September 2007 - mathfun@mailman.xmission.com

[math-fun] Compute a Function From its Fourier Series?
by Bill Gosper 26 Sep '07

26 Sep '07

Bob Baillie asked >One of the first examples of Fourier series that a student encounters is something like this: f(x) = x/2, for -Pi < x < Pi. The Fourier series is Sum[ (-1)^(n+1)/n * Sin[n x] ]. A second, more complicated example is: f(x) = -Pi/2, for -Pi < x < 0, f(x) = Pi/2, for 0 < x < Pi. This Fourier series is Sum[ (1 - (-1)^n)/n * Sin[n x] ]. Can one work backwards, from the coefficients to the function? About all you can do is replace sin(x) with Im(e^ix) and treat the problem as a hypergeometric series, of which you can only do the easiest cases. Rarely one might use a bibasic series with q=e^ix and p=1. To see the hopelessness of the general case, note that the Snowflake and more general fractals have fairly nice Fourier coefficients: http://gosper.org/fst.dvi, with (poorly scanned) figures http://gosper.org/fst1.png, http://gosper.org/fst1.png, http://gosper.org/fst1.png, and http://gosper.org/fst1.png . I have even seen a fairly simple Fourier series for the boundary of the Mandelbrot set. Just imagine writing a closed form for Im(Snowflake). That paper also gives the Fourier series for an arc repeated around the edges of a regular polygon or "star", and by letting the dimension parameter d=1, the Fourier series for the polygon itself. By studying these, we can learn to recognize the series for piecewise linear functions. --rwg PS, on the Lisp Machine I once had a demo where Macsyma computed in closed form the F.s. of a point alternately tracing an equilateral triangle and its inscribed circle, which I then animated as a sum of rotating vectors, using that machine's superior color graphics primitives. --------------------------------- Take the Internet to Go: Yahoo!Go puts the Internet in your pocket: mail, news, photos & more.

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[math-fun] Re: math-fun Digest, Vol 55, Issue 12
by Steve Witham 26 Sep '07

26 Sep '07

>From: Bill Gosper <rwmgosper(a)yahoo.com> > > >) rabbits may be any of warren, nest, colony, bevy, bury, drove, or trace. > > Plague. > > JZ>Oddly, a quick perusal uncovers no specific word for a collection of > >collective nouns. I have to assume I've just missed it. And a > >collection of those words would be...? > > Plague. Yeah, as in, "a plague of lexicologists." --Steve

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[math-fun] Mathematical cosmos article by Max Tegmark
by Dan Asimov 26 Sep '07

