February 2012 - mathfun@mailman.xmission.com

[math-fun] Extreme close-ups create a Klein Bottle.
by Ray Tayek 24 Feb '12

24 Feb '12

http://www.thebigquestions.com/2012/02/23/who-expected-that/ Published by <http://www.thebigquestions.com/author/landsburg/>Steve Landsburg on February 23, 2012 in <http://www.thebigquestions.com/category/coolstuff/>Cool Stuff, <http://www.thebigquestions.com/category/empirical-puzzles/>Empirical Puzzles and <http://www.thebigquestions.com/category/math/>Math . <http://www.thebigquestions.com/2012/02/23/who-expected-that/#comments>14 Comments Suppose you go around taking extremely close-up black-and-white pictures of randomly chosen natural and unnatural objects (rocks, trees, streams, buildings, etc.). What do they look like? Well, each one looks like a patch of varying shades of gray, of course. But do some patches arise more than others? If each of your close-ups is, say, three pixels by three pixels, Which would you expect to see more of: This? [] Or this? [] Well, each 3 x 3 square of pixels is essentially a list of nine numbers (one for the darkness of each pixel, so that a pure black pixel is a 1 , a pure white pixel is a 0 , a nearly-black pixel is, say, a .9 , etc.). A list of nine numbers specifies a location in nine-dimensional euclidean space (just as a list of three numbers specifies a location in the three-dimensional space in which we appear to live). So the question becomes: Where in nine-dimensional space do close-up patches of photos tend to live? A natural expectation is that theyre scattered randomly and at first blush that appears to be accurate. But it turns out that if you take a closer look, using a mathematical lens that lets you see more clearly into nine dimensions, theyre not scattered randomly at all. Instead, theyre clustered around (of all things!) a <http://en.wikipedia.org/wiki/Klein_bottle>Klein bottle, which is a two-dimensional surface that cant be squeezed into three dimensions, but fits perfectly well in nine (or for that matter in four). (Actually its a somewhat thickened Klein bottle, and hence four-dimensional instead of two-dimensional, much as a solid hula hoop is a somewhat thickened circle and hence three-dimensional instead of one-dimensional.) In other words, if you randomly photograph 10,000 objects, randomly choose 3-by-3 pixel patches, and plot the corresponding points in nine-dimensional space, what youll see (insofar as you can see in nine dimensions) is a somewhat blurry Klein bottle. Whoda thunk? <http://www.dam.brown.edu/people/mumford/Papers/DigitizedVisionPapers--forNo…>Here is some of the original research and <http://www.math.upenn.edu/%7Eghrist/preprints/barcodes.pdf>here is an expository (but pretty high-tech) paper on this and related matters. Edited to add: The original computational work is <http://www.springerlink.com/content/pm1718435n685294/>here. A survey article by Gunnar Carlsson, one of the original researchers, is <http://www.ams.org/journals/bull/2009-46-02/S0273-0979-09-01249-X/>here<http://comptop.stanford.edu/>. The research groups webpage is --- co-chair http://ocjug.org/

1 0

[math-fun] (further) generalized Lambert series with Theta-convergence
by Bill Gosper 23 Feb '12

