the blazing speed of 300 bits per second displayed several limit sets. The reaction to the limit curves wiggling their way across the screen was very positive, and several mathematicians there also undertook the construction of various computer programs to study different aspects of Kleinian groups.

That left us with the task of writing an explanation of our algorithms and computations. However, at that point it was certainly past time for me to complete my thesis. Around 1981, I had the very good fortune of chatting with a new grad student at Harvard by the name of Curt McMullen who had intimate knowledge of the computer systems at the Thomas J. Watson Research Center of IBM, thanks to summer positions there. After roping Curt in, and at the invitation and encouragement of Benoit Mandelbrot, Curt and David M. made a set of extremely high quality and beautiful black-and-white graphics of limit sets. I would like to express my gratitude for Curt’s efforts of that time and his friendship over the years; he has had a deep influence on my own efforts on the project.

Unfortunately, as we moved on to new and separate institutions, with varying computing facilities, it was difficult to maintain the programs and energy to pursue this project. I would like to acknowledge the encouragement I received from many people including my friend Bill Goldman while we were at M.I.T., Peter Tatian and James Russell, who worked with me while they were undergraduates at M.I.T., Al Marden and the staff of the Geometry Center, Charles Matthews, who worked with me at Oklahoma State, and many other mathematicians in the Kleinian groups community. I would also like to thank Jim Cogdell and the Southwestern Bell Foundation for some financial support in the final stages. The serious and final beginning of this book took place when Caroline agreed to contribute her own substantial research work in this area and her expository gifts, and also step into the middle between the first and third authors to at least moderate their tendency to keep programming during our sporadic meetings to find the next cool picture. At last, we actually wrote some text.

We have witnessed a revolution in computing and graphics during the years of this project, and it has been difficult to keep pace. I would also like to thank the community of programmers around the world for creating such wonderful free software such as TEX, Gnu Emacs, X Windows and Linux, without which it would have been impossible to bring this project to its current end.

During the years of this project, the most momentous endings and beginnings of my life have happened, including the loss of my mother Elizabeth, my father William, and my grandmother and family’s matriarch Elizabeth, as well as the birth of my daughters Julie and Alexandra. I offer my part in these pictures and text in the hope of new beginnings for those who share our enjoyment of the human mind’s beautiful capacity to puzzle through things. Programming these ideas is both vexing and immensely fun. Every little twiddle brings something fascinating to think about. But for now I’ll end.

The first year was one of frustration, staring at pictures like the ones in Chapter 9 without being able to get any real handle on what was going on. Then one morning one of us woke up with an idea. We tried a few hand calculations and it seemed promising, so we asked Dave W. to draw us a picture of what we called the ‘real trace rays’. What came back was a rudimentary version of the last picture in this book the one we have called ‘The end of the rainbow’. For me it was more like ‘The beginning of the rainbow’, one of the defining moments of my mathematical life. Here we were, having made a total shot in the dark, having no idea what the rays could mean, but knowing they had absolutely no right to be arranged in such a nice way. It was obvious we had stumbled on something important, and from that moment, I was hooked.

For another year we struggled to fit the rays into the one mathematical straight-jacket we could think of, but it just didn’t quite work. One day, I ran into Curt McMullen and mentioned to him what we were playing with. ‘Real trace’, he pondered, ‘That’s the convex hull boundary’.’ And with that clue, we were off. What Curt had told us was that to understand the two dimensional pictures we had to look in threedimensional non-Euclidean space, real Thurston stuff, as you might say. Finally we were able to verify at least most of the two Dave’s conjectures theoretically.

When the 19th century mathematician Mary Somerville received a letter inviting her to make a translation, with commentary, of Laplace’s great book Mécanique Céléste, she was so surprised she almost returned the letter thinking there must have been some mistake. I suppose I wasn’t quite so surprised to get a letter from David M. asking me to help them write about their pictures, but it wasn’t quite an everyday occurrence either. Although I may perhaps write another book, I am unlikely ever again to have the chance to work on one which will be so much trouble and so much fun.

