Dear Categorists - I've attempted to improve my web version of "week202" based on Steve Schanuel's comments. Here's the next issue, which contains two different constructions by Robin Houston, both of which give a "golden object" - i.e. an object G in some rig category, equipped with an isomorphism between G and G^2 + 1. Best, jb ....................................................................... Also available at http://math.ucr.edu/home/baez/week203.html February 24, 2004 This Week's Finds in Mathematical Physics - Week 203 John Baez Last week I posed this puzzle: to find a "Golden Object". A couple days ago I got a wonderful solution from Robin Houston, a computer science grad student at the University of Manchester. So, I want to say a bit more about the golden number, then describe his solution, and then describe how he found it. Supposedly the Greeks thought the most beautiful rectangle was one such that when you chop a square off one end, you're left with a rectangle of the same shape. If your original rectangle was 1 unit across and G units long, after you chop a 1-by-1 square off the end you're left with a rectangle that's G-1 units across and 1 unit long: G ......................... . . . . . . . . . . . . 1 . . . 1 . . . . . . . 1 . G-1 . ......................... So, to make the proportions of the little rectangle the same as those of the big one, you want "1 is to G as G-1 is to 1" or in other words: 1/G = G - 1 or after a little algebra, G^2 = G + 1 so that G = (1 + sqrt(5))/2 = 1.618033988749894848204586834365... while 1/G = 0.618033988749894848204586834365... and G^2 = 2.618033988749894848204586834365... (At this point I usually tell my undergraduates that the pattern continues like this, with G^3 = 3.618... and so on - just to see if they'll believe anything I say.) These days, the number G is called the Golden Number, the Golden Ratio, or the Golden Section. It's often denoted by the Greek letter Phi, after the Greek sculptor Phidias. Phidias helped design the Parthenon - and supposedly packed it full of golden rectangles, to make it as beautiful as possible. The golden number is a great favorite among amateur mathematicians, because it has a flashy sort of charm. You can find it all over the place if you look hard enough - and if you look too hard, you'll find it even in places where it's not. It's the ratio of the diagonal to the side of a regular pentagon! If you like the number 5, you'll be glad to know that 5 + sqrt(5) G = sqrt[-------------] 5 - sqrt(5) If you don't, maybe you'd prefer this: G = exp(arcsinh(1/2)) My favorite formulas for the golden number are G = sqrt(1 + sqrt(1 + sqrt(1 + sqrt(1 + sqrt(1 + sqrt(1 + ... and the continued fraction: 1 G = 1 + --------- 1 + 1 -------- 1 + 1 ------- 1 + 1 ------ 1 + 1 ---- 1 + 1 ---- . . . These follow from the equations G^2 = G + 1 and G = 1 + 1/G, respectively. If you chop off the continued fraction for G at any point, you'll see that G is also the limit of the ratios of successive Fibonacci numbers. See "week190" for a very different proof of this fact. However, don't be fooled! The charm of the golden number tends to attract kooks and the gullible - hence the term "fool's gold". You have to be careful about anything you read about this number. In particular, if you think ancient Greeks ran around in togas philosophizing about the "golden ratio" and calling it "Phi", you're wrong. This number was named Phi after Phidias only in 1914, in a book called _The Curves of Life_ by the artist Theodore Cook. And, it was Cook who first started calling 1.618... the golden ratio. Before him, 0.618... was called the golden ratio! Cook dubbed this number "phi", the lower-case baby brother of Phi. In fact, the whole "golden" terminology can only be traced back to 1826, when it showed up in a footnote to a book by one Martin Ohm, brother of Georg Ohm, the guy with the law about resistors. Before then, a lot of people called 1/G the "Divine Proportion". And the guy who started *that* was Luca Pacioli, a pal of Leonardo da Vinci who translated Euclid's Elements. In 1509, Pacioli published a 3-volume text entitled Divina Proportione, advertising the virtues of this number. Some people think da Vinci used the divine proportion in the composition of his paintings. If so, perhaps he got the idea from Pacioli. Maybe Pacioli is to blame for the modern fascination with the golden ratio; it seems hard to trace it back to Greece. These days you can buy books and magazines about "Elliot Wave Theory", a method for making money on the stock market using patterns related to the golden number. Or, if you're more spiritually inclined, you can go to workshops on "Sacred Geometry" featuring talks about the healing powers of the golden ratio. But