November 2012 - mathfun@mailman.xmission.com

Re: [math-fun] when and why was it agreed that > 0.999...=1?
by Henry Baker 25 Nov '12

25 Nov '12

"Carrying On with Infinite Decimals" is an interesting read. As I was reading it, I thought about acyclic ("FIR" ?) _circuits_ for performing addition, which in real life are more-or-less asynchronous; a circuit element at depth N from a leaf has to wait until all of its inputs have stabilized before becoming stable itself. Within this circuit framework, different methods of carry propagation can be discussed, including the traditional right-to-left "ripple" carry method taught in grade school, and the "carry-lookahead" method taught in Digital Logic 101. Interestingly, I believe that the "carry lookahead" method solves the problem of these infinitely rippling carries. See also the recursive binary tree format for positional notation; a number representation is no longer a _sequence_ but a _tree_, where the sequence of leaves of the tree (the "fringe") is the traditional digit sequence. Vuillemin, Jean. "Efficient Data Structure and Algorithms for Sparse Integers, Sets and Predicates". IEEE Symp. on Computer Arith., Portland, OR, June, 2009. ISBN:: 978-0-7695-3670-5. http://www.ac.usc.es/arith19/sites/default/files/3670a007-session1-paper1.p… When this binary tree representation is amended with the "quadtree hack", wherein any subtree all of whose digits are identical is replaced with that single digit, carry-lookahead addition becomes _the_ obvious/canonical method for addition. See the Lisp code at the following link for more details. http://home.pipeline.com/~hbaker1/rb.lsp --- Some infinite acyclic circuits can be modeled by circuits with cycles (e.g., "feedback", or the completely opaque "infinite impulse response"/IIR, circuits). Perhaps you can expand your paper along this line ? At 06:36 PM 11/18/2012, James Propp wrote: >If one is willing to live without subtraction, then there is a consistent >mathematical structure in which .9999... is different from 1; see my >preprint "Carrying On with Infinite Decimals" ( >http://jamespropp.org/carrying.pdf) which my students presented at a >recent Gathering for Gardner, and which I will eventually get around to >publishing in one of the G4G books. (If any of the editors of the series >are reading this, please feel free to gently hassle me about this, now or >at some future time.)

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[math-fun] mathematics of clothing design (patterns) ??
by Henry Baker 24 Nov '12

24 Nov '12

Humans & their predecessors have been sewing clothing from cut-out pieces of hide or cloth for perhaps 250k - 500k years. There must be some part of mathematics that deals with pattern parts & how they sew together to form 3D surfaces. Most clothing materials are flexible, but more-or-less flat. The only way to generate a 3D curved surface is by sewing the parts together in complex ways. I'm particularly interesting in trousers -- especially those for athletic endeavors (biking, running, rowing). The idea is to develop clothing that doesn't bind, but also has seams in places that aren't going to rub your skin raw. Any of you who have ridden bicycles in "street clothes" will know precisely what I mean. While an undergrad at MIT, I used to ride my bike upwards of 7 miles each way in Levis jeans, which used to have _rivets_, including one in a particularly sensitive location, which you could also appreciate by standing too near a campfire. The only saving grace was that I rode most of the way standing up. I've recently had difficulty finding athletic clothing that _doesn't_ have seams in the wrong places. I'm wondering if the problem is mathematically insoluble -- perhaps the design of trousers requires seams that end up in the wrong places. Can anyone on math-fun provide pointers to the appropriate mathematical literature? Thanks in advance.

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[math-fun] Can someone "shew that"
by Bill Gosper 24 Nov '12

24 Nov '12

EllipticPi[1/3 (7 + 4 I √2), 1/81 (17 + 56 I √2)] == 1/9 (-7 + 4 I √2) EllipticPi[1/3 (7 - 4 I √2),1/81 (17 - 56 I √2)] and EllipticPi[1/27 (7 + 4 I √2), 1/81 (17 + 56 I √2)] == 1/108 (81 I √2 EllipticK[1/9] - 27 I √2 EllipticK[1/81 (17 + 56 I √2)] + 27 (4 - I √2) EllipticPi[1/27 (7 - 4 I √2), 1/81 (17 - 56 I √2)] + (-64 + 25 I √2) EllipticPi[ 1/3 (7 - 4 I √2), 1/81 (17 - 56 I √2)]) ?--rwg

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Re: [math-fun] Triple denests while inner fails
by Bill Gosper 24 Nov '12

