May 2011 - mathfun@mailman.xmission.com

Re: [math-fun] true self-referential sentences with false contrapositives
by James Propp 12 May '11

12 May '11

Adam Goucher's sentence "If 3 is a prime, then the contrapositive of this sentence is false" and its formal contrapositive "If the contrapositive of this sentence is true, then 3 is not prime" may contradict one another, but what I want is a provably true sentence whose formal contrapositive is provably false. Jim Propp

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[math-fun] Geometric proof of Wilson's Theorem
by Adam P. Goucher 11 May '11

11 May '11

Greetings! I was contemplating card games, where there is initially a shuffled deck of playing cards. I decided to cycle through my deck of Penrose tiling playing cards, drawing a walk on the Cartesian plane, such that a spade corresponds to moving down, a heart corresponds to moving up, etc. Now, this created a cyclic walk of length 52 on a Cartesian grid. I then replaced the four cardinal directions (one for each suit) with 52 unit vectors (one for each card). I soon realised that the mathematics with p cards, where p is prime, is much more interesting: Associate each of the p cards with a unit vector with argument 2*pi*k/p radians (0 <= k < p). They have a sum of zero, so a walk produced by adding these vectors in any order is necessarily cyclic, returning to its original starting point. There are p! different permutations of p vectors, but (as we are only considering cycles) the start point is immaterial, so there are only (p-1)! distinct walks using this technique. Imagine the walks as (possibly self-intersecting) polygons with directed edges. It is clear that these polygons can only have order-1 or order-p rotational symmetry, as any automorphisms must preserve adjacency of vertices, and thus be cyclic permutations. The polygons with order-p rotational symmetry are the regular p-gons and star p-gons, of which there are p-1 options. There are thus (p-1)!-(p-1) walks with no (order-1) symmetry. Clearly, any rotation by a multiple of 2*pi/p radians (equivalent to rotating each of the vectors on the cards) will transform any valid walk into another. Consequently, there is an equivalence relation on the asymmetric walks, partitioning them into sets of p directly congruent walks. Hence, p divides (p-1)! - (p-1) (the number of asymmetric walks) (p-1)! - p + 1 is congruent to 0 (mod p) (p-1)! - p is congruent to -1 (mod p) (p-1)! is congruent to -1 (mod p) This is the statement of Wilson's Theorem, so the proof is complete. I suppose this is one of the rare examples where a proof of a serious mathematical theorem arises from considering something as frivolous as playing card games, where the shuffled deck imitates a random number generator. If you're (un?)lucky, I might explore PRNGs and playing cards in a future e-mail! (If it leads to proofs like this, I should play card games with fellow mathematicians in the Manchester branch of Starbucks more often... : ) Sincerely, Adam P. Goucher

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Re: [math-fun] true self-referential sentences with false contrapositives
by Bill Gosper 11 May '11

11 May '11

1. The importance of this discussion should not be underestimated. 2. The importance of this discussion can not be underestimated. --rwg On Wed, May 11, 2011 at 2:13 AM, Adam P. Goucher <apgoucher(a)gmx.com <http://gosper.org/webmail/src/compose.php?send_to=apgoucher%40gmx.com>> wrote:> "If 3 is a prime, then the contrapositive of this sentence is false.">> The contrapositive is:>> "If the contrapositive of this sentence is true, then 3 is not prime.">> which is false. (The contrapositive of the contrapositive is the> original statement, which is true.) You claim that the first sentence is true and the second is false. Why is this any more correct than the claim that the first statement is false and the second statement is true? It's similar to statements like 1. Statement 2 is false 2. Statement 1 is false You can assign the truth values (T, F) or (F, T) without contradiction. But I'm not sure it's really correct to say that these are meaningful statements that are true or false at all. Is the statement "This statement is true" true or false? I don't find this any more paradoxical than the trio 1. The statement directly below this one is false. 2. The statement directly below this one is false. 3. 2 + 3 + 5 Statements 1 and 2 are the same statement, yet they have different truth values. A more troublesome paradox: What about the statements: 1. 2 + 2 = 4 2. You will mail me a check for a thousand dollars tomorrow. 3. An odd number of these three statements are true. Which of these 3 statements are true, and which are false? Andy

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Re: [math-fun] Klein quartic
by Dan Asimov 10 May '11

10 May '11

I just learned from Wikipedia that the article by Noam Elkies in "The Eightfold Way" -- "The Klein quartic in number theory" -- contains a short exposition of a proof (by the mathematician Monsur Kenku) that uses the Klein quartic to demonstrate Heegner's theorem (that was conjectured by Gauss): The only imaginary quadratic number fields Q(sqrt(-n)) such that the corresponding ring of algebraic integers has unique factorization are those for n in {1, 2, 3, 7, 11, 19, 43, 67, 163}. (see p. 93 of the book). Elkies's article uses mathematics that is far beyond my current level of understanding. --Dan Sometimes the brain has a mind of its own.

