June 2011 - mathfun@mailman.xmission.com

[math-fun] Slanted coverage
by Bill Gosper 07 Jul '11

07 Jul '11

In those ubiquitous little tables of area and volume formulae in handbooks and textbooks, there's often a picture of a skewed cone next to the right cone to emphasize that they have the same volume formula. Did you ever wonder why there's never the skewed cone area formula? For the special case with the apex directly above a point on the circumference, I get pi h r 3F2[-1/2,1/4,3/4; 1/2, 1; -4 r^2/h^2], which doesn't even converge for h < 2 r. The area element for the general x,y,z parametric surface is just Out[85]= (1,0) (0,1) Norm[Cross[({x, y, z}) , ({x, y, z}) ]] (I don't recall this from 18.03.) E.g., for the torus with radii r and R, In[116]:= % /. {x :> (Cos[#2]*(R + r*Sin[#1]) &), y :> (Sin[#2]*(R + r*Sin[#1]) &), z :> (r*Cos[#1] &)} Out[116]= Norm[Cross[({Cos[#2] (R + r Sin[#1]) & , Sin[#2] (R + r Sin[#1]) & , (1,0) r Cos[#1] & }) , ({Cos[#2] (R + r Sin[#1]) & , (0,1) Sin[#2] (R + r Sin[#1]) & , r Cos[#1] & }) ]] In[117]:= % /. f_[L_List] :> f /@ L /. Abs[w_] -> w Out[117]= Sqrt[Power[-(-r Sin[#1] & ) (Cos[#2] (R + r Sin[#1]) & ) + (0 & ) (r Cos[#1] Sin[#2] & ), 2] + Power[-(0 & ) (r Cos[#1] Cos[#2] & ) + (-r Sin[#1] & ) (-(R + r Sin[#1]) Sin[#2] & ), 2] + Power[(r Cos[#1] Cos[#2] & ) (Cos[#2] (R + r Sin[#1]) & ) - (r Cos[#1] Sin[#2] & ) (-(R + r Sin[#1]) Sin[#2] & ), 2]] In[119]:= % /. (f_ &) :> (f &)[t, u] Out[119]= 2 2 2 2 Sqrt[r Cos[u] Sin[t] (R + r Sin[t]) + 2 2 2 2 r Sin[t] (R + r Sin[t]) Sin[u] + 2 2 2 (r Cos[t] Cos[u] (R + r Sin[t]) + r Cos[t] (R + r Sin[t]) Sin[u] ) ] In[120]:= FullSimplify[%] Out[120]= 2 2 Sqrt[r (R + r Sin[t]) ] In[125]:= Assuming[R > r > 0, Integrate[%120, {t, 0, 2*Pi}, {u, 0, 2*Pi}]] Out[125]= 2 4 Pi r R Did Archimedes know this? --rwg

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Re: [math-fun] Interesting orbital facts
by Tom Duff 30 Jun '11

30 Jun '11

"Adam P. Goucher" <apgoucher(a)gmx.com> writes: > I wonder what happens if we have a solar system in the real projective > plane, where all conics are indistinguishable? Then there is no definition > of 'escaping from orbit'. Designing a set of physical laws invariant under > projective transform would be difficult, though. I'm pretty sure there's no physics in projective spaces. Certainly there's no native notion of distance since there are no parallel lines. So you can't gin up an inverse square law in any obvious way, nor even notions of velocity and acceleration. -- Tom Duff. Crop it, flop it or drop it.