Greek texts seem remarkably quiet about this number. The first recorded hint of it is Proposition 11 in Book II of Euclid's "Elements". It also shows up elsewhere in Euclid, especially Proposition 30 of Book VI, where the task is "to cut a given finite straight line in extreme and mean ratio", meaning a ratio A:B such that A:B::(A+B):A (i.e., "A is to B as A+B is to A") This is later used in Proposition 17 of Book XIII to construct the pentagonal face of a regular dodecahedron. Of course, Euclid wasn't the first to do all these things; he just wrote them up in a nice textbook. By now it's impossible to tell who discovered the golden ratio and just what the Greeks thought about it. For a sane and detailed look at the history of the golden ratio, try this: 1) J. J. O'Connor and E. F. Robertson, The Golden Ratio, http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Golden_ratio.html While I'm at it, I should point out that you that Theodore Cook's book introducing the notation "Phi" is still in print: 2) The Curves of Life: Being an Account of Spiral Formations and Their Application to Growth in Nature, to Science, and to Art: with Special Reference to the Manuscripts of Leonardo da Vinci, Dover Publications, New York, 1979. If you want to see what Euclid said about the golden ratio, you can also pick up a cheap copy of the Elements from Dover - but it's probably quicker to go online. There are a number of good places to find Euclid's Elements online these days. Topologists know David Joyce as the inventor of the "quandle" - an algebraic structure that captures most of the information in a knot. Now he's writing a high-tech edition of Euclid, complete with Java applets: 3) David E. Joyce's edition of Euclid's Elements, http://aleph0.clarku.edu/~djoyce/java/elements/toc.html Joyce is carrying on a noble tradition: back in 1847, Oliver Byrne did a wonderful edition of Euclid complete with lots of beautiful color pictures and even some pop-up models. You can see this online at the Digital Mathematics Archive: 4) Oliver Byrne's edition of Euclid's Elements, online at the Digital Mathematics Archive, http://www.sunsite.ubc.ca/DigitalMathArchive/ The most famous scholarly English translation of Euclid was done by Sir Thomas Heath in 1908. You can find it together with an edition in Greek and a nearly infinite supply of other classical texts at the Perseus Digital Library: 5) Thomas L. Heath's edition of Euclid's Elements, online at The Perseus Digital Library, http://www.perseus.tufts.edu/ But I'm digressing! My main point was that while G = (1 + sqrt(5))/2 is a neat number, it's a lot easier to find nuts raving about it on the net than to find truly interesting mathematics associated with it - or even interesting references to it in Greek mathematics! The cynic might conclude that the charm of this number is purely superficial. However, that would be premature. First of all, there's a certain sense in which G is "the most irrational number". To get the best rational approximations to a number you use its continued fraction expansion. For G, this converges as slowly as possible, since it's made of all 1's: 1 G = 1 + --------- 1 + 1 -------- 1 + 1 ------- 1 + 1 ------ 1 + 1 ---- 1 + 1 ---- . . . We can make this more precise. For any number x there's a constant c(x) that says how hard it is to approximate x by rational numbers, given by lim inf |x - p/q| = c(x)/q^2 q -> infinity where q ranges over integers, and p is an integer chosen to minimize |x - p/q|. This constant is as big as possible when x is the golden ratio! It'd be ironic if the famously "rational" Greeks, who according to legend even drowned the guy who proved sqrt(2) was irrational, chose the most irrational number as the proportions of their most beautiful rectangle! But, it wouldn't be a coincidence. Their obsession with ratios and proportions led them to ponder the situation where A:B::(A+B):A, and this proportion instantly implies that A and B are incommensurable, since if you assume A and B are integers and try to find their greatest common divisor using Euclid's algorithm, you get stuck in an infinite loop. Euclid even mentions this idea in Proposition 2 of Book X: If, when the less of two unequal magnitudes is continually subtracted in turn from the greater that which is left never measures the one before it, then the two magnitudes are incommensurable. He doesn't explicitly come out and apply it to what we now call the golden ratio - but how could he not have made the connection? For more info on the Greek use of continued fractions and the Euclidean algorithm, check out the chapter on "antihyphairetic ratio theory" in this book: 6) D. H. Fowler, The Mathematics of Plato's Academy: A New Reconstruction, Oxford U. Press, Oxford, 