23 Feb '12

Joerg>From the department of formulas that could possibly be found in some paper from 200 years ago. >Semi-recently I made the observation (and posted it here) that sum(n>=0, x*q^n/(1-x*q^n) can be computed as sum(n>=0, q^(n^2+n)*x^(n+1)*(1 - x*q^(2*n+1)) / ((1 - x*q^n)*(1 - q*q^n))) and this allows the fast computation of sums of inverse Fibs (and infinitely more sums of reciprocal order-two linrecs) without splitting into even and odd part (yes, RWG, AIM304 exists). YOW, I'd forgotten that stunt! >After much effort I found that sum(n>=0, t^n / (1-x*q^n) ) Random aside: Per my recent (generalized) contiguous Heine mail, with c=a and d=b q = x q and z=t, sum(((t^n)/(1 - q^n * x)),n,0,inf) = ((a * (1 - q * t) * x * sum(((q^(2 * n) * t^n)/(1 - q^n * x)),n,0,inf) - ( - q * t * x + q * x - a * q * t + a) * sum(((q^n * t^n)/(1 - q^n * x)),n,0,inf) - q + a)/(q * (t - 1))) independent of a! Joerg>can be computed as sum(n=0, S, (1-x*t*q^(2*n))*(x*t)^n*q^(n^2) / ( (1-x*q^n)*(1-t*q^n)) ) (which btw. is symmetric in x and t). This is driving me nuts. The 3x3 system mentioned in that Heine mail refuses to triangularize for this Lambert case, which is degenerate enough to permit discarding two parameters from the Heine N matrix. One of these, call it k, can be via a reversible (unit determinant) "recoord" (our new name for "transformal"). But k then persists in the other matrices, and K, the k-bumpng matrix, does not become the 3x3 identity! Thus, wandering around in the k-n plane is a no-op, because, by path invariance, N(k,n).K(k,n+1) = K(k,n).N(k+1,n), but N(k,n)=N(k+1,n)! There must be some use for this. Rolling back to a 1998 Heine 2x2 system (that was born triangular), then replacing n+1 by n+h by shifting i,j,k,n by 1-h,1-h,1-h,h-1, then specializing h to i*t (and discarding the I matrix), then q^t->t, gives the J,K,N matrices {{{-(((1 - q^(n + j))*(1 - 1/(q^(k - j - 2)*t))*t)/(q^(n + j + 1)*(1 - 1/q^(k - j - 1)))), -((1 - q^(n + k - 1))/ (q^(n + k - 1)*(1 - 1/q^(k - j - 1))))}, {0, 1}}, {{-(((q^(n + k - 1)*(1 - q^(k - j))*t)/(1 - q^(n + k)))*(1 - q^(k - j - 1)*t)), 1/(1 - q^(k - j - 1)*t)}, {0, 1}}, {{(q^(k - j - 1)*(1 - q^(n + j))*t)/(1 - q^(n + k)), 1}, {0, 1}}} Then pegging j at x and closing a rectangle based at k=x+1,n=0, then q^x->x gives a *different* theta-convergent series symmetric in t and x: Sum[t^n/(1 - q^n*x), {n, 0, Infinity}] == Sum[((-1)^k*q^(k^2/2 - k/2)*QPochhammer[q, q, k]*t^k*x^k)/ (QPochhammer[t, q, k + 1]*QPochhammer[x, q, k + 1]), {k, 0, Infinity}] (Dialog) In[649]:= Simplify[ FunctionExpand[Series[%[[2]] /. \[Infinity] -> 6, {t, 0, 6}]]] (Dialog) Out[649]= SeriesData[t, 0, {(1 - x)^(-1), (1 - q x)^(-1), ( 1 - q^2 x)^(-1), (1 - q^3 x)^(-1), (1 - q^4 x)^(-1), ( 1 - q^5 x)^(-1), (1 - q^6 x)^(-1)}, 0, 7, 1] Like I said: nuts. I'll bet yours|Fine's is in there somewhere. --rwg >Now this one turns out to be easily obtained by setting a := -b in Fine's "versatile" relation (14-1) (p.15 in "Basic Hypergeometric Series and Applications"). Anybody with one bit of interest in q-things: Buy this book, it's a marvel.

3 15

Re: [math-fun] Suggestions requested for a recreational math book for a bright 15-year-old
by Dan Asimov 23 Feb '12

23 Feb '12

Thanks everyone for the suggestions, especially the recreational math books. I don't need any more suggestions, since it's now down to either the updated version of Gardner's first collection of columns, or The Book of Numbers by Conway/Guy. --Dan ________________________________________________________________________________________ It goes without saying that .

