April 2012 - mathfun@mailman.xmission.com

Re: [math-fun] math-fun Digest, Vol 110, Issue 3 - polygon pair tessellations of the plane
by Stuart Errol Anderson 06 Apr '12

06 Apr '12

Too easy, evidently. 9,11, 13 can produce equilateral polygon pairs of minimum size with different shapes. Looking at 15 there is at least one other way of doing it (with 3 symmetrical arms) but that pair won't tile the plane, a third tile would be needed. Stuart > > > 3/ Is the minimum m-sided equilateral polygon shape unique for given n? > > Stuart > > > ------------------------------ > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > End of math-fun Digest, Vol 110, Issue 3 > ****************************************

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Re: [math-fun] Areals of circular points
by Dan Asimov 05 Apr '12

05 Apr '12

Adam wrote: << Oh, sorry about the lack of explanation. Areals (also known as barycentric coordinates) are a way of assigning a triple of values to any point in the (projective) plane, based on some arbitrary reference triangle. The vertices of the triangle (A, B and C) are assigned the areals (1,0,0), (0,1,0) and (0,0,1). All scalar multiples are considered equivalent, so (x,y,z) = (kx,ky,kz). The centroid of the triangle (G) is (1,1,1). The line at infinity is defined by x + y + z = 0. When the projective plane is extended so that (x,y,z) are complex instead of real, it can be shown that all circles pass through two points on the line at infinity, known as 'circular points'. Indeed, a circle can be defined as any conic passing through those two points; note that the degrees of freedom are reduced from five to three. >> If I'm understanding, you are referring (in 2nd paragraph) to what topologists call CP^2, the complex projective plane. For any homogeneous polynomial P in C[x,y,z], the equation P(x,y,z) = 0 will define a surface in CP^2 (or "curve" to algebraic geometers), which is smooth under certain natural nonsingularity conditions. Assuming P(x,y,z) = 0 is smooth, then for deg(P) = d, the genus of that surface will be g = (d-1)(d-2)/2. The polynomial P(x,y,z) := x+y+z gives a smooth surface of genus g = 0, i.e., topologically the sphere S^2. CP^2 is kind of tricky to visualize, but one way to think of it is a 4-ball D^4 with an identification on its boundary bd(D^4) == S^3 == {(z,w) in C^2 : |z|^2 + |w|^2 = 1} via the Hopf map S^3 -> S^2, i.e., identifying (z,w) and (z',w') in S^3 if and only if (z',w') = (uz,uw) where u is a complex number with |u| = 1. (In fact no matter which point Q of CP^2 you start with, the points L(r) at a distance r > 0 from Q will form a 3-sphere S^3 as r increases, until it reaches the maximum distance from Q (= the diameter of CP^2), upon which L(r) will be a 2-sphere S^2. So I think what Adam (as well as maybe algebraic geometers in general) calls the "line at infinity" is topologically a 2-sphere S^2. But I'm puzzled by the comment that all circles pass through the "line at infinity" in two points, since one example of a circle (in the usual sense) in CP^2 is the set {[exp(it), 0, 1] in CP^2 : t in [0,2pi)}, which intersects x+y+z in only one point, [-1,0,1]. Or am I missing something? --Dan ________________________________________________________________________________________ It goes without saying that .

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Re: [math-fun] Areals of circular points
by Dan Asimov 05 Apr '12

05 Apr '12

Adam wrote: << The circular points at infinity have the following (unnormalised) areals: x = a^2 (2a^2 - b^2 - c^2) - (c^2 - b^2)(c^2 - b^2 + delta) y = b^2 (2b^2 - c^2 - a^2) - (a^2 - c^2)(a^2 - c^2 + delta) z = c^2 (2c^2 - a^2 - b^2) - (b^2 - a^2)(b^2 - a^2 + delta) where delta = ± sqrt(a^4 + b^4 + c^4 - a^2 b^2 - b^2 c^2 - c^2 a^2) = ± sqrt((-a-b-c)(-a+b+c)(a-b+c)(a+b-c)) = ± 4i [ABC], where [ABC] is the area of the reference triangle. The sign choice for delta determines which of the two circular points is being specified. >> Sorry if I missed something, but what is a circular point at infinity? And what is an areal? Thanks, Dan ________________________________________________________________________________________ It goes without saying that .