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Re: [math-fun] Redefine Halphen's constant?
by Bill Gosper 10 May '11

10 May '11

For some reason, the original message >From rwg(a)sdf.lonestar.org Mon Aug 18 05:49:22 2008 is missing from my inbox, but the followup is there, and both are in the archive, so hopefully the rest of math-fun got both. These were about the family of closed curves described parametrically by (theta_2,theta_1}, which if 3D plotted for 0<q<.9, draws a "space plane" whose fuselage cross sections range from a tiny circle through a near square to nearly a plus-sign. ParametricPlot3D[{EllipticTheta[1, t, q], EllipticTheta[2, t, q], q}, {t, 0, 2*\[Pi]}, {q, 0, .9}, BoxRatios -> {1, 1, 1}] The q values maximizing area and "squareness" are respectively ParametricPlot[{{EllipticTheta[1,t, .4393996289909801220193544165615151], EllipticTheta[2, t, .4393996289909801220193544165615151]}, {EllipticTheta[1, t, .32810656687497825302548], EllipticTheta[2, t, .32810656687497825302548]}}, {t, 0, 2*Pi}] What's new tonight is an exact formula for the cross-sectional area 2 Pi EllipticThetaPrime[1, 0, q^2] (which can be written in Macsyma as 4 * pi * eta^3(q^4) ) and volume 2 pi^2 * sech(pi/sqrt(2)) . Before I found the area formula, Julian labored mightily to get 30 digits of the q maximizing area, difficult because Mma 8.0 seems to be missing the numerics: In[1854]:= Derivative[0, 1, 1][EllipticTheta][1, 69, .105] Out[1854]= (0,1,1) EllipticTheta [1, 69, 0.105] I'm puzzled by this apparently nondescript q and the corresponding maximal shape. I found no relations other than the Halphen's result . --rwg Cc: rwg(a)sdf.lonestar.org <I should have gotten two!> Subject: [math-fun] Redefine Halphen's constant? X-List-Received-Date: Mon, 18 Aug 2008 11:49:22 -0000 If you try to approximate e^-x on [0,oo) with a ratio of polynomials of degree n, then instead of vanishing, the error for large x will approach the ratio of the leading coefficients, which will also be the ripple height in the best (minimax) approximation. For successively larger n, this ripple height goes down by a factor which was erroneously conjectured to be 1/9, but is more like .107653919, and is somewhat wryly called the One Ninth constant. But it is sometimes also called Halphen's constant, because its square root coincidentally solves the equation 0=Theta_2''(0,i q), posed by Halphen. This square root (.32810656687497825302548) also has nice geometric properties w.r.t. the family of curves (or surface) whose parametric equation is (x,y) = (Theta_2(t), Theta_1(t)), 0<q<1. For small q this is a tiny "circle" which enlarges and grows "squarer" as q -> .328, in the manner of |x|^n + |y|^n = r^n, but rotated by pi/4, so that the incircle is tangent at +-x = +-y (t= (2k+1) pi/4). At q=.328, the incircle is maximal (r = 1.34883916748787) and the curvature vanishes at the four tangencies. The "CRT screen" exponent is n=5.419, a very close fit, but x^n+y^n has nonvanishing curvature at the axes. For q>.328, the curve develops concavities, eventually approaching an infinite "plus" sign. The equation stating the vanishing curvature at t=pi/4 is Theta_2'(pi/4) Theta_1''(pi/4) = Theta_1'(pi/4) Theta_2''(pi/4). But Theta_2'(pi/4) = Theta_1'(pi/4), and obviously doesn't vanish, and since Theta_2''(pi/4) = Theta_1''(pi/4) for all q, the Theta'' must vanish on both sides of the eqn. (I.e., the parametric curves must have their inflections at pi/4.) Maximality of the inradius is d Theta_2(pi/4)/dq = 0, but d Theta_s(z)/dq = -Theta_s''(z)/4/q, which for z=pi/4 is equivalent to the vanishing curvature. Alternative expressions for d Theta_2(pi/4)/dq are d Theta_2(0,i q)/dq /sqrt(2)/i^(1/4) = - Theta_2''(0,i q)/4/sqrt(2)/i^(1/4)/q = - Theta_1''(pi/4,q)/4/q . The area is maximal (7.40051953747301) at q = 0.43939962899098. Adding a z coordinate of q produces a surface resembling a squat, blunt-nosed bomb with four infinitely spreading tailfins, volume = 4.231765651557. Anyway, howsabout we honor Halphen with .328 and relieve "One ninth" of its extra, though more decorous synonym? Plouffe's Inverter might be the most influential current reference using "Halphen's" to mean "One ninth", which is (fortunately) discouraged in the current edition of Finch's (more influential) Constants book. The Weissopedia sez "See One Ninth constant", but doesn't come right out and say they're the same. --rwg (This exercise exposed many deficiencies in my Macsyma Theta function package (http://www.tweedledum.com/rwg/thet.gif, not with I.E.), but surprisingly more in Mma 6.0.) Great, politically incorrect Mythbusters question: Is it actually possible to break a (healthy) camel's back with straw? >From rwg(a)sdf.lonestar.org Tue Aug 19 06:22:27 2008 From: rwg(a)sdf.lonestar.org To: rwg(a)sdf.lonestar.org User-Agent: SquirrelMail/1.4.9a Cc: math-fun(a)mailman.xmission.com, rwg(a)sdf.lonestar.org Subject: Re: [math-fun] Redefine Halphen's constant? X-List-Received-Date: Tue, 19 Aug 2008 12:22:27 -0000 > Adding a z coordinate of q produces a surface resembling a squat, blunt-nosed > bomb with four infinitely spreading tailfins, volume = 4.231765651557. > Actually, it looks rather hypersonic if you crop the infinite tail and stretch it a bunch: http://gosper.org/sst.png . >volume = 4.231765651557. Per Macsyma. Mma is still working on the twelve digits I asked for yesterday. >This exercise exposed many deficiencies in my Macsyma Theta function package Mostly in the area of simplifying special values and derivatives. Analogous to trig (under control of the same (%piargs) switch), it now translates out quarterperiods and quarterquasiperiods (which can get messy): [...] <Let me know if you want them re-sent.> [...] %pi 3 8 thetaderiv (---, 1, q) = 4 eta (q ) 4 4 The 9th derivative wrt q at q=0 of the nth derivative wrt z of theta[3](z,q): (c436) thetaderiv[3](z,n,0,9) n %pi n (d436) 725760 6 cos(6 z + -----) 2 >I wonder if WRI would like one of these.-) > --rwg