And don’t think this book is the end of the story. If you flick through you will see cartoons of a rather portly character gluing up pieces of rubber into things like doughnuts. In fact all our present tale revolves about ‘one-holed doughnuts with a puncture’. For the last few years, I have been trying to understand what happens when the doughnuts acquire more holes. The main thing I can report is - it’s a lot more complicated! But the same wonderful structures, yet more intricate and inviting, are out there waiting to be tamed.

I would like to thank the EPSRC for the generous support of a Senior Research Fellowship, which has recently allowed me to devote much time to both the mathematical and literary aspects of this challenging project.

This is a book which can be read on many levels. Like most mathematics books, it builds up in sequence, but the best way to read it may be skipping around, first skimming through to look at the pictures, then dipping in to the text to get the gist and finally a return to understand some of the details. We have tried to make the first part of each chapter relatively simple, giving the essence of the ideas and postponing the technicalities until later. The more technical parts of the discussion have been relegated to the Notes and can be skipped as desired. Material important for later reference is displayed in Boxes.

The first two chapters, on Euclidean symmetries and complex numbers respectively, contain material which may be partially familiar to many readers. We have aimed to present it in a form suited to our viewpoint, at the same time introducing as clearly as possible and with complementary graphics the mathematical terminology which will be used throughout the book. Chapter 3 introduces the basic double spiral maps, called Möbius symmetries, on which all of our later constructions rest. From then on, we build up ever more complicated ways in which a pair of Möbius maps can interact, generating more and more convoluted and intricate fractals, until in Chapters 10 and 11 we actually reach the frontiers of current research. The entire development is summarised in the Road Map on the final page.

Words which have a precise mathematical meaning are in bold face the first time they appear. We have not always spelled out the intricacies of the precise mathematical definition, but we have also tried not to say anything which is mathematically incorrect. We have used a small amount of our own terminology, but in so far as possible have stuck to standard usage. Non-professional readers will therefore have to forgive us such terms as quasifuchsian and modular group, while readers with a mathematical training should be able to follow what we mean.

The book is written as a guide to actually coding the algorithms which we have used to generate the figures. A vast set of further explorations is possible for those readers who invest the time to program. This is prime hacking country! Because we hope for a wide variety of readers with many different platforms at their disposal, we have sketched each step in ‘pseudo-code’, the universal programming pidgin.

Inevitably we have suppressed a good deal of relevant mathematics and anyone wishing to pursue these ideas seriously will doubtless sooner or later have to resort to more technical works. Actually there are no very accessible books about Kleinian group limit sets’, but there are plenty of texts which discuss the basics of symmetry and complex numbers. Some complex analysis books touch on Möbius maps and there is more in modern books on two-dimensional hyperbolic geometry. In the later part of the book we have cited a rather random collection of recent research papers which have important bearing on our work. These are absolutely not meant to be exhaustive, but should serve to help professional readers find their way round the literature.

Finally our Projects need some comment. They can be ignored: we aren’t going to grade them or supply answers! Rather, we intend them as ‘explorations’ to tempt you if you enjoy the material and want to take it further. Some are fairly straightforward extensions or elucidations of material in the text and some involve open-ended questions for which there is no definite answer. A few are definitely research problems. Others again explain details which are needed for full understanding or verification of the more technical points in our story. We have to leave it to the reader to pick and choose which ones suit their taste and mathematical experience.

We thank especially our cartoonist Larry Gonick for his uncanny ability to translate a complicated three-dimensional manipulation into an immediately evident cartoon. For historical background we are indebted to the St. Andrews History of Maths web site, tempered with many erudite details and healthy doses of scholarly scepticism from our friends David Fowler and Paddy Patterson. (All remaining errors, are, of course, our own.) Klein’s own book Entwicklung der Mathematikim 19. Jahrhundert has also been an important source. We have read the Hua-Yen Sutra in the translation The Flower Ornament Scripture by Thomas Cleary, Shambhala Publications, 1993, and quotations are reproduced here with thanks. We should like to thank the Mathematics Departments of Brown, Oklahoma State, Warwick, Harvard and Minnesota for their hospitality. We should like to thank the NSF through its grant to the Geometry Center and EPSRC from their Public Understanding of Science budget for financial support. Finally we should like to thank our publisher David Tranah of Cambridge University Press, without whose constant prodding and encouragement this book would almost certainly never have seen the light of day.