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Re: [math-fun] Exact N-body simulations?
by Phil Carmody 30 Jun '11

30 Jun '11

From: Veit Elser <ve10(a)cornell.edu> > This reminds me of a game we played in high school on graph > paper called "racetrack". ... > (2) The change in the displacement vector of any > player may change, from one move to the next, by at most > one unit horizontally *or* vertically. From: Hans Havermann <pxp(a)rogers.com> > http://chesswanks.com/txt/RaceTrack.html I notice that the two are variants, at the point emphasised above. I'm not sure which version would be harder to find an optimal strategy in. Phil

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[math-fun] tennis and randomness
by James Propp 30 Jun '11

30 Jun '11

Someone (I've forgotten who, alas!) mentioned to me a few weeks ago that researchers have studied patterns in tennis service, and have found that when it comes to deciding where to serve, top-ranked tennis players do a better job of "acting like a coin" than lower-ranked players. Can anyone provide information on this? Thanks! Jim Propp

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[math-fun] Interesting orbital facts
by Henry Baker 30 Jun '11

30 Jun '11

Earth's velocity in orbit ~ 30 km/sec ~ 10^-4 c http://en.wikipedia.org/wiki/Earth%27s_orbit If we increase the speed, the orbit becomes eccentric (e>0): v/vcircular = sqrt(1+e) But if the eccentricity e equals or exceeds 1, then the Earth escapes from orbit. So, the maximum speed for such an Earth is sqrt(2)*30 km/sec ~ 42.43 km/sec ~ 1/7000 c (so we shouldn't need Einstein). Now, the same calculations apply to an object of _any_ mass, so long as it isn't large relative to the Sun's mass. http://www.eso.org/public/outreach/eduoff/aol/market/experiments/hb/ In particular, we can have a highly eccentric comet that comes into the Earth's orbit at its closest point; we know that the speed of this comet at its closest point will lie between 30 and 42.43 km/sec. Note that if our comet is orbiting in the same plane as the Earth, and in the same direction, any potential collision with the Earth will occur at a significant speed, but far less than if it had a retrograde orbit, or an orbit perpendicular to the Earth's orbital plane; or a hyperbolic (extra-solar system) "orbit". This highly eccentric comet can have just about any semi-major axis that we choose; let's choose a semi-major axis of ~ 10,000 AU; this will give the comet an orbital period of ~ 1 million years. Thus, our comet may have revolved around the Sun ~ 4600 times during the lifetime of the solar system. This is a very small number, relative to the current upper limit estimates of 150,000,000 years (150 million Earth orbits) for the main body of the solar system to "relax" into its current configuration, during which our comet would have entered the inner solar system only 150 times. We also note that our comet (or some like it) may have run out of luck 65.5 million years ago (4600 minus 65 revolutions) and possibly caused the K-T event. http://en.wikipedia.org/wiki/Cretaceous%E2%80%93Tertiary_extinction_event Such comets need not have come from outside the solar system -- they might very well have been produced by close encounters of early small planets/planetoids that nearly got ejected from the solar system. Current N-body models of the early solar system seem to ignore bodies that have been "ejected"; once they achieve certain speeds and/or distances, they are ignored for future time-steps of the models. While ignoring them is appropriate for figuring out what happens to what's left in the solar system, it isn't when you are interested in what might happen to such bodies that didn't get completely ejected.

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[math-fun] Algorithms for Solving Rubik's Cubes
by Hans Havermann 30 Jun '11

30 Jun '11

"Erik Demaine, an associate professor of computer science and engineering at MIT; his father, Martin Demaine, a visiting scientist at MIT’s Computer Science and Artificial Intelligence Laboratory; graduate student Sarah Eisenstat; Anna Lubiw, who was Demaine’s PhD thesis adviser at the University of Waterloo; and Tufts graduate student Andrew Winslow showed that the maximum number of moves required to solve a Rubik’s cube with N squares per row is proportional to N^2/log N. 'That that’s the answer, and not N^2, is a surprising thing', Demaine says." The MIT-news backgrounder titled 'The math of the Rubik’s cube' is here: http://web.mit.edu/newsoffice/2011/rubiks-cube-0629.html The to-be-presented paper (in pdf-format) is here: http://arXiv.org/pdf/1106.5736v1

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[math-fun] Exact N-body simulations?
by Henry Baker 29 Jun '11