26 Sep '07

Rich, Here is the article I mentioned. If you see fit to send it on to math-fun, feel free to go ahead and do so, as a forward from me. (Obviously, the miscellanea at the end of the article could be deleted if you think it should be.) --Dan ____________________________________________________________________________________________________________________________________________________________ Mathematical cosmos: Reality by numbers 14 September 2007 Max Tegmark Mathematical structures of the universe --------------------------------------- What is the meaning of life, the universe and everything? In the sci-fi spoof The Hitchhiker's Guide to the Galaxy, the answer was found to be 42; the hardest part turned out to be finding the real question. Indeed, although our inquisitive ancestors undoubtedly asked such big questions, their search for a "theory of everything" evolved as their knowledge grew. As the ancient Greeks replaced myth-based explanations with mechanistic models of the solar system, their emphasis shifted from asking "why" to asking "how". Since then, the scope of our questioning has dwindled in some areas and mushroomed in others. Some questions were abandoned as naive or misguided, such as explaining the sizes of planetary orbits from first principles, which was popular during the Renaissance. The same may happen to currently trendy pursuits like predicting the amount of dark energy in the cosmos, if it turns out that the amount in our neighbourhood is a historical accident. Yet our ability to answer other questions has surpassed earlier generations' wildest expectations: Newton would have been amazed to know that we would one day measure the age of our universe to an accuracy of 1 per cent, and comprehend the microworld well enough to make an iPhone. Mathematics has played a striking role in these successes. The idea that our universe is in some sense mathematical goes back at least to the Pythagoreans of ancient Greece, and has spawned centuries of discussion among physicists and philosophers. In the 17th century, Galileo famously stated that the universe is a "grand book" written in the language of mathematics. More recently, the physics Nobel laureate Eugene Wigner argued in the 1960s that "the unreasonable effectiveness of mathematics in the natural sciences" demanded an explanation. Here, I will push this idea to its extreme and argue that our universe is not just described by mathematics - it is mathematics. While this hypothesis might sound rather far-fetched, it makes startling predictions about the structure of the universe that could be testable by observations. It should also be useful in narrowing down what an ultimate theory of everything could look like. The foundation of my argument is the assumption that there exists an external physical reality independent of us humans. This is not too controversial: I would guess that the majority of physicists favour this long-standing idea, though it is still debated. Metaphysical solipsists reject it flat out, and supporters of the so-called Copenhagen interpretation of quantum mechanics may reject it on the grounds that there is no reality without observation (New Scientist, 23 June, p 30). Assuming an external reality exists, however, physics theories aim to describe how it works. Our most successful theories, such as general relativity and quantum mechanics, describe only parts of this reality: gravity, for instance, or the behaviour of subatomic particles. In contrast, the holy grail of theoretical physics is a theory of everything - a complete description of reality. My personal quest for this theory begins with an extreme argument about what it is allowed to look like. If we assume that reality exists independently of humans, then for a description to be complete, it must also be well defined according to non-human entities - aliens or supercomputers, say - that lack any understanding of human concepts. Put differently, such a description must be expressible in a form that is devoid of human baggage like "particle", "observation" or other English words. In contrast, all physics theories that I have been taught have two components: mathematical equations, and words that explain how the equations are connected to what we observe and intuitively understand. When we derive the consequences of a theory, we introduce concepts - protons, stars, molecules - because they are convenient. However, it is we humans who create these concepts. In principle, everything could be calculated without this baggage: a sufficiently powerful supercomputer could calculate how the state of the universe evolves over time without interpreting it in human terms. All of this raises the question: is it possible to find a description of external reality that involves no baggage? If so, such a description of objects in this reality and the relations between them would have to be completely abstract, forcing any words or symbols to be mere labels with no preconceived meanings whatsoever. Instead, the only properties of these entities would be those embodied by the relations between them. This is where mathematics comes in. To a modern logician, a mathematical structure is precisely this: a set of abstract entities with relations between them. Take the integers, or geometric objects like the dodecahedron, a favourite of the Pythagoreans (see Diagram). This is