1987. Anyway, it's actually important in physics that the golden number is so poorly approximated by rationals. This fact shows up in the Kolmogorov- Arnold-Moser theorem, or "KAM theorem", which deals with small perturbations of completely integrable Hamiltonian systems. Crudely speaking, these are classical mechanics problems that have as many conserved quantities as possible. These are the ones that tend to show up in textbooks, like the harmonic oscillator and the gravitational 2-body problem. The reason is that you can solve such problems if you can do a bunch of integrals - hence the term "completely integrable". The cool thing about a completely integrable system is that time evolution carries states of the system along paths that wrap around tori. Suppose it takes n numbers to describe the position of your system. Then it also takes n numbers to describe its momentum, so the space of states is 2n-dimensional. But if the system has n different conserved quantities - that's basically the maximum allowed - the space of states will be foliated by n-dimensional tori. Any state that starts on one of these tori will stay on it forever! It will march round and round, tracing out a kind of spiral path that may or may not ever get back to where it started. Things are pretty simple when n = 1, since a 1-dimensional torus is a circle, so the state *has* to loop around to where it started. For example, when you have a pendulum swinging back and forth, its position and momentum trace out a loop as time passes. When n is bigger, things get trickier. For example, when you have n pendulums swinging back and forth, their motion is periodic if the ratios of their frequencies are rational numbers. This is how it works for any completely integrable system. For any torus, there's an n-tuple of numbers describing the frequency with which paths on this torus wind around in each of the n directions. If the ratios of these frequencies are all rational, paths on this torus trace out periodic orbits. Otherwise, they don't! KAM theory says what happens when you perturb such a system a little. It won't usually be completely integrable anymore. Interestingly, the tori with rational frequency ratios tend to fall apart due to resonance effects. Instead of periodic orbits, we get chaotic motions instead. But the "irrational" tori are more stable. And, the "more irrational" the frequency ratios for a torus are, the bigger a perturbation it takes to disrupt it! Thus, the most stable tori tend to have frequency ratios involving the golden number. As we increase the perturbation, the last torus to die is called a "golden torus". You can actually *watch* tori breaking into chaotic dust if you check out the applet illustrating the "standard map" on this website: 7) Takashi Kanamaru and J. Michael T. Thompson, Introduction to Chaos and Nonlinear Dynamics, http://www.sekine-lab.ei.tuat.ac.jp/~kanamaru/Chaos/e/Standard/ The "standard map" is a certain dynamical system that's good for illustrating this effect. You won't actually see 2d tori, just 1d cross-sections of them - but it's pretty fun. For more details, try this: 8) M. Tabor, Chaos and Integrability in Nonlinear Dynamics: An Introduction, Wiley, New York, 1989. In short, the golden number is the best frequency ratio for avoiding resonance! Some audiophiles even say this means the best shaped room for listening to music is one with proportions 1:G:G^2. I leave it to you to find the flaw in this claim. For more dubious claims, check out the ad for expensive speaker cables at the end of this article. KAM theory is definitely cool, but we shouldn't rest content with this when skeptics ask if the golden number is all it's cracked up to be. I figure it's part of our job as mathematicians to keep on discovering mind-blowing facts about the golden number. A small part, but part: we shouldn't give up the field to amateurs! Penrose has done his share. His "Penrose tiles" take crucial advantage of the self-similarity embodied by the golden number to create nonperiodic tilings of the plane. This helped spawn a nice little industry, the study of "quasicrystals" with 5-fold symmetry. Here's a good introduction for mathematicians: 9) Andre Katz, A short introduction to quasicrystallography, in From Number Theory to Physics, eds. M. Waldschmit et al, Springer, Berlin, 1992, pp. 496-537. By the way, this same book has some nice stuff on the role of the golden number in KAM theory and the theory of iterated maps from the circle to itself: 10) Predrag Cvitanovic, Circle maps: irrationally winding, in From Number Theory to Physics, eds. M. Waldschmit et al, Springer, Berlin, 1992, pp. 631-658. 