24 Nov '12

just out of curiosity, is there a point of de-nesting if it leads to more square roots? bob baillie --- Warut Roonguthai wrote:> Another example, sqrt(3+sqrt(5+2*sqrt(3))) => 1/2 + sqrt(3)/2 + sqrt((4-sqrt(3))/2).> > If (a^2-b)^2-c is a perfect square,> then sqrt(a+sqrt(b+sqrt(c))) can be (partially) denested.> > Warut> > > On Sat, Nov 24, 2012 at 4:22 PM, Bill Gosper <billgosper(a)gmail.com <https://ma.sdf.org/sm/src/compose.php?send_to=billgosper%40gmail.com>> wrote:>> It's clear that √(a+√(b+√c)) can't completely denest unless √(b+√c) does.>> Failing that, I'd've expected nothing better than √(b+√c) + √d + √e + ...,>> i.e., the √(b+√c) would act atomically, like √integer. But>>>> Sqrt[1/2 - Sqrt[-2 + Sqrt[5]]] ==>> 1/2 Sqrt[-1 + Sqrt[5]] + 1/4 (Sqrt[2] - Sqrt[10])>> --rwg First off, denestings (of not just square roots) are interesting, mysterious, pretty, and (Algebra I) accessible. Next, Bruce Berndt once wrote me "In the several years that I devoted to calculating Ramanujan's class invariants and singular moduli, I often needed to denest radicals. Here is a very mild contribution that I made in this direction. www.math.uiuc.edu/~berndt/articles/*radicals*.pdf " Also, denesting sums of radicals can permit additional simplification. What is Sqrt[1/2 - Sqrt[-2 + Sqrt[5]]] - 1/2 Sqrt[-1 + Sqrt[5]] ? Also, denesting radicals inside transcendental functions can permit the invocation of addition formulas. Also, denested expressions are canonical, promoting recognition. Minimal polynomial notation is even more canonical, but conceals essential relationships. But for evaluation, it usually beats both nested and unnested radicals. Finally, denesting can re-express surds with large coefficients as powers of surds with small coefficients: (Sqrt[3] - Sqrt[2])^16 has more square roots than 46099201 - 18819920 Sqrt[6]. Which do you like? --rwg Subj: Re: Boy do we need a good radical simplifier On Thu, Jan 6, 2011 at 11:19 PM, Bill Gosper <billgosper(a)gmail.com> wrote: > (Apologies if my suspicion of having already sent this is correct.) > http://en.wikipedia.org/wiki/Theta_function gives several slightly > unsimplified valuations of theta_3(0,e^-(n pi)), and then > > EllipticTheta[3, 0, E^-(6*\[Pi])] -> > 3^(5/8)*(3*Sqrt[2] + 3^(5/4) + 2*Sqrt[3] - 3^(3/4) + 12^(3/4) - > 4)^(1/3)/6/(1 + Sqrt[6] - Sqrt[2] - Sqrt[3])^(1/6)*\[Pi]^(1/4)/ > Gamma[3/4] > > a cube root over a 6th root instead of just a square root: > (Sqrt[1+Sqrt[2]+3^(1/4)+Sqrt[3]]*Pi^(1/4))/(2^(3/4)*3^(3/8)*Gamma[3/4]) > i.e. > 1/4 1/4 > -6 Pi Sqrt[1 + Sqrt[2] + 3 + Sqrt[3]] Pi > EllipticTheta[3, 0, E ] -> ---------------------------------------- > 3/4 3/8 3 > 2 3 Gamma[-] > 4 > This is a respectable approximation to 1: > N[%, 22] > > 1.000000013024824272160 -> 1.000000013024824272160 > > Mathematica has the requisite machinery, but requires constant manual > struggling > against its thuggish nesting of radicals and reversion to Root notation. > It needs > to distinguish Radical Number from Algebraic Number. > > It also needs some way to decache generated polynomials, which seem to be > causing > a storage leak. > --rwg >

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[math-fun] Triple denests while inner fails
by Bill Gosper 24 Nov '12

24 Nov '12

It's clear that √(a+√(b+√c)) can't completely denest unless √(b+√c) does. Failing that, I'd've expected nothing better than √(b+√c) + √d + √e + ..., i.e., the √(b+√c) would act atomically, like √integer. But Sqrt[1/2 - Sqrt[-2 + Sqrt[5]]] == 1/2 Sqrt[-1 + Sqrt[5]] + 1/4 (Sqrt[2] - Sqrt[10]) --rwg

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[math-fun] Can anyone crack the WWII pigeon code?
by Stuart Anderson 24 Nov '12