The lacy web in Figure 7.1 is called the Apollonian gasket. Usually, it is constructed by a simple geometric procedure, dating back to those most famous of geometers, the ancient Greeks. We shall start by explaining the traditional construction, but as we shall disclose shortly, the gasket also represents another remarkable way in which the Schottky dust can congeal. The pictures you see here were actually all drawn using a refinement of the DFS algorithm for tangent Schottky circles.

The starting point of the traditional construction is a chain of three non-overlapping disks, each tangent to both of the others. A region between three tangent disks is a ‘triangle’ with circular arcs for sides. This shape is often called an ideal triangle: the sides are tangent at each of the three vertices so the angle between them is zero degrees.

The gasket is activated by the fact that in the middle of each ideal triangle there is always a unique ‘inscribed disk’ or incircle, tangent to the three outer circles. It is really better to think of the gasket as a construction on the sphere. Insides and outsides don’t matter any more, so we may as well start with any three mutually tangent circles. You can see lots of disks and incircles in Figure 7.2.

In the figure, we show two initial configurations of three tangent blue disks. When you take out the three blue disks you are left with two red ideal triangles. Each red ideal triangle has a yellow incircle. See how each yellow incircle divides the red triangle into three more triangles.

For repetitive people (a necessary quality in this subject, you might say), it is only natural to draw the incircles in these new triangles, resulting, of course, in even more triangles of the same kind. The bottom frame shows this subdivision carried out twice more, with green and then even smaller purple disks. In The Cat in the Hat Comes Back,’ the cat takes off his hat to reveal Little Cat, who then removes his hat and releases Little Cat, who then uncovers Little Cat, and so on. Now imagine there are not one but three new cats inside each cat’s hat. That gives a good impression of the explosive proliferation of these tiny ideal triangles. Carry out this process to infinity, and Voom, the Apollonian Gasket appears.

The Apollonian gasket is indeed very pretty, but the reason for introducing it here is that, remarkably, it is also the limit set of a Schottky group made by pairing tangent circles. Exactly the same intricate mathematical object can created by completely different means! You can see better how this works in the beautiful glowing version in Figure 7.3. The solid red circles in this picture are the initial Schottky circles in a very special configuration which we will look at closely in the next section. The glowing yellow limit set can be recognized as the same as the Apollonian gasket of Figure 7.1. The picture was made by pairing four tangent circles arranged in the configuration shown in Figure 7.4. The four circles are tangent not only in a chain; there are also extra tangencies between \(C_a\) and \(C_A\), and between \(C_b\) and \(C_B\).

As you iterate the pairing transformations \(a\) and \(b\), the extra tangency proliferates, with the effect that inside each disk \(D\) you see three further Schottky disks tangent to \(D\) and each of the other two. In our version, the circles have been coloured depending on their level, starting with red at the first or lowest level, gradually changing to yellow, green and then blue. The small yellow and blue circles pile up, highlighting the limit set with a mysterious glow.

In this chapter, we shall be exploring various features of the gasket. Notwithstanding the extra tangency, it turns out that each limit point is still associated to exactly one or two infinite words in the generators \(a,b,A\) and \(B\). You will be able to make your own version of the glowing gasket by running our DFS algorithm for the group generated by the transformations \(a\) and \(b\). The algorithm draws this complicated lacework as a continuous curve, which is hard to imagine until you see it in progress on a computer screen. The curve snakes its way through the gasket, apparently leaving one region for quite a while until finally weaving its way back. Animation is the true reward of successfully implementing the program we have been learning to build.

The configuration of tangent circles which produced the gasket is shown in the right frame of Figure 7.4. The picture has been arranged so that \(C_a\) goes through \(\infty\), hence it appears in the figure as a straight line. In addition, \(C_A\) and \(C_a\) are tangent at 0 and \(C_B\) and \(C_b\) are tangent at \(-i\). You can see how this picture is made by creating extra tangencies among a kissing chain of four circles by comparing with the nearby arrangement of four circles in the left hand frame.