11) Jean-Christophe Yoccoz, Introduction to small divisors problems, in From Number Theory to Physics, eds. M. Waldschmit et al, Springer, Berlin, 1992, pp. 659-679. Conway and Sloane are also pulling their weight. Starting from the relation between the golden ratio, the isosahedron, and the 4-dimensional big brother of the icosahedron (the "600-cell"), they've described how to construct the coolest lattices in 8 and 24 dimensions using "icosians" - which are certain quaternions built using the golden ratio. I discussed this circle of ideas in "week20", "week59" and "week155". But if you want some really scary formulas involving the golden ratio, Ramanujan is the one to go to. Check these out: 1 -------------- 1 + exp(-2pi) ------------- 1 + exp(-4pi) = exp(2pi/5) [sqrt(G sqrt(5)) - G] ------------ 1 + exp(-6pi) ----------- 1 + exp(-8pi) --------- . . . and 1 + exp(-2pi sqrt(5)) ------------------- 1 + exp(-4pi sqrt(5)) ----------------- 1 + exp(-6pi sqrt(5)) ------------------ 1 + exp(-8pi sqrt(5)) ------------------ . . . sqrt(5) = exp(2pi/5) [ ------------------------------------- - G] 1 + [5^{3/4} (G - 1)^{5/2} - 1]^{1/5} These are special cases of a monstrosity called the Rogers-Ramanujan continued fraction, which is a kind of "q-deformation" of the continued fraction for the golden ratio. For details, start here: 12) Eric W. Weisstein, Rogers-Ramanujan Continued Fraction, http://mathworld.wolfram.com/Rogers-RamanujanContinuedFraction.html The golden number also shows up in the theory of quantum groups. I talked about this in "week22" so I won't explain it again here. But, I can't resist mentioning that Freedman, Larsen and Wang have subsequently shown that a certain topological quantum field theory called Chern-Simons theory, built using the quantum group SU_q(2), can serve as a universal quantum computer when the parameter q is a fifth root of unity. And, this is exactly the case where the spin-1/2 representation of SU_q(2) has quantum dimension equal to the golden number! 13) Michael Freedman, Michael Larsen, Zhenghan Wang, A modular functor which is universal for quantum computation, available at quant-ph/0001108. But don't get the wrong idea: it's not that some magic feature of the golden number is required to build a universal quantum computer! It's just that the 5 seems to be the *smallest* number n such that SU_q(2) Chern-Simons theory is computationally universal when q is an nth root of 1. That's pretty much everything I know about the golden number. So now, what about this "Golden Object" puzzle? Basically, the problem was to find an object that acts like the golden number. The golden number has G = G^2 + 1, so we want to find a object G equipped with a nice isomorphism between G and G^2 + 1. If G is just a set, this means we want a nice one-to-one correspondence between pairs of elements of G, and elements of G together with one other It doesn't matter what that other thing is, so let's call it "@". (You may be wondering about the word "nice". The point is, the problem is too easy if we don't demand that the solution be nice in some way - some way that I don't feel like making precise.) Here's Robin Houston's answer: Define a "bit" to be either 0 or 1. Define a "golden tree" to be a (planar) binary tree with leaves labelled by 0, 1, or *, where every node has at most one bit-child. For example: /\ is a golden tree, but /\ is not. /\ 1 /\ * 0 * 0 1 Let G be the set of golden trees. We define an isomorphism f: G^2 -> G + {@} as follows. First we define f(X, Y) when both X and Y are golden trees with just one node, this node being labelled by a bit. We can identify such a tree with a bit, and doing this we set f(0, 0) = 0 f(0, 1) = 1 f(1, 0) = * f(1, 1) = @ In the remaining case, where the golden trees X and Y are not just bits, we set f(X, Y) = /\ X Y There are different ways to show this function f is a one-to-one correspondence, but the best way is to see how Houston came up with this answer! He didn't just pull it out of a hat; he tackled the problem systematically, and that's why his solution counts as "nice". It's easy to find a set S equipped with an isomorphism S = P(S) where P is some polynomial with natural number coefficients. You just use the fixed-point principle described in "week108". Namely, you start with the empty set, keep hitting it with P forever, and take a kind of limit. This is how I built the set of binary trees last week, as a solution of T = T^2 + 1. The problem is that the isomorphism we seek now: G^2 = G + 1 (1) is not of this form. So, what Houston does is to make a substitution: G = H + 2 Given this, we'd get (1) if we had H^2 + 4H + 4 = H + 3 (2) and we'd get (2) if we had H^2 + 4H + 1 = H (3) which is of the desired form. We can rewrite (3) as H = 1 + H^2 + 2H + H2 and in English this says "an element of H is either a *, or a pair