24 Nov '12

http://www.bbc.co.uk/news/uk-20456782 Britain's top code-breakers say they are stumped by a secret code found on the leg of a dead pigeon. The remains of the bird were found in a chimney in Surrey with a message from World War II attached. Experts at the intelligence agency GCHQ have been struggling to decipher the message since they were provided with it a few weeks ago. They say it may be impossible to decode it without more information - some of which could come from the public. The message was discovered by David Martin when he was renovating the chimney of his house in Surrey. Among the rubbish, he found parts of a dead pigeon including a leg. Attached to the leg was a red canister. Inside the canister was a thin piece of paper with the words "Pigeon Service" at the top and 27 handwritten blocks of code. AOAKN HVPKD FNFJW YIDDC RQXSR DJHFP GOVFN MIAPX PABUZ WYYNP CMPNW HJRZH NLXKG MEMKK ONOIB AKEEQ WAOTA RBQRH JRZCQ TPZEH LKXGH RGGHT DJOFM FNKTQ KLDTS FQIRW AOAKN 27 1525/6 ------------------------------------------------------------ Speculation is it may be encrypted with a one time code pad. Stuart

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[math-fun] brachistochrone for rolling ball
by Bill Gosper 22 Nov '12

22 Nov '12

Warren S> The underlying formula for a sliding frictionless bead was V = sqrt(2*h*g) where g=gravity acceleration, h=vertical height dropped so far, V=speed. For a ball (or other rotationally symmetric) rolling object with zero slip, a constant fraction of the energy is in rotational, and a constant fraction in translational, energy. Hence V=const*sqrt(2*h*g). This is the same formula, just with an altered value of g. Therefore, I conclude the brachistochrone, isochrone, etc will have unaltered (still cycloid) shape, although of course you want for the center of mass of the ball to move along the cycloid, not the contact point of ball with track -- hence track would need to be a "1-radius-expanded" version of cycloid. ------- Aha, so Galileo was right in the limit of large balls! You've preempted Veit's Mechanics homework. --rwg It is unnecessary to make balls have foamy outside and dense inside to reduce their moment of inertia. Any MoI is fine. Galileo was not aware of brachist. or iso property of cycloid, but he was aware of, or at least conjectured, cycloid as best possible shape for thin beam arch (constant beam width) -- which I have discussed here, the cycloid has uniform compressive stress but not uniform bending stress. Apparently Galileo invented the cycloid. The notion of "uniform stress" is something that can be defined without needing calculus -- probably for somebody as ingenious as Galileo, pure Euclidean geometry would suffice. (Archimedes similarly computed area & volume of sphere without needing calculus.)

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[math-fun] Galileo and the cycloid
by Bill Gosper 22 Nov '12

22 Nov '12

I find unanimity on the web that Galileo thought the semicircle was tautochronous, i.e., that the pendulum period was amplitude-invariant. I find this incredible. He was a great experimenter. One site excuses his error by quoting some small % difference between low and high amplitudes. But I calculate that for a string length of 1.00544 ft = g/32 sec^2, the small amplitude period is 2 π/√32 = 1.107 sec, whereas the theoretical period for a full semicircular swing is ¼!^2 √(8/π) = 1.311 sec. A related mystery is why Galileo was fascinated by the cycloid. He *named* it. He couldn't compute its area, so he made a metal one and weighed it. Could he have at least suspected it was the tautochrone? Neil points out that this image http://www.scitechantiques.com/cycloidhtml/images/chrisy_cycloid_SML.jpg from a slightly dodgy website, is probably photoshopped. The page (with a broken video) claims that this beautiful piece of cabinetry admirably demonstrates tautochronicity. How could it, with rolling spheres? Their centers of mass do not follow a cycloid, and they have angular momentum. An interesting problem would be to manufacture heavy balls with very low moment of inertia. Neil suggests instead a lightweight cylinder with an osmium spindle. To reduce friction, maybe taper the ends slightly and run it on a pair of thin rails. A challenge to Veits' Mechanics geeks might be to fudge the museum cycloid to make a true tautochrone curve for a sphere of specific mass and diameter. Would the the mass actually matter? Hard to see how, given the Leaning Tower results. This exhibit would convince all the average kids, and confuse the really smart ones. How would George Hart handle this moral dilemma? --rwg

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Re: [math-fun] poisoned wine bottles - a greedy search
by W. Edwin Clark 22 Nov '12