The generating matrices for the gasket are quite simple: \[a=\begin{pmatrix}1&0\\-2i&1\end{pmatrix}\quad\text{and}\quad b=\begin{pmatrix}1-i&1\\1&1+i\end{pmatrix}.\]

We shall have more to say about how we arrived at these particular formulas later on. Note that \(\mathop{\mathrm{Tr}}{a}=\mathop{\mathrm{Tr}}{b}=2\), so \(a\) and \(b\) are parabolic. Looking at the arrangement of Schottky circles in Figure 7.4, you see the fixed point of \(a\) is 0, the tangency point of the circles \(C_a\) and \(C_A\). In Figure 7.3, you can see two chains of tangent circles nesting down on 0 from above and below. The same phenomenon occurs at \(-i\), the tangency point of \(C_B\) and \(C_b\) and the fixed point of \(b\). Notwithstanding extra tangencies, the generators \(a\) and \(b\) still pair opposite circles in the initial tangent chain \(C_a,C_b,C_A\) and \(C_B\). This means that for nesting circles we still need the commutator condition \(\mathop{\mathrm{Tr}}{abAB}=2\), which is not hard to check.

We have been speaking as if there is only one Apollonian gasket, but could we not get different gaskets by starting with different tangent chains? Not really, because it turns out that any chain of three tangent circles can be conjugated to any other three. As you can work out in Project 7.1, this stems from the fact that there is always a Möbius map carrying any three points to any other three. Since the gasket is activated by its initial ideal triangle, and since the procedure at each step consists in adding incircles, a Möbius map which conjugates one ideal triangle to another carries the whole gasket along in its wake.

This explains why it makes sense to talk about the Apollonian gasket, because up to conjugation by Möbius maps there is really only one.

Figure 7.6 is a wonderful picture of what happened when we introduced Dr. Stickler to Apollonius! It is a pretty intricate arrangement, so let’s take a bit of time understanding what has happened to the tiles. To get a grasp on the situation, look back at the three pictures in Figure 7.4, and watch the progression across the three frames. On the left the limit set is a loop or quasicircle, so the ordinary set - what is left when you take away the limit set - has two parts, a pink inside and a white outside. In the central picture, the pink part has collapsed into a myriad of tangent disks, and the red Schottky circles \(C_a\) and \(C_A\) touch at 0. On the right, the gasket group, the ‘horns’ of the pink region have also come together, causing the white outside to fracture into disks as well. Notice how the memory of which was inside and which was outside still persists, because what were the ‘inside’ disks are pink while the ‘outside’ ones are white.

In each picture, the initial Schottky circles are blue and red. Watch them to follow the fate of the tiles. On the left, as usual for a kissing Schottky group, they surround the central inner four sided tile. If we transported this tile around by the group, we would see a tessellation　of the pink region similar to the one in Figure 6.6. (There is also an outer tile, the region outside the four Schottky circles, which as usual you can see more clearly by imagining it on the Riemann sphere.) The inner and outer parts of the ordinary set are invariant under the group, so if you apply any transformation of the group to any tile in the pink region ‘inside’ the limit set, you get another tile which is also ‘inside’.

In the central picture, where \(a\) has become parabolic, the inner tile has been pinched into two halves. Each half-tile is an ideal triangle, with two red sides and one blue. You should think of this pair of triangles as one composite two-part tile. Moved around by the group, the composite tile will cover all the pink circles. There is an outer tile in this picture too, which (on the Riemann sphere) remains in one piece.

On the right, in the gasket group, both \(a\) and \(b\) have been pinched so that now \(C_b\) and \(C_B\) also tuch at \(-i\). Now there are four basic half-tiles. The two pink ones will produce a tiling of the pink circles and the white ones will make a tiling of the white circles. In the glowing gasket picture, these four tiles are black. The upper two ‘horizontal’ ideal triangles are the remnants of the inner Schottky tile, while the lower ‘vertical’ triangle is a remnant of the outer one. If you look carefully, you can just see its twin peeping out in the bottom centre of the page.

Now we can go back to the picture of Dr. Stickler meeting Apollonius. The party is taking place in the remnants of the ‘pink’ circles. If you compare with the half-tiles in Figure 7.4, something rather odd has happened to Dr. Stickler - when the original tile split in two, his head ended up in the green half-tile and his feet in the blue one. Fortunately, there is a transformation of the group (namely \(B\)) which carries the blue Stickler to the green one, moving the blue half-tile containing the blue feet to the yellow half-tile containing the green feet. Had we not pointed out his difficulties you might not even have noticed that anything was wrong. After gluing the yellow half-tile to the green halftile, the relieved (but still slightly greenish) Dr. Stickler stands reunited in a new and complete tile whose images under the group map to all the Sticklers in the picture.