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[math-fun] Polygon pair tessellations of the plane
by Stuart Anderson 05 Apr '12

05 Apr '12

As everyone knows, only the triangle, square and hexagon tile the plane by themselves. However for other regular polygons it seems we can obtain tilings of the plane composed of pairs of tiles, one of which is a regular polygon and the other an equilateral polygon where tilings are edge to edge with all edges in both tiles are the same size. * A regular polygon is a polygon that has all sides equal and all interior angles equal. * An 'equilateral' polygon is a polygon that has all sides equal but not all interior angles equal. (by this definition an equilateral triangle is not an 'equilateral' polygon but a regular polygon) Definitions from; www.mathopenref.com/polygonregular.html www.mathopenref.com/polygonirregular.html 1/ What is the smallest m-sided equilateral polygon for a given n-sided regular polygon in a tiling of the plane? Tyler from http://www.superliminal.com/geometry/tyler/Tyler.htm by Melinda Green and Don Hatch is ideal for experimenting with these kinds of tiles. Using Tyler I obtained these values for n and m. Polygon pair tessellations n-sided m-sided regular equilateral polygon polygon n m graphics of tilings -- --- ------------------------------------------- 3 0 www.squaring.net/gfx/3n.png 4 0 www.squaring.net/gfx/4n.png 5 4 www.squaring.net/gfx/5n-4m.png 6 0 www.squaring.net/gfx/6n.png 7 8 www.squaring.net/gfx/7n-8m.png 8 6* www.squaring.net/gfx/8n-4m.png, www.squaring.net/gfx/8n-6m.png, 9 12 www.squaring.net/gfx/9n-12m.png 10 6 www.squaring.net/gfx/10n-6m.png 11 16 www.squaring.net/gfx/11n-16m.png 12 8* www.squaring.net/gfx/12n-3m.png, www.squaring.net/gfx/12n-8m.png 13 20 www.squaring.net/gfx/13n-20m.png 14 10 www.squaring.net/gfx/14n-10m.png 15 24 www.squaring.net/gfx/15n-24m.png 16 12 www.squaring.net/gfx/16n-12m.png 17 28 www.squaring.net/gfx/17n-28m.png 18 6 www.squaring.net/gfx/18n-6m.png 19 32 www.squaring.net/gfx/19n-32.png 20 16 www.squaring.net/gfx/20n-16.png 21 36 www.squaring.net/gfx/21n-36.png 22 18 www.squaring.net/gfx/22n-18.png 23 40 www.squaring.net/gfx/23n-40.png 24 9 www.squaring.net/gfx/24n-9.png 25 44 www.squaring.net/gfx/25n-44.png The pairs satisfy these formulae except for 8 6* and 12 8* where the formula gives 8 4 and 12 3. 4 and 3 are regular polygons, not equilateral polygons as defined above. if n = 0(mod 6), then m = n/2-3 else if n = 0(mod 2), then m = n-4 else if n = 1(mod 2), then m = 2n-6 2/ Is it always possible to tile the plane with these 2 prototiles; An n-sided regular polygon and an m-sided equilateral polygon pair for all n? It appears that in the 3 cases above the Conway criterion can always be applied to do so. 3/ Is the minimum m-sided equilateral polygon shape unique for given n? Stuart

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[math-fun] Fwd: David L. Waltz, Computer Science Pioneer, Dies at 68
by rcs＠xmission.com 04 Apr '12