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Re: [math-fun] Convex polyominoes
by Robert Munafo 09 May '11

09 May '11

My program is correctly generating the polyominoes (verified by comparing counts to A000105), but I'm stuck (in my mind) on how to derive the convex hull and how to test other points in the grid to see of they are inside or outside the convex hull. Once I know which points define the vertices of the convex hull, I could use the Jordan Curve theorem for testing other points, but when a point falls right on the boundary I have to worry about round-off error. Any hints? Allan Wechsler wrote: > On a previous occasion, I talked about enumerating "disklike" polyominoes, polyominoes which occurred as the intersection between a disk and a square lattice. Today I tried counting polyominoes which occurred as the intersection between any convex figure and the lattice. This implies that the convex hull of the polyomino (viewed as a set of lattice points) includes no additional lattice points. > > I call these polyominoes "convex", but apparently the term has already been taken by a different class, so I don't know what we should really call them. > > Starting from the monomino, the counts seem to be 1, 1, 2, 5, 10, 25. The sequence is not in OEIS, though I confess some uncertainty about the 25 convex hexominoes, and would like confirmation. -- Robert Munafo -- mrob.com Follow me at: fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com- youtube.com/user/mrob143 - rilybot.blogspot.com

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Re: [math-fun] Convex polyominoes
by Dan Asimov 09 May '11

09 May '11

Given a closed & bounded convex figure K: Does "intersection between any convex figure and the lattice" mean the union of grid squares that intersect K ? Or the union of grid squares that are each entirely contained in K ? (Or maybe both definitions yield the same set of polyominoes?) * * * Allan's definitions of disklike polyomino, and the one below, suggest another one: * A polyomino that is topologically a closed disk. (E.g., this would exclude the ringlike heptomino O_7.) This would include all n-ominoes through n = 6, but then things start to get interesting. (A perhaps nicer variant would be to require only that the *open* polyomino be a topological disk. This would rescue O_7.) * * * I like hexagonal polyominoes -- say, "heximoes" -- since no two cells can overlap on just a vertex, thus avoiding borderline cases like H_7. Here the topological-disk ones exclude n-heximoes only for n >= 6. QUESTION: Is there a simple asymptotic expression H(n) for the number of topological-disk n-heximoes? (In 3-space one could likewise ask for topological-3-disks tesssellated by truncated octahedral tiles, and an asymptotic formula for their counts, etc.) --Dan --------------------------------------------------------------- Allan wrote: << On a previous occasion, I talked about enumerating "disklike" polyominoes, polyominoes which occurred as the intersection between a disk and a square lattice. Today I tried counting polyominoes which occurred as the intersection between any convex figure and the lattice. This implies that the convex hull of the polyomino (viewed as a set of lattice points) includes no additional lattice points. >> Sometimes the brain has a mind of its own.