What happened to the tiles in the last section, has, of course, also an interpretation in terms of surfaces. Looking back to the picture on p. 190 which showed how tiles were glued up in a kissing Schottky group, we can work out what happens when we bring the four circles together to make the gasket. It takes a bit of stretching and squeezing to do this, which we have illustrated in Figure 7.7.

The result, shown in the last panel, is our old friend the pretzel with three circles pinched to points or cusps: the waist as in the last chapter and, in addition, loops around the top and bottom tori. Both top and bottom are now ‘spheres’ with three cusps or punctures each. One pair of cusps on each sphere are joined together like ‘horns’, and these two ‘horned spheres’ are themselves joined together at the last two cusps.

The gasket group is called doubly cusped because we have pinched two extra loops, \(a\) and \(b\). It is also sometimes called maximally cusped, because, after all this squeezing, there are no more curves left to pinch. In Chapter 9, we shall see that you can make many variants of the gasket group by imposing more complicated relationships between the curves we choose to pinch on the top and bottom halves of the pinched pretzel.

Figure 7.7. Pinching curves. How gluing up the gasket configuration of tangent circles leads to a pair of triply-punctured spheres. The \(a\) and \(b\) curves we have to shrink are are marked \(L\) and \(M\). Instead of pulling the upper and lower partially glued-up cylinders logether right away, as we did in Figure 6.16, it now takes some effort first to twist them relative to each other in such a way that when we glue-up, the dotted loops are in their proper position ready to be shrunk.

图 7.7. 捏合曲线。如何通过粘合垫片的切圆得到一对带有三穿孔球面。需要收缩的 \(a\) 和 \(b\) 曲线分别标记为 \(L\) 和 \(M\)。与 图 6.16 中直接粘合上下圆柱体的操作不同，此时需要先使两者相对扭转，确保粘合时虚线环处于正确位置以便后续收缩。

Figure 7.6 is made up of lots of disks full of Dr. Sticklers, each tiled by ideal triangles shown in grey. These disks are the remnants of the pink region in Figure 7.4. For most of the rest of this chapter, we shall be occupied with the tiling of just one of these disks. The same tiling fills out the insides of each of the glowing circles in Figure 7.3. The group of symmetries which goes with this very special disk tessellation is called the modular group and has been the well-spring of a huge body of mathematics.

Since the tiling in each disk is the same, we may as well focus on the large disk through -1 and 0, shown in yellow in Figure 7.8. To understand how these ideal triangle tiles cover the yellow disk we need to find the subgroup of all the transformations in the gasket group which map the inside of this disk to itself. This subgroup (which is of course also a group in its own right), or any of its conjugates, is what we call the modular group. The basic tile is made up of two ideal triangles, the ones coloured green and yellow in Figure 7.6. The two triangles together form one of our familiar four-sided pinched-off tiles with four circular arc sides. Moved around by the modular group, they tile the whole yellow disk.

We worked out the labels of the boundary circles \(C_a, C_A, C_{BA}\) and \(C_{Ba}\) in Figure 7.8 of the four-sided tile by going to part of the level-two Schottky chain for the gasket group. (You may find it easiest to check the arrangement in a picture like the left frame of Figure 7.4 without all the extra gasket tangencies first.) Notice the four tangency points of these circles are all on the boundary of the yellow disk. As you can see, the four circles form a new chain of tangent circles. As usual, \(a\) pairs \(C_A\) to \(C_a\). In addition, \(BAb\) pairs \(C_{Ba}\) to \(C_{BA}\) because:

\[BAb\left(C_{Ba}\right) = BAb\left(B\left(C_a\right)\right) = BA\left(C_a\right) = B\left(C_A\right) = C_{BA}.\]

Inside the gasket group we have found another mini-chain of four tangent circles, together with a pair of transformations which match them together in pairs!