29 Jun '11

All this talk of Gaussian integers reminded me of a problem that I'd dearly love to see solved. Consider N bodies, each with integer masses. Start them in initial positions in a 2D (for the moment, there's plenty of complexity in 2D) which are on integer coordinates. The attractive Newtonian gravitation force is proportional to m*m'/(dx^2+dy^2), so this force is rational. We need to compute new positions for each of the bodies in such a way that all the conservation laws are preserved exactly (if this is possible), yet the number of bits required doesn't grow too quickly (see below). We need to preserve the total energy, we need to preserve momentum & angular momentum, & we might like to preserve information (the system is reversible). Note that we don't necessarily require that delta-t be constant, although if it isn't, we may need to have another counter to represent the "clock" -- i.e., the global time. I don't know how to do this even in the 2-body case, but the Minsky circle hack may provide some guidance -- if we could get the two bodies to each draw a Minsky circle while revolving around one another, then we'd have a good start on the problem. In addition to solving the problem of finite arithmetic, we also need to figure out what happens in collisions. Taking a cue from Fredkin, we should probably have 100% elastic collisions in order to ensure reversibility. How does this square with PoincarÃ© ? Well, I'm assuming that my N-body problem for finite N is Turing complete, so with enough bodies and enough "tape" (empty space), we should be able to compute any computable function. Minsky already provides one potential representation with his "counter machines". So we want a reasonably "efficient" Turing Machine (or Counter Machine) simulation so that we aren't wasting more than a fixed fraction of the bits required to represent the state (N pairs of integers or N gaussian integers). Has anyone attempted this sort of exact simulation before?

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[math-fun] Continuous Urinal Protocol
by Adam P. Goucher 28 Jun '11

28 Jun '11

You may have encountered the (discrete) Urinal Protocol on the journal section of the xkcd website: http://blog.xkcd.com/2009/09/02/urinal-protocol-vulnerability/ I propose the following alternative, the Continuous Urinal Protocol: We have a unit disk, and a small real number d. A number of points are placed on the disk according to the following rules: 1. The first point is typically placed on the circumference of the disk. 2. Subsequent points are placed so as to maximise the minimum distance to any other point. (To maintain the bathroom analogy, this could be a circular shower room.) The circle fills up like so: * Point 1 is placed on the circumference (Rule 1); * Point 2 is placed diametrically opposite; * Points 3 and 4 are placed to complete an inscribed square; * Point 5 is placed in the centre, forming a quincunx pattern; * Points 6, 7, 8 and 9 are placed at the vertices of a square, such that a regular octagon is described. Beyond this point, some subset of the circumcentres of the eight triangles are filled up (3 or 4, depending on how the triangles are chosen). So, the resulting pattern is (very slightly) non-deterministic, as choices between two equally optimal points can influence the entire configuration. A stricter variant is to order the distances to each point in ascending order (to form an n-tuple), then choose the lexicographically greatest value. (i.e., if the closest points are tied, consider the next closest, ad infinitum.) It is left as an exercise to the reader to demonstrate that this will necessarily result in patterns with order-4 rotational symmetry for all n = 4k + 1, k > 0. [Hint: Use Luke Betts' Theorem.] Can we prove that this procedure is necessarily deterministic, up to symmetry? Sincerely, Adam P. Goucher

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[math-fun] One Math Museum, Many Variables
by Henry Baker 28 Jun '11