2 1

Re: [math-fun] Suggestions requested for a recreational math book for a bright 15-year-old
by Stuart Anderson 23 Feb '12

23 Feb '12

Here's my list; Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics - John Derbyshire ( some of it may be beyond a 15 year old, but then again ...) Fermat's Last Theorem - Simon Singh The Code Book - Simon Singh The Equation that Couldnt be Solved - Mario Livio The Golden Ratio: The Story of Phi, the World's Most Astonishing Number - Mario Livio Poincares Prize - George G Szpiro Four Colors Suffice - Robin Wilson Tracking the Automatic Ant - David Gale Finding Moonshine, A Mathematician's Journey Through Symmetry - Marcus Du Sautoy, Five Golden Rules - John Casti Labyrinths - Luis Borges -(not strictly mathematics - but there is 'The Unimaginable Mathematics of Borges Library of Babel' - William Bloch) The Tinkertoy Computer - A.K. Dewdney Cogwheels of the Mind, The Story of Venn Diagrams - AWF Edwards Flatterland - Ian Stewart, or any of his other recreational math books The Man Who Loved Only Numbers , The Story of Paul Erdös and the Search for Mathematical Truth - Paul Huffman Any/some of Clifford Pickovers books - If he or she likes art as well as maths; Tilings and Patterns - Grunbaum and Shephard MC Escher - Sandra Forty

2 1

Re: [math-fun] isosceles triangle dissections (Stuart Anderson)
by Bill Gosper 23 Feb '12

23 Feb '12

Ouch. I never imagined that an "oversized incircle" could isoscelize all seven subtriangles! Had I merely sliced off the largest possible 45-67.5-67.5 and repeated on the remaining 45-22.5-112.5 the process shown here<http://gosper.org/acutefail.png>, I would have found at least -Floor[Tan[69]] solutions instead of Round[Tand[69]]. Many thanks, Hans! --rwg hans> > So, can someone scan in (the picture(s) from) that '61 article? http://chesswanks.com/txt/AcuteIsoscelesDissectionOfAnObtuseTriangle.png

2 1

[math-fun] fun programs by Roice Nelson - Gravitation3D & Magictile
by Stuart Anderson 23 Feb '12

23 Feb '12

Some fun software ;) On magictile; http://www.gravitation3d.com/magictile/ "This program aims to support twisty puzzles based on regular polygonal tilings having Schlafli symbols of the form {p,3} for any p>=2. That is, all regular tilings of polygons with two or more sides, where three tiles (puzzle faces) meet at a vertex. The Rubik's cube is the special case where faces are squares (p=4). The other familiar special cases are the Megaminx (p=5) and the Pyraminx (p=3), although you'll discover the last takes a slightly different form under this abstraction (akin to Jing's Pyraminx). All the other puzzles are new as far as I know, and some may be surprising, e.g. the puzzles based on digons (p=2). Each 2D tiling admits a particular constant curvature (homogeneous) geometry. The geometry is Spherical for p=2 to p=5, Euclidean (flat) for p=6, and Hyperbolic for p>=7. Since you can't "isometrically embed" the entire hyperbolic plane in 3-space, I have a connection to hyperpuzzling even though I'm talking about 2D tilings! " On Gravitation3D ;http://www.gravitation3d.com/ "Gravitation3D, an elegant and beautiful shareware program aimed at providing an intuitive understanding of the laws of gravity and having fun. Gravitation3D is an OpenGL based shareware program written for Microsoft Windows PCs. It started as a hobby project in July 2002, and quickly I could see from the excitement in friends and colleagues that it would be something worthwhile to share with others. After some more concentrated development work, it was first published on the web April 29th, 2003. The program provides a way for you to play the creator. You construct unique solar systems from scratch by adding gravitational bodies and configuring their starting properties. You then start your simulation and watch your systems evolve. Gravitation3D can provide hours of fun without even realizing you are learning! I already consider it a success because of the subtle insights I've gained from it. Gravitation3D is geared towards the physics, space, and math hobbyist. It is also an excellent educational tool, and would be perfect for generating interest in astrophysics, or for the high school or college physics teacher who wanted to develop lesson plans around it." "When it came to presenting the puzzles, what I chose to do was stereographically project them onto a flat plane from their respective geometries." Stuart Anderson