This construction shows that the modular group is a new kind of ‘necklace group’, made by disregarding all the rest of the gasket and looking only at the disks produced by acting with \(a\) and \(BAb\) on the four circles which bound the new tile. The new group is generated by the transformations \(a\) and \(BAb\). Indeed in Figure 7.3, you can actually pick out chains of image disks nicely shrinking down onto the glowing limit circle through -1 and 0 . The only difference from the kissing Schottky groups we met in the last chapter is that the two generators pair not opposite circles but adjacent ones. As we shall explain in more detail on p. 213 ff., the image circles shrink because \(a,BAb\) and their product \(aBAb\) are all parabolic.

The same pattern of pairing circles is repeated all over the gasket. Every pink disk is the image of the yellow one under some element in the gasket group, which conjugates our modular group to another ‘modular group’ acting in the new disk. The white disks are different from the pink ones, because you can never get from pink to white using transformations in the gasket group. However you can still find a chain of four tangent circles matched in the same pattern, as described in Project 7.4.

You might well imagine that we should be set to repeat everything we did in the last chapter. By taking four tangent circles and pairing them in this new pattern we should presumably get a whole new lot of quasifuchsian groups. Not so! It turns out that the rigours imposed by specifying that the two generators and their product are all parabolic actually ‘freeze’ the group. Without any mention of circle chains, we prove in Note 7.1 the remarkable fact that all groups made with pairing conditions like this are, up to conjugation, ‘the same’. What this means in more detail is this. Suppose that \(U\) and \(V\) are any two parabolic Möbius transformations with the property that \(UV\) is also parabolic, and such that the fixed points \({\rm Fix}\,U\) and \({\rm Fix}\,V\) of \(U\) and \(V\) are not the same. Then there is always a conjugating map \(M\) such that: \[ MUM^{-1} = \begin{pmatrix} 1 & 0 \\ -2 & 1 \end{pmatrix}, \quad MVM^{-1} = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}. \] This explains why there are so many circles in the gasket group, and why you get an identical tiling pattern in each one.

The result just discussed shows that the modular group is conjugate to a very famous group of great importance in number theory. It is made by arranging the four Schottky circles with their tangency points at \(-1,0,1\) and \(\infty\). You can see these, coloured red and green, in the left frame of Figure 7.9. Since one of the tangency points is the point at infinity, two of the circles show up as green vertical lines. These green lines are paired by \(b=\begin{pmatrix}1&2\\0&1\end{pmatrix}\), while the two red circles tangent at 0 are paired by \(a=\begin{pmatrix}1&0\\-2&1\end{pmatrix}\). Notice how \(a\) and \(b\) match adjacent circles in the chain in exactly the pattern of the red and green arrows in Figure 7.8. In fact, as you can easily calculate, \(ab\) is the parabolic transformation \(\begin{pmatrix}1&2\\-2&-3\end{pmatrix}\).

It is no coincidence that the entries of these three matrices are integers. The right frame of Figure 7.9 is a more complicated picture which shows all the symmetries of the tiling on the left. Each ideal triangle has been subdivided into three hatched and three unhatched sub-triangles. (The sub-triangles are not quite ideal, because only one of their angles is 0 .) The group of symmetries of this more complicated tiling is, from the point of view of Möbius maps, the simplest group of all: just the set of all \(2\times 2\) matrices \(\begin{pmatrix}p&q\\r&s\end{pmatrix}\) with integer entries \(p,q,r\) and \(s\) and determinant \(ps-qr\) equal to 1. To distinguish from the group of the left frame, we sometimes call it the full modular group. The matrices in the (smaller) modular group of the left picture are just those matrices with integer entries for which \(q\) and \(r\) are even and \(p\) and \(s\) are odd.

There is a very beautiful connection between the modular tessellation and fractions: the points where the ideal triangles meet the real axis are exactly the rational numbers. Although it is something of a digression here, we want to take the time to explain the pattern, which turns out to be indispensible when we come to map making in Chapter 9.