in stark contrast to the way most of us first perceive mathematics - either as a sadistic form of punishment, or as a bag of tricks for manipulating numbers. Like physics, maths has evolved to ask broader questions. Modern mathematics is the formal study of structures that can be defined in a purely abstract way. Think of mathematical symbols as mere labels without intrinsic meaning. It doesn't matter whether you write "two plus two equals four", "2 + 2 = 4" or "dos m√°s dos igual a cuatro". The notation used to denote the entities and the relations is irrelevant; the only properties of integers are those embodied by the relations between them. That is, we don't invent mathematical structures - we discover them, and invent only the notation for describing them. So here is the crux of my argument. If you believe in an external reality independent of humans, then you must also believe in what I call the mathematical universe hypothesis: that our physical reality is a mathematical structure. In other words, we all live in a gigantic mathematical object - one that is more elaborate than a dodecahedron, and probably also more complex than objects with intimidating names like Calabi-Yau manifolds, tensor bundles and Hilbert spaces, which appear in today's most advanced theories. Everything in our world is purely mathematical - including you. If that is true, then the theory of everything must be purely abstract and mathematical. Although we do not yet know what the theory would look like, particle physics and cosmology have reached a point where all measurements ever made can be explained, at least in principle, with equations that fit on a few pages and involve merely 32 unexplained numerical constants (Physical Review D, vol 73, 023505). So the correct theory of everything could even turn out to be simple enough to describe with equations that fit on a T-shirt. Before discussing whether the mathematical universe hypothesis is correct, however, there is a more urgent question: what does it actually mean? To understand this, it helps to distinguish between two ways of viewing our external reality. One is the outside overview of a physicist studying its mathematical structure, like a bird surveying a landscape from high above; the other is the inside view of an observer living in the world described by the structure, like a frog living in the landscape surveyed by the bird. One issue in relating these two perspectives involves time. A mathematical structure is by definition an abstract, immutable entity existing outside of space and time. If the history of our universe were a movie, the structure would correspond not to a single frame but to the entire DVD. So from the bird's perspective, trajectories of objects moving in four-dimensional space-time resemble a tangle of spaghetti. Where the frog sees something moving with constant velocity, the bird sees a straight strand of uncooked spaghetti. Where the frog sees the moon orbit the Earth, the bird sees two intertwined spaghetti strands. To the frog, the world is described by Newton's laws of motion and gravitation. To the bird, the world is the geometry of the pasta. A further subtlety in relating the two perspectives involves explaining how an observer could be purely mathematical. In this example, the frog itself must consist of a thick bundle of pasta whose structure corresponds to particles that store and process information in a way that gives rise to the familiar sensation of self-awareness. Fine, so how do we test the mathematical universe hypothesis? For a start, it predicts that further mathematical regularities remain to be discovered in nature. Ever since Galileo promulgated the idea of a mathematical cosmos, there has been a steady progression of discoveries in that vein, including the standard model of particle physics, which captures striking mathematical order in the microcosm of elementary particles and the macrocosm of the early universe. The hypothesis also makes a much more dramatic prediction: the existence of parallel universes. Many types of "multiverse" have been proposed over the years, and it is useful to classify them into a four-level hierarchy. The first three levels correspond to non-communicating parallel worlds within the same mathematical structure: level I simply means distant regions from which light has not yet had time to reach us; level II covers regions that are forever unreachable because of the cosmological inflation of intervening space; and level III, often called "many worlds", involves non-communicating parts of the Hilbert space of quantum mechanics into which the universe can in a sense "split" during certain quantum events. Level IV refers to parallel worlds in distinct mathematical structures, which may have fundamentally different laws of physics. Today's best estimates suggest that we need a huge amount of information, perhaps 10100 bits, to fully describe our frog's view of the observable universe, down to the positions of every star and grain of sand. Most physicists hope for a theory of everything that is much simpler than this and can be specified in few enough bits to fit in a book, if not on a T-shirt. The mathematical universe hypothesis implies that such a simple theory must predict a multiverse. Why? Because this theory is by definition a complete description of reality: if it lacks enough bits to completely specify