1 0

[math-fun] Suggestions requested for a recreational math book for a bright 15-year-old
by Dan Asimov 23 Feb '12

23 Feb '12

A bright friend of mine just turned 15. They are rather advanced in math for their age and a good chess-player. At age 12 I was ecstatic to receive the first book in Martin Gardner's series of collected columns, around 1959. My friend has not seen anything by Gardner yet, so those are the first thing that comes to mind. Yet I wonder if at this point they might seem somewhat dated. All suggestions and opinions welcome. --Dan ________________________________________________________________________________________ It goes without saying that .

11 10

Re: [math-fun] isosceles triangle dissections (Stuart Anderson)
by Bill Gosper 23 Feb '12

23 Feb '12

Ah, thank you both. But alas, the problem has all along been to do it with acute *isosceles* triangles. I believe Neil has a different Gardner book claiming the answer is still nine, but no picture. I'm beginning to wonder if it's an error. --rwg Stuart> I am advised (but cant verify); It's a little further along (page 258) in the answers section. ;) Hans> http://chesswanks.com/txt/Acute&IsoscelesTriangles.pdf

2 2

[math-fun] Polyjogs
by ed pegg 22 Feb '12

22 Feb '12

Derrick Schneider uses the four trijogs to make his puzzle Square Dance. fit the four pieces into either a 7x9 or 8x8 rectangle. Both solutions are unique. The puzzle was a prize-winner in the Nob Yoshigahara Puzzle Design competition. George Sicherman has enumerated polyjogs.http://www.recmath.org/PolyCur/polyjogs/ The 17 tetrajogs are easy to place into Burr Tools, and they can fit into a 20x20 square. A 19x20 rectangle looks very doable, but Burr Tools wants a few hundred thousand years for that analysis. Can anyone place the 17 tetrajogs into a 19x20 rectangle by hand? --Ed

1 0

Re: [math-fun] (further) generalized Lambert series with Theta-convergence (news from planet Gosper)
by Bill Gosper 22 Feb '12

22 Feb '12

joerg> * Bill Gosper <billgosper(a)gmail.com <http://gosper.org/webmail/src/compose.php?send_to=billgosper%40gmail.com>> [Feb 18. 2012 07:42]:> [...] > which is somewhat less peculiar when properly simplified:> > Sum[t^n/QPochhammer[x, q, 1 + n], {n, 0, Infinity}] ==> Sum[(q^j*QPochhammer[q^(1 + j), q])/(QPochhammer[q^j*t, q]*> QPochhammer[q^j*x, q]), {j, 0, Infinity}]> DLMF 17.6.12 http://dlmf.nist.gov/17.6 --rwg joerg>From the equation I get ( N := \infty ) qbin(t,q,N) * qbin(x,q,N) / qbin(q,q,N) * sum(n=0,N, t^n / qbin(x,q,n+1) ) == sum(j=0,N, ( q^j * qbin(t,q,j) * qbin(x,q,j) ) / ( qbin(q,q,j) ) ) Now the second one is invariant against exchange of x and t, and so is the factor of the first one preceding the sum. Hence we have: sum(n=0,N, t^n / qbin(x,q,n+1) ) == sum(n=0,N, x^n / qbin(t,q,n+1) ) Now this a nice counterpart of (my/Osler's) sum(n=0,N, t^n / (1 - x*q^n) ) == sum(n=0,N, x^n / (1 - t*q^n) ) > equivalently> QHypergeometricPFQ[{q, 0}, {x}, q, t] ==> (QHypergeometricPFQ[{t, x/q}, {0}, q, q]*QPochhammer[q, q])/> (QPochhammer[t, q]*QPochhammer[x, q]) Which may well be in http://dlmf.nist.gov/ If so, can you point to it? > [...] So now there's a \section{News from planet Gosper}

2 1