Figure 7.11 shows the first few levels in the modular tessellation. The basic tile is the four-sided region, called an ideal quadrilateral, which was bounded by the coloured lines in Figure 7.9. Images of this ideal quadrilateral are shown bounded by solid arcs. The dotted arcs divide them into the two ideal triangles which we saw, half hatched and half white, on the left in Figure 7.9. As the group acts on the basic tile, we get more and more smaller and smaller tiles nesting down to the real axis. The vertices of all these tiles meet the real axis in points which are all fractions. Several things can be read off from a careful examination of this intricate pattern:

• All vertices of the tiles are rational numbers \(p/q\).
• If \(r/s\) and \(p/q\) are two vertices of the same tile, then \(ps - rq = \pm 1\).
• If \(r/s < p/q\) are the outer two vertices of a tile, then the third vertex between them is \((r + p)/(s + q)\).

Check this out! For instance, between \(2/3\) and \(1/2\), we get \((2 + 1)/(3 + 2)\), that is \(3/5\).

It’s easy to see why this happens. As we have seen, a typical matrix in the modular group will look like

\[M = \begin{pmatrix} p & q \\ r & s \end{pmatrix}\]

where \(p, q, r\) and \(s\) are integers and \(ps - rq = 1\). If \(M\) acts on the vertices of the initial triangle with vertices \(0 = 0/1, 1 = 1/1\) and \(\infty = 1/0\), then we get the new triangle with vertices \(M(0) = r/s, M(\infty) = p/q\) and \(M(1) = (p + r)/(q + s)\). Assuming all four entries are positive, we have \(r/s < (p + r)/(q + s) < p/q\) (you can see this by multiplying out). This is just what we found in Figure 7.11. If \(p, q, r, s\) are not all positive, there are half a dozen other cases in which the order of the points \(M(0), M(1)\) and \(M(\infty)\) is different but we get the same result. The same thing happens if we start from the other triangle \(M(-1), M(0), M(\infty)\).

Any two fractions \(r/s\) and \(p/q\) such that \(ps - qr = \pm 1\) are called neighbours. Thus any two vertices of an ideal triangle in the modular tessellation are neighbours. If \(p/q\) is a fraction, then, as we explain in Project 7.5, the process of finding its neighbours is essentially Euclid’s two thousand year old algorithm for finding the highest common factor of two numbers, surely one of the most useful and clever algorithms of all time. The rule for finding the ‘next’ point \(\frac{p+r}{q+s}\) between two neighbours is every student’s dream of what addition of fractions should be. This simple form of fraction ‘addition’ is sometimes called Farey addition’, which one might want to symbolise with a funny symbol like:

\[ \frac{p}{q} \oplus \frac{r}{s} = \frac{p + r}{q + s} \]

Farey addition gives a neat way of organising the rational numbers. Instead of the usual way of arranging them in increasing order (which is difficult, because you never know which one should come ‘next’), fractions can be described by a sequence of left or right moves, reflecting the choice at each stage of whether we choose the new pair of neighbours to the right, or the pair of neighbours to the left.

For positive fractions, the starting point are the two fractions \(0/1\) and \(1/0\), which we can regard as special honourary neighbours because they are connected by a side of our initial triangle, the vertical imaginary axis. Farey addition gives the in-between fraction \(0/1\oplus 1/0=1/1\).

Now we have a choice: go to the ‘left’ and look in the interval between 0 and 1, or go to the ‘right’ and look in the interval between 1 and \(\infty\). Suppose we are aiming for the fraction \(3/5\). Then we turn to the left and apply the Farey addition \(0/1 \oplus 1/1 = 1/2\). At the next stage, we choose the right interval and Farey add to get \(1/2 \oplus 1/1 = 2/3\). Finally, we choose the left interval and Farey add \(1/2 \oplus 2/3 = 3/5\). An exactly similar procedure could be applied to home in on any fraction \(p/q\). Our choice of left-right turns is a driving map: \(3/5\) is given by the instructions ‘left, right, left’. This arrangement of fractions and sequence of right-left moves is closely related to a way of writing fractions as what are called continued fractions, explained in Note 7.2.

The modular group is a new kind of ‘necklace group’. It is still made by pairing four tangent circles, and the only difference from the kissing Schottky groups we met in the last chapter is that the generators pair not opposite circles but adjacent ones. Whenever we have an arrangement of paired tangent circles like this, something like the necklace condition on p. 168 must still be true, but because we are pairing the circles in a different pattern, we can expect that different elements must be parabolic to cause the image circles to shrink.