28 Jun '11

FYI -- http://www.nytimes.com/2011/06/28/science/28math.html June 27, 2011 One Math Museum, Many Variables By KENNETH CHANG For everyone who finds mathematics incomprehensible, boring, pointless, or all of the above, Glen Whitney wants to prove you wrong. He believes that tens of thousands of visitors will flock to his Museum of Mathematics, to open in Manhattan next year, and leave invigorated about geometry, numbers and many more mathematical notions. ÂWe want to expose the breadth and the beauty of mathematics,Â said Mr. Whitney, a former math professor who parlayed his quantitative skills into a job at a Long Island hedge fund. He quit in late 2008 with connections to deep pockets and a quest to make math fun and cool. Two years ago, he and his team built a carnival-like traveling exhibit called the Math Midway, a proof-of-concept for the coming museum. It includes a tricycle with square wheels of different sizes that visitors can ride smoothly around a circular path ridged like a flowerÂs petals. An accompanying sign explains why: The undulating circular surface rises and falls exactly to offset the odd shape of the wheels, so that the tricycleÂs axles Â and the rider Â remain at the same height as they move. Mr. Whitney hopes that colorful, interactive props will help his cause. ÂIf we just pluck people in the street Â ÂWhat adjectives would you use to describe math?Â Â very few of them would say, Âbeautiful,Â Â Mr. Whitney said. His vision has enticed large contributions. The museum, which will be at 11 East 26th Street, has raised $22 million, including $2 million from Google and a lot from individual donors (yes, thereÂs some hedge fund money in there). It remains to be seen whether a math museum can succeed. There are currently zero math museums in the United States, and the one small one that did exist, on Long Island, closed in 2006. There are plenty of science museums that cover math topics, but Mr. WhitneyÂs museum, nicknamed MoMath, will be devoid of dinosaurs and planetarium shows and will instead focus on the abstract. ÂTheyÂre a dedicated bunch of idealists,Â Sylvain E. Cappell, a New York University mathematician on the museum advisory council, said of Mr. Whitney and his staff. Without a museum yet, Mr. Whitney periodically gives walking tours to point out the mathematical wonders that can be seen around Manhattan. At 42, he exudes a boyish, geeky enthusiasm as he talks about how the branches of ginkgo trees intersect at right angles more often than those of other trees, or points out that the bolts that open and close New York fire hydrants are pentagonal, rather than the usual six-sided variety. Three years ago, he was working with algorithms at Renaissance Technologies, a private investment firm that uses mathematical models to figure out where to put money. But after a decade there, he was looking for a new career path with a Âmore direct socially redeeming value,Â he said. Then, he heard that the math museum on Long Island, Goudreau, had closed. He started thinking that there should be a math museum and that he should be the one to build it. ÂI really felt that I found my calling,Â Mr. Whitney said. ÂI donÂt mean to be grandiose, but it was something that felt like it really fit with my lifetime of experiences and abilities and likes and so on.Â Under his vision, MoMath will be one small way to bolster mathematics education in the United States. For years, American students have performed in the middle range on international comparisons of math skills, and an oft-heard worry is that the United States might lose its technological prowess. While Mr. Whitney cites these dynamics as a reason for his quest, he is also a realist. Yes, the museum could serve as an intellectual catalyst and teaching resource, but it alone is not going to raise math scores. ÂIÂm certainly not holding my breath for that,Â he said. Rather, he said, the museumÂs mission is to shape cultural attitudes and dispel the bad rap that most people give math. ÂItÂs the only field you can go to a cocktail party and talk to people with pride about how lousy you are,Â Mr. Whitney said. He hopes the museum can inspire at least a few to plunge into math more deeply. He imagines breaking down a piece of cutting-edge math research into pieces that enthusiastic visitors could help solve. ÂWe want to be a place where that spark can ignite,Â he said. For Mr. Whitney, the spark came after a broken collarbone. When he was 14, he attended a math camp at Ohio State University Â he saw it as a chance to get away from home for the summer, he said, not to learn math, a tedious subject that he found easy. During a soccer game, he collided with someone bigger, leaving him injured. With nothing else to do, he looked over the problem sets he had been ignoring. The problems were different from the ones from school, spanning different branches of math and highlighting the connections among them. ÂI fell in love with mathematics that summer, and IÂve had a lifetime love affair with it ever since,Â he said. After majoring in math at Harvard and earning a Ph.D. at the University of California, Los Angeles, he taught at the University of Michigan before joining Renaissance Technologies. He commutes to a modest office in Midtown scattered with math puzzles and sculptures, where he and a team of about 20 brainstorm about exhibits for the museum, which right now is 19,000 empty square feet. One idea is a large cube with square holes punched through each side, a structure known as a Menger sponge. When a visitor pulls the cube apart diagonally, the holes turn into six-sided stars. ÂItÂs like a Âgosh, thatÂs really coolÂ kind of emotion people have,Â said George Hart, the museumÂs chief of content. ÂItÂs a very nice example of how mathematics can give you these big surprises.Â The opening is more than a year away, but Mr. Whitney is already dreaming bigger: a larger museum, a palpable cultural impact. ÂThere are all sorts of myths about mathematics out there,Â he said Â math is hard, math is boring, math is for boys, math doesnÂt matter in real life. ÂAll these are cultural myths that we want to blow apart.Â