consisting of two guys that are either bits or elements of H - but not both bits". So, a guy in H is a golden tree! But, if it has just one node, that node can only be labelled by a *, not a 0 or 1. This means there are precisely 2 golden trees not in H. So, G = H + 2 is the set of all golden trees, and our calculation above gives an isomorphism G^2 = G + 1. Voila! Note that to derive (3) from (1) we need to subtract, which in general is not allowed in this game. Here we are subtracting constants, and Houston says that's allowed by the "Garsia-Milne involution theorem". I don't know this theorem, so I'll make a note to myself to learn it. But luckily, we don't really need it here: we only need to derive (1) from (3), and that involves addition, so it's fine. Part of what makes Houston's solution "nice" is that it suggests a general method for turning polynomial equations into recursive definitions of the form S = P(S). Another nice thing is that his trick delivers a structure type G(X) that reduces to G when X = 1. To get this, first use the fixed-point method to construct a structure type H(X) with an isomorphism H(X) = (H(X) + X)^2 + 2H(X) Then, define G(X) = H(X) + X + 1 and note that this gives G(X)^2 = G(X) + X which reduces to G^2 = G + 1 when X = 1. As if this weren't enough, Houston also gave another solution to the puzzle. He showed that James Propp's proposed Golden Object, described last week, really is a Golden Object! Maybe Propp already knew this, but I sure didn't. The idea of the proof is pretty general. Suppose we've got a category that's a "2-rig" in the sense of "week191". And, suppose we've got an object X equipped with an isomorphism X = 1 + 2X (4) so that X acts like "-1". For example, following Schanuel and Propp, we can take the category of "sigma-polytopes" and let X be the open interval: then isomorphism (4) says (0,1) = (0,1/2) + {1/2} + (1/2,1) Or, following Houston, we can take the category of sets and let X be the set of finite bit-strings. Then (4) says "a finite bit-string is either the empty bit-string, or a bit followed by a finite bit-string". The relation between these two examples is puzzling to me - if anyone understands it, let me know! But anyway, either one works. Now let G be the object of "binary trees with X-labelled leaves": G = X + X^2 + 2X^3 + 5X^4 + 14X^5 + 42X^6 + ... where the coefficients are Catalan numbers. Let's show that G is a Golden Object. To do this, we'll use (4) and this isomorphism: G = G^2 + X (5) which says "a binary tree with X-labelled leaves is a pair of such trees, or a degenerate tree with just one X-labelled node". The formula for G involving Catalan numbers is really just the fixed-point solution to this! Here is Houston's fiendishly clever argument. Suppose Z is any type equipped with an isomorphism Z = Z' + X for some Z'. Then Z + X + 1 = Z' + 2X + 1 = Z' + X = Z This applies to Z = G^2, since G^2 = (X + G^2)^2 = (2X + 1 + G^2)^2 has a X term in it when you multiply it out, so it's of the form Z' + X. Therefore we have an isomorphism G^2 = G^2 + X + 1 But we also have an isomorphism G + 1 = G^2 + X + 1 by (5). Composing these, we get our isomorphism G^2 = G + 1. Golden! I'll stop here. Quote of the week: "As a high-end cable manufacturer, Cardas Audio strives to address every detail of cable and conductor construction, no matter how small. An elegant solution deals with quality, not quantity. Cable geometry problems are resolved in the cable's design, not after the fact with filters. George Cardas received U.S. Patent Number 4,628,151 for creating Golden Section Stranding Audio Cable. It is truly unique. George introduced the concept of Golden Section Stranding to high-end audio, but the Golden Ratio, 1.6180339887... : 1, is as old as nature itself. The Golden Ratio is the mathematical proportion nature uses to shape leaves and sea shells, insects and people, hurricanes and galaxies, and the heart of musical scales and chords. "Discovered" by the Greeks, but used by the Egyptians in the Great Pyramid centuries before, man has employed the Golden Ratio to create his most beautiful and naturally pleasing works of art and architecture." - Cardas Audio speaker cable advertisement ----------------------------------------------------------------------- Previous issues of "This Week's Finds" and other expository articles on mathematics and physics, as well as some of my research papers, can be obtained at http://math.ucr.edu/home/baez/ For a table of contents of all the issues of This Week's Finds, try http://math.ucr.edu/home/baez/twf.html A simple jumping-off point to the old issues is available at http://math.ucr.edu/home/baez/twfshort.html If you just want the latest issue, go to http://math.ucr.edu/home/baez/this.week.html