our universe, then it must instead describe all possible combinations of stars, sand grains and such - so that the extra bits that describe our universe simply encode which universe we are in, like a multiversal phone number. Thus, describing a multiverse can be simpler than describing one universe. Pushed to its extreme, the hypothesis implies the level-IV multiverse. If there is a mathematical structure that is our universe, and its properties correspond to our physical laws, then each mathematical structure with different properties is its own universe with different laws. Indeed, the level-IV multiverse is compulsory, since these structures are not "created" and don't exist "somewhere" - they just exist. Stephen Hawking once asked, "What is it that breathes fire into the equations and makes a universe for them to describe?" For the mathematical cosmos, there is no fire-breathing required, since the point is not that a mathematical structure describes a universe, but that it is a universe. The existence of the level-IV multiverse also answers a confounding question emphasised by the physicist John Wheeler: even if we found equations that describe our universe perfectly, then why these particular equations, not others? The answer is that the other equations govern parallel universes, and that our universe has these particular equations because they are statistically likely, given the distribution of mathematical structures that can support observers like us. It is crucial to ask whether parallel universes are within the purview of science, or are merely speculation. They are not a theory in themselves, but rather a prediction made by certain theories. For a theory to be falsifiable, we need not be able to test all its predictions, merely at least one of them. So here's a testable prediction: if we exist in many parallel universes, then we should expect to find ourselves in a typical one. Suppose we succeed in computing the probability distribution for some number, say the dark energy density or the number of dimensions of space, measured by a typical observer in a mathematical structure where this number has meaning. If we find that this distribution makes the value measured in our own universe highly atypical, it would rule out the multiverse, and hence the mathematical universe hypothesis. Ultimately, why should we believe the mathematical universe hypothesis? Perhaps the most compelling objection is that it feels counter-intuitive and disturbing. I personally dismiss this as a failure to appreciate Darwinian evolution. Evolution endowed us with intuition only for those aspects of physics that had survival value for our distant ancestors, such as the parabolic trajectories of flying rocks. Darwin's theory thus makes the testable prediction that whenever we look beyond the human scale, our evolved intuition should break down. We have repeatedly tested this prediction, and the results overwhelmingly support it: our intuition breaks down at high speeds, where time slows down; on small scales, where particles can be in two places at once; and at high temperatures, where colliding particles change identity. To me, an electron colliding with a positron and turning into a Z-boson feels about as intuitive as two colliding cars turning into a cruise ship. The point is that if we dismiss seemingly weird theories out of hand, we risk dismissing the correct theory of everything, whatever it may be. If the mathe matical universe hypothesis is true, then it is great news for science, allowing the possibility that an elegant unification of physics and mathematics will one day allow us to understand reality more deeply than most dreamed possible. Indeed, I think the mathematical cosmos is the best theory of everything that we could hope for: it would mean no aspect of reality is off-limits from our scientific quest to uncover regularities and make quantitative predictions. However, it would also shift the ultimate question once again. We might abandon as misguided the question of which particular mathematical equations describe all of reality, and instead ask how to compute the frog's view of the universe - our observations - from the bird's view. That would determine whether we have uncovered the true structure of our universe, and help us figure out which corner of the mathematical cosmos is our home. -----end----- Related Articles ---------------- Impossible things for breakfast, at the Logic Caf√© http://www.newscientist.com/article.ns?id=mg19425991.400 April 2007 The Big Questions: Will we ever have a theory of everything? http://www.newscientist.com/article.ns?id=mg19225780.074 November 2006 The theory of everything: Are we nearly there yet? http://www.newscientist.com/article.ns?id=mg18624971.500 April 2005 e: the mystery number http://www.newscientist.com/article.ns?id=mg19526131.600 July 2007 Weblinks -------- Max Tegmark, Massachusetts Institute of Technology http://space.mit.edu/home/tegmark/index.html Eugene Wigner, Nobel laureate http://nobelprize.org/nobel_prizes/physics/laureates/1963/wignerbio.html Platonic solids, Wolfram MathWorld http://mathworld.wolfram.com/PlatonicSolid.html Ultimate ensemble theory of the universe http://www.arxiv.org/abs/gr-qc/9704009 -------------------------------------------------------------------------- [From issue 2621 of New Scientist magazine, 14 September 2007, page 38-41]