22 Nov '12

Found by a slightly randomized greedy search algorithm after about a million tries, the following unique union code for n = 10 of size 20: (Not containing {} and it cannot be added.) {2}, {3, 10}, {4, 6}, {7, 9}, {1, 3, 5}, {1, 4, 9}, {1, 6, 10}, {2, 4, 7}, {2, 5, 9}, {3, 4, 8}, {5, 6, 8}, {1, 2, 3, 7}, {1, 2, 5, 6}, {1, 6, 7, 8}, {1, 8, 9, 10}, {2, 3, 8, 9}, {2, 6, 8, 10}, {3, 4, 5, 10}, {3, 6, 7, 9}, {5, 6, 7, 10} Converted to binary strings: 0000000010 0000010101 0000101000 0000110011 0001000111 0001001010 0010001100 0010110000 0011100001 0100001001 0100010010 0101000000 0101100100 0110000110 1000000100 1000011100 1000100001 1001110000 1010100010 1110000001 Perhaps with more patience and a faster computer one could find larger unique codes for n = 10 by this method. Distribution of sizes found of maximal unique union codes for n = 10 in 1102809 tries by randomized greedy search: 3, 17597 4, 5792 5, 132469 6, 18133 7, 14628 8, 126673 9, 286749 10, 35964 11, 38246 12, 120017 13, 165637 14, 86386 15, 33268 16, 16729 17, 4137 18, 364 19, 19 20, 1 21, 0 On Wed, Nov 21, 2012 at 9:14 AM, Veit Elser <ve10(a)cornell.edu> wrote: > > On Nov 20, 2012, at 10:35 PM, Thomas Colthurst <thomaswc(a)gmail.com> > wrote: > > > I found an "unique union" code for n=10 with size 21, just by taking > > 0000000000 along with all of the circular shifts of 0000001011 and > > 0001100101. > > > > Since 1000 / 21 = 47.6 ..., this gives a rats solution that loses at most > > 2*48 = 96 bottles. > > > > -Thomas C > > Wow! I was looking for exactly this object. How did you succeed? > > I noticed a pattern among the optimal codes for n=4,6,8 : > > 0 0 0 0 > 1 0 0 0 > 0 1 0 0 > 0 0 1 0 > 0 0 0 1 > > 0 0 0 0 0 0 > 1 1 0 0 0 0 > 0 0 1 1 0 0 > 1 0 1 0 1 0 > 0 1 0 1 1 0 > 0 1 1 0 0 1 > 1 0 0 1 0 1 > 0 0 0 0 1 1 > > 0 0 0 0 0 0 0 0 > 1 1 0 0 0 0 0 0 > 0 0 1 1 0 0 0 0 > 1 0 1 0 1 0 0 0 > 0 1 1 0 0 1 0 0 > 0 0 0 1 1 1 0 0 > 1 0 0 1 0 0 1 0 > 0 1 0 0 1 0 1 0 > 0 0 0 0 0 1 1 0 > 0 1 0 1 0 0 0 1 > 0 0 0 0 1 0 0 1 > 1 0 0 0 0 1 0 1 > 0 0 1 0 0 0 1 1 > > If we think of the columns as codewords (of codes of length 4, 6, 8), > these are all constant weight, constant distance: (w,d)=(1,2), (3,4), > (4,6). I have no idea why the columns should have this symmetrical > structure, it is just an observation. My next step was going to be to look > for a constant weight, constant distance code of size 10, with codewords of > length 21. But you beat me to it! Your code (transpose) has constant weight > 7 and constant distance 10. > > BTW, for the poisoned bottle-pair application one of the codewords of the > "unique union" code must be the zero codeword. But all the optimal codes so > far have this property. > > -Veit > > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >

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Re: [math-fun] Coordination sequence for 3.3.4.3.4 lattice
by Dan Asimov 21 Nov '12

21 Nov '12

Funny, I was just thinking about that Archimdean tiling the day before, since it's dual to the "Cairo" tiling by (non-regular) pentagons. In fact of all the Archimedean tilings of the plane, I find that one to be the least appealing, aesthetically. And I was asking myself why I have that reaction. I think it's because of two contradictory first impression of those squares and triangles is that they're chaotic. And the second impression is that with so many triangles arranged edge-to-edge, it looks unduly tame. On the other hand, the dual tiling is perhaps the most beautiful Archimedean dual. http://en.wikipedia.org/wiki/Cairo_pentagonal_tiling . << There's a pretty Archimedian tiling with vertex figure 3.3.4.3.4 . . . >> Daniel Asimov Visiting Scholar Department of Mathematics University of California Berkeley, California

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