04 Apr '12

[A fair number of funsters are AILab alumni, and knew Dave. -- Rich] ----- Forwarded message from hbaker1(a)pipeline.com ----- From: Henry Baker <hbaker1(a)pipeline.com> http://www.nytimes.com/2012/03/24/science/david-l-waltz-computer-science-pi… March 23, 2012 David L. Waltz, Computer Science Pioneer, Dies at 68 By JOHN MARKOFF David L. Waltz, a computer scientist whose early research in information retrieval provided the foundation for todays Internet search engines, died on Thursday in Princeton, N.J. He was 68. The cause was brain cancer, his wife, Bonnie Waltz, said. He died at the University Medical Center at Princeton. During his career as a teacher and a technologist at start-up companies as well as large corporate laboratories, Dr. Waltz made fundamental contributions to computer science in areas ranging from computer vision to machine learning. One signal achievement was the development of a basic technique that makes it possible for computers to render three-dimensional scenes accurately. As part of his Ph.D. dissertation at the Massachusetts Institute of Technology, he developed an algorithm that could extract a rich three-dimensional understanding of a scene from two-dimensional line drawings with shadows. The 3-D research was seminal in the fields of computer vision and artificial intelligence. Known as constraint propagation, the technique is now used in industry for solving problems like route scheduling, package routing and construction scheduling. At M.I.T., Dr. Waltz was taught by Marvin Minsky, a pioneer in artificial intelligence. Dr. Waltz graduated in 1972, then taught computer science at the University of Illinois at Urbana-Champaign and, later, at Brandeis University in Massachusetts. But it was as a member of a group of researchers at the Thinking Machines Corporation, in Cambridge, Mass., that Dr. Waltz made his breakthrough in information retrieval. Thinking Machines was an early maker of massive, parallel supercomputers, and by joining the company, in 1984, Dr. Waltz gained access to computers that by 80s standards held vast amounts of fast random-access memory, up to 512 megabytes. For the first time it was possible to use simple algorithms with lots and lots of data, said Brewster Kahle, a computer scientist who directs the Internet Archives and was one of the Thinking Machines researchers. Access to that database was crucial to Dr. Waltzs development of a technique known as memory, or case based, reasoning. It revolutionized the way computers recognized characters, words, images and later, even voices. Before, a computer had to follow a set of programmed rules to arrive at recognition (its an i if theres a dot, for example). Now it could comb through its vast memory and deduce what the image was by comparing it to what had been stored there. The technique transformed the field of artificial intelligence and also greatly advanced voice recognition and machine vision technology. And it led directly to the big data and data-science approaches that are essential tools for search engines, allowing them to sift through large collections of information to improve accuracy and relevance. He was a real pioneer, said Peter Norvig, Googles director of research. The two main changes that got us modern A.I. were probabilistic reasoning and using memory rather than rules. I dont know if Larry and Sergey read his papers directly, he added, referring to Googles founders, Larry Page and Sergey Brin, but the idea, filtered through however many people, was certainly a key. While at the University of Illinois, Dr. Waltz turned to the field of natural language understanding, a component of artificial intelligence involving the interpretation of language. With support from the Office of Naval Research, he built a question-answering system called Planes and explored the use of neural networks in language processing. In another early project, a Thinking Machines group led by Dr. Waltz designed an information retrieval system that made it possible for a remote user to gain access to a supercomputer and then be able to search through large volumes of documents. The system, known as Wide Area Information Server, or WAIS, and designed in cooperation with the Dow Jones Corporation, Apple Computer and KPMG Peat Marwick, was not the first information retrieval system. But it was innovative in enabling the user to uncover connections between seemingly disparate documents. For example, the WAIS system was able to give an early warning of the Chernobyl disaster in 1986 after it discovered a report of an abnormal radiation reading in Scandinavia, according to W. Daniel Hillis, the co-founder of Thinking Machines. WAIS also introduced techniques to narrow a document search. It was followed by other search systems, like Veronica, Gopher and Archie, which predated the search engines offered today by Google, Microsoft, Yahoo and other companies. After leaving Thinking Machines in 1993, Dr. Waltz joined the NEC Research Institute in Princeton, where he was president from 2000 to 2002. He left to help create the Center for Computational Learning Systems at Columbia, where he was director. The center has worked with Con Edison of New York in developing systems that can predict power failures and thus enhance maintenance of the electric power grid. Researchers there are also working on creating a computer-based system to give people with epilepsy early warnings of seizures. The technique involves mining data generated by electrodes implanted in patients. Dr. Waltz earlier was instrumental in establishing interdisciplinary research centers: the Beckman Institute at the University of Illinois, and the Volen National Center for Complex Systems at Brandeis. David Leigh Waltz was born in Boston on May 28, 1943, to Maynard C. Waltz and the former Lubov Leonovich. His father, a physicist, worked at M.I.T.s Radiation Laboratory during World War II and later at Bell Labs. Dr. Waltz obtained both undergraduate and graduate degrees at M.I.T. in electrical engineering. He lived in Princeton. Besides his wife, he is survived by a brother, Peter; a son, Jeremy; a daughter, Vanessa Waltz, and a granddaughter. ----- End forwarded message -----