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[math-fun] Kissing number question for an optimal cylinder
by Dan Asimov 26 Sep '07

26 Sep '07

Define the "optimal cylinder" as the finite cylinder whose diameter equals its height.* Let C_0 := { (x,y,z) | x^2 + y^2 <= 1 and |z| <= 1 }. QUESTION: How many congruent copies of C_0 -- call them C_1,...,C_n -- can simultaneously touch C_0, subject to the RULE: Any two cylinders among C_0,...,C_n are allowed to overlap on their boundaries but must have disjoint interiors. (I don't know the answer, but for comparison's sake the largest n that I know works is listed far below. Perhaps this number can be proved maximum.) --Dan ________________________________________________________________________________________ * the cylinder with the most volume for a given surface area 20

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[math-fun] Atle Selberg
by Schroeppel, Richard 25 Sep '07

25 Sep '07

I noticed a long obit in the Los Angeles Times last week for Atle Selberg, who provided the first(*) elementary proof of the Prime Number Theorem. http://www.latimes.com/news/printedition/california/la-me-selberg22aug22 ,1,4873005.story?coll=la-headlines-pe-california free, short: http://www.norwaypost.no/cgi-bin/norwaypost/imaker?id=96351 Rich ----- * Modulo a priority dispute with Erdos.

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Re: [math-fun] Spiders and regular webs
by Dan Asimov 25 Sep '07

25 Sep '07

Olivier Gerard wrote: << I just spotted an old article of National Geographic http://news.nationalgeographic.com/news/2004/06/0624_040624_tvspider.html titled "New" Spider Species Weaves Uncommonly Regular Webs Reading the article, it seems the spider makes regular cells and put them together to match the place it wants to cover. . . . . . . >> I did not get the impression that a spider "spider makes regular cells and put them together . . .". Rather, from that & some other spider websites, I think all that "regular" means here is that the radii of an "orb" web tend to be approximately equally spaced -- i.e., at equal angles from neighboring radii. (In orb webs, the threads that are roughly perpendicular to the radii are described as spirals.) For reasons of structural efficiency, it seems plausible that the radii be at equal angles from their neighbors, and that the spiral be equiangular. --Dan

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Re: [math-fun] Spiders and regular webs
by Dan Asimov 25 Sep '07

25 Sep '07

Olivier Gerard wrote: << I just spotted an old article of National Geographic http://news.nationalgeographic.com/news/2004/06/0624_040624_tvspider.html titled "New" Spider Species Weaves Uncommonly Regular Webs Reading the article, it seems the spider makes regular cells and put them together to match the place it wants to cover. . . . . . . >> I did not get the impression that a spider "spider makes regular cells and put them together . . .". Rather, from that & some other spider websites, I think all that "regular" means here is that the radii of an "orb" web tend to be approximately equally spaced -- i.e., at equal angles from neighboring radii. (In orb webs, the threads that are roughly perpendicular to the radii are described as spirals.) For reasons of structural efficiency, it seems plausible that the radii be at equal angles from their neighbors, and that the spiral be equiangular. --Dan

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[math-fun] Collectives
by Bill Gosper 25 Sep '07

25 Sep '07

JZ>On Sep 23, 2007, at 8:48 AM, greenwald(a)cis.upenn.edu wrote: >> (Unrelated to math-fun: Is "herd" right? It's the first term that >> came to my mind, but what's *really* the right term for a group of >> rabbits? a flock?), JZ>According to Wikipedia (http://en.wikipedia.org/wiki/List_of_animal_names >) rabbits may be any of warren, nest, colony, bevy, bury, drove, or trace. Plague. JZ>Oddly, a quick perusal uncovers no specific word for a collection of >collective nouns. I have to assume I've just missed it. And a >collection of those words would be...? Plague. --rwg --------------------------------- Catch up on fall's hot new shows on Yahoo! TV. Watch previews, get listings, and more!

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[math-fun] Spiders and regular webs
by Olivier Gerard 25 Sep '07

25 Sep '07

Hello, I just spotted an old article of National Geographic http://news.nationalgeographic.com/news/2004/06/0624_040624_tvspider.html titled "New" Spider Species Weaves Uncommonly Regular Webs Reading the article, it seems the spider makes regular cells and put them together to match the place it wants to cover. Nowhere is an example image of such "Uncommonly Regular" web to be seen in the article itself. I have searched in vain such an image on the WWW. One of my trouble is that the only identification mentionned in the article was that the spider was of the Ochyroceratidae family and found in Peru. But there is more than 150 species in that family. Perhaps can math-fun members more versed in arachnology give me more information on the subject ? Olivier

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[math-fun] menger sponge slicing
by Thane Plambeck 25 Sep '07

25 Sep '07

via Sebastien Perez-Duarte: What happens when you slice the Menger sponge diagonally through the center? Take a look at this graphic of the sponge first: http://www.flickr.com/photos/sbprzd/1364226559/ Then look at Seb's answer here: http://flickr.com/photos/sbprzd/1432723128/ -- Thane Plambeck tplambeck(a)gmail.com http://www.plambeck.org/ehome.htm

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