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[math-fun] Regularization of gravitational 2-body problem ??
by Henry Baker 27 Jun '11

27 Jun '11

A number of people (including Newton himself) have noticed that Newton's elliptical solution for orbits in an inverse square central force field (F ~ 1/r^2) is related to elliptical solutions for orbits in a linear central force field (F ~ r) -- i.e., a simple harmonic oscillator following Hooke's Law. This connection is most elegantly demonstrated when the orbit is in the complex plane, and the harmonic orbit is the simple _square_ of the inverse square law orbit [Needham1993] (equivalently, the inverse square orbit is (one of) the "square root(s)" of the harmonic orbit). This "trick" was also used by Levi-Civita [1920] to "regularize" the solutions to the 2-body problem to gracefully handle the case where the bodies would collide. There was apparently no 3-dimensional version of the Levi-Civita "trick", but Kustaanheimo and Stiefel [KS1965] (also [Waldvogel1972]) found a 4-dimensional version, which has been reinterpreted using quaternions by a number of people, including Vrbik [Vrbik2003] and Saha [Saha2009]. My problem: isn't the extension to the quaternion case completely trivial? After all, the 2-body system is inherently planar, so regardless of the orientation of the plane in 3-space, a simple rigid coordinate rotation should bring the complex plane into alignment with the plane of the 2-body problem. Thus, as Hamilton himself said many times (but not in these words!) in his Elements of Quaternions, the plane spanned by 1 and a "pure" unit quaternion q (pure <=> scalar part=0) is isomorphic to the complex plane, and hence any two quaternions lying in this (1,q) plane commute with one another, so the Levi-Civita algebra all carries over without having to worry about non-commutativity. I realize that the quaternion mapping used by both Vrbik and Saha may not work this way, but the trivial mapping suggested above should work just fine. What am I missing here? (Aarseth's NBODYx codes utilize a KS mapping locally for 2 bodies who are perilously close, so these kinds of mappings are part of the inner workings of some of the best n-body simulation codes in existence.) ----------------- (I was able to find all of these papers on the web w/o having to log in, except for the Levi-Civita and KS papers.) Saha, Prasenjit. "Interpreting the Kustaanheimo-Stiefel transform in gravitational dynamics. Arxiv 0803.4441v3 [astro-ph] 21 Aug 2009, pp1-5. Vrbik, Jan. "A novel solution to Kepler's problem". Eur. J. Phys 24 (2003), 575-583. Aarseth, Sverre J. "From NBODY1 to NBODY6: The Growth of an Industry". Astron. Soc. of the Pacific 111:1333-1346, Nov. 1999. Needham, Tristan. "Newton and the Transmutation of Force". Amer. Math. Monthly 100, 2 (Feb. 1993), 119-137. Waldvogel, Joerg. "A New Regularization of the Planar Problem of Three Bodies". Celestial Mechanics 6 (1972), 221-231. Kustaanheimo, P., Stiefel, E. J. Reine Angew. Math., 218, 204, 1965. Levi-Civita, T. Acta Math., 42, 99, 1920.

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