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[math-fun] Areals of circular points
by Adam P. Goucher 04 Apr '12

04 Apr '12

The circular points at infinity have the following (unnormalised) areals: x = a^2 (2a^2 - b^2 - c^2) - (c^2 - b^2)(c^2 - b^2 + delta) y = b^2 (2b^2 - c^2 - a^2) - (a^2 - c^2)(a^2 - c^2 + delta) z = c^2 (2c^2 - a^2 - b^2) - (b^2 - a^2)(b^2 - a^2 + delta) where delta = Â± sqrt(a^4 + b^4 + c^4 - a^2 b^2 - b^2 c^2 - c^2 a^2) = Â± sqrt((-a-b-c)(-a+b+c)(a-b+c)(a+b-c)) = Â± 4i [ABC], where [ABC] is the area of the reference triangle. The sign choice for delta determines which of the two circular points is being specified. Sincerely, Adam P. Goucher

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[math-fun] Bessel at High Precision?
by Bill Gosper 04 Apr '12

04 Apr '12

*Quad>* Hey all. Does anyone have any ideas on computing Bessel functions to high precision (over the complex plane)? Googling gives [1] and [2]. Sorry, [1] is behind a paywall. I am wondering if one of you has a secret magical sum or product that just magically works well for high precision computation. If not, maybe we can develop some strategies! Thanks, Robert Smith [1] dl.acm.org/citation.cfm?id=1466254 [2] web.cs.dal.ca/~jborwein/bessel.pdf *------ You're of course welcome to my Macsyma code. At the time, I used (a tiny fraction of) Watson's Treatise, but now there's dlmf*.nist.gov/10 . My only novelty was using matrix products to do the series--merely a small constant speedup. I may not even have gotten around to handling the asymptotics. Do you suspect Mathematica knows something we don't? --rwg

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[math-fun] Amazon's new delivery service
by Henry Baker 02 Apr '12

02 Apr '12

And people gave _me_ grief over my idea to deliver UPS/Fedex packages to people via GPS-guided "smart" bombs right into their back or front yards... http://aws.typepad.com/aws/2012/04/new-amazon-fresh-servers.html

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Re: [math-fun] Asperger syndrome, Autism, & math talent
by Bill Gosper 02 Apr '12

02 Apr '12

E.g., hyperlexia <http://en.wikipedia.org/wiki/Hyperlexia> and Jake: Math prodigy proud of his autism <http://www.youtube.com/watch?v=OR36jrx_L44> --rwg (NOT a fan of 60 Minutes) -------- On Monday 26 March 2012 20:19:52 Warren Smith wrote: > Supposedly, Asperger syndrome (which allegedly no longer "exists," > having been merged into the "Autism spectrum"), is linked to > "mathematical talent." I thought that was already pretty much the conventional wisdom. -- g[areth]

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[math-fun] Hamiltonian paths
by Fred lunnon 02 Apr '12

02 Apr '12

Would somebody who knows a little more graph theory than I do please take a look at http://en.wikipedia.org/wiki/Hamiltonian_path and give an opinion? The section entitled "Bondy–Chvátal theorem" looks to be seriously mangled; in particular, flatly contradicted by the accompanying circuit on a dodecahedron; later theorems appear to have "smaller" substituted for "greater". Has a troll been at work, I wonder? Fred Lunnon

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