With the notation of the figure beside Box 20, we have \(a(P) = R\) and \(b(R) = P\), so that the four tangency points of the circles are \(S = \text{Fix}(a)\), \(Q = \text{Fix}(b)\), \(P = \text{Fix}(ba)\), and \(R = \text{Fix}(ab)\). By similar reasoning to that in Chapter 6, in order for the image circles near \(S\) and \(Q\) to shrink, the generators \(a\) and \(b\) must be parabolic. Moreover, \(ba\) must also be parabolic, to make the circles shrink at \(P\). Notice that \(ab\) and \(ba\) are conjugate (since \(b(ab)b^{-1} = ba\)), so saying that \(ab\) or \(ba\) must be parabolic is really one and the same thing. The wonderful thing is, that as we proved in Note 7.1, all groups with these three elements parabolic are automatically conjugate. This is so important to us that we summarize it in Box 20.

Because the pattern of pairing circles is different, so is the arrangement in which the labelled circles are laid down in the plane. The Schottky circles in Figure 7.11 are labelled according to our usual rules, so for example, \(C_{ba}\) still means the image of circle \(C_a\) under the map \(b\). However, if you look carefully, you will see that the order of the circles along the line is not the same as our original order round the boundary of the word tree on p. 104. The labels can be read off in their correct order from the revised version in Figure 7.12. (To see this you will have to twiddle the diagram around so the arrows from the vertex you are interested in are pointing ‘down’ rather than ‘up’.) There is a subtle difference from our original word tree, because there the cyclic order round a vertex was \(a,B,A,b\) while now it is \(a,A,b,B\). The ramifications of this seemingly minor change propagate down the tree.

I could spin a web if I tried.’ said Wilbur, boasting. ‘Ive just never tried.’

‘Let’s see you do it,’ said Charlotte…

‘OK.’ replied Wilhur. ‘You coach me and I’t’ spin one. It must be a lot of fun to spill a web. How do I start?

“要是我愿意，我也能织网。”威尔伯吹嘘道，“只是我从来没试过。”

“那你来织一个给我们看看吧。”夏洛特说。

“好啊。”威尔伯答道，“你来指导我，我就织一个。织网一定很好玩。我该怎么开始呢？”

As any mathematician who has revealed his (or her) occupation to a neighbour on a plane flight has discovered, most people associate mathematics with something akin to the more agonizing forms of medieval torture. It seems indeed unlikely that mathematics would be done at all, were it not that a few people discover the play that lies at its heart. Most published mathematics appears long after the play is done, cloaked in lengthy technicalities which obscure the original fun. The book in hand is unfortunately scarcely an exception. Never mind; after a fairly detailed introduction to the art of creating tilings and fractal limit sets out of two very carefully chosen Möbius maps, we are finally set to embark on some serious mathematical play. The greatest rewards will be reaped by those who invest the time to set up their own programs and join us charting mathematical territory which is still only partially explored.

All the limit sets we have constructed thus far began from a special arrangement of four circles, the Schottky circles, grouped into two pairs. For each pair, we found a Möbius map which moved the inside of one circle to the outside of the other. Our initial tile was the region outside these four circles. By iterating, we produced a tiling which covered the the plane minus the limit set, near which the tiles shrank to minute size. Depending on how we chose the initial Schottky circles, the limit set was either fractal dust, a very crinkled fractal loop we called a quasicircle or, in certain very special cases, a true circle.

The problem with this approach is that it is just too time-consuming to set up the circles and maps which pair them. Free-spirited play shouldn’t be ruined by too much preparation. Why not throw the Schottky circles away, take any pair of \(2\times 2\) matrices for our generators \(a\) and \(b\), run our limit point plotting program, and see what we get?

Hold on though - how exactly will this work? The shrinking disks were so reassuring, and the limit set was so comfortably nestled within them, that it is hard to see why we won’t get chaos in their absence. No matter, the worst that is likely to happen is that the hard disk crashes, so why not give it a try? Luckily, on p. 182 ff. we already upgraded the DFS code to remove the calculation of Schottky disks from the branch termination procedure. All we need do is take the plunge and run the very same algorithm for any pair of transformations \(a\) and \(b\).