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[math-fun] How Math Comes to Mind
by Hans Havermann 09 May '11

09 May '11

A Princeton University podcast (258 MB) How Math Comes to Mind: Intuition, Visualization, and Teaching http://www.princeton.edu/WebMedia/media/lectures/20110427_publect_dehaene_s… The abstract: Stanislas Dehaene is director of the INSERM-Cognitive Neuroimaging Unit and a professor at the College de France. He is a neuroscientist. The most recent edition of his book The Number Sense will be published in April. Steven Strogatz is a professor and director of the Center for Applied Mathematics at Cornell University. His most recent book, published in 2009, is The Calculus of Friendship. In this free- wheeling conversation between a cognitive scientist and a mathematician, Professors Dehaene and Strogatz will exchange ideas about how people learn math and gain intuition about it, whether the human mind creates math or discovers it, where math is born in the brain, and what the best ways may be to lead students (and their parents) to embrace math and understand it. 79 minutes in length, the latter part of the video suffers from the audience questions being (largely) inaudible. Nevertheless, I feel that most subscribers to this list will find something worthwhile in the presented discussion. If you replace the mp4 terminus of the link with mp3, you'll get instead a 36 MB audio version.

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[math-fun] Polymath Project throws mathematical conundrums open to all to tackle
by Henry Baker 09 May '11

09 May '11

FYI -- http://www.guardian.co.uk/science/2011/may/08/welcome-to-wikimaths Welcome to WikiMaths Â home of hard sums The Polymath Project throws mathematical conundrums open to all to tackle Matt Parker guardian.co.uk, Sunday 8 May 2011 20.00 BST Now unprecedented numbers of people can work on the same problem online. Mathematicians are not known as a social bunch, but a new "WikiMaths" project is allowing anyone to join in their cutting-edge research. A study into the effectiveness of the world's first virtual mathematics project will be released this week. It all started in 2009, when Cambridge mathematician Tim Gowers wrote about the possibility of an open online group allowing unprecedented numbers of people to work on the same problem, hopefully solving conundrums much more quickly. He suggested the "HalesÂJewett theorem" as a good first target. Analagous to a complicated game of noughts and crosses played on a 4x4 cube in five dimensions, the theorem shows how many squares you would need to block to make it impossible to complete any straight lines. On a 3x3 grid, you can do this by blocking three squares; in five dimensions, things are a bit more complicated. This theorem had already been proven, but the solution was long and complicated and no one had found a much-needed basic proof. Contributions poured in Â a staggering 1,228 significant comments across 14 blog posts with 39 people providing meaningful contributions. Within six weeks the answer had been found. It was published under the collective pseudonym "DHJ Polymath". But was the process truly collaborative? Researchers at Carnegie Mellon University in Pittsburgh, think so. Much of the work was done by professional mathematicians, but a number of smaller, vital contributions came from those without serious credentials. The 39 contributors to the HalesÂJewett theorem solution ranged from the world's top mathematicians to secondary school maths teachers. Several seminal ideas came from inexperienced mathematicians. Which all means that the exercise could redefine who is considered a mathematician Â and offer new insight into unsolved problems. The researchers are presenting their results in Vancouver next week, while the "Polymath Project" as it now known, continues to work on seven different problems with more than 5,000 comments from 275 unique contributors. Why not join in at polymathprojects.org? guardian.co.uk Â© Guardian News and Media Limited 2011

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[math-fun] true self-referential sentences with false contrapositives
by James Propp 09 May '11

09 May '11

The sentence "If two plus two equals four, then the premise of this sentence contains numbers" is true, while its literal contrapositive "If the premise of this sentence does not contain numbers, then two plus two does not equal four" is false. Can anyone find a more elegant example of a true self-referential sentence of the form "If ..., then ..." whose literal contrapositive is false? One might limit oneself to sentences that do not explicitly refer to their own structure as English sentences, but only refer to their own logical structure. Then the search for a true sentence with a false contrapositive could be set up as a "logical programming" problem on the space of all sentences of the form "If P then Q" where P and Q are arbitrary 6-variable Boolean functions of "this sentence", "the premise of this sentence", "the conclusion of this sentence", "the inverse of this sentence", "the converse of this sentence", and "the contrapositive of this sentence"; but since there are (2^(2^6))^2 = 2^128 or about 10^40 possibilities, a brute-force search of the space won't be feasible. I raised a version of this question in some forum (sci.math.research?) a number (20?) of years ago, and I think someone gave an answer that ruled out candidate solutions of a particular form (perhaps "If A implies B then C implies D"), but I can't remember any details. (Can anyone else?). I also sent the question to Raymond Smullyan and I think to Douglas Hofstadter as well, but I don't recall either of them replying. Can anyone come up with such a sentence or show that none exists? Jim Propp

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