April 2016 - mathfun@mailman.xmission.com

[math-fun] turtleneck sweater for octopi
by Bill Gosper 14 Apr '16

14 Apr '16

http://home.btconnect.com/GavinTheobald/HTML/Octagon.html#Hexadecagon I don't recall curved cuts in a polygonal dissection. Can't the circular segments be replaced by segments of hexadecagons? --rwg Are curves essential to any known minimal dissection?

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[math-fun] pointless chess/go Lasker Lange trivia
by Warren D Smith 12 Apr '16

12 Apr '16

Edward & Emanuel Lasker were related but only distantly. They also knew each other and both played go and chess. Also, there were two famous "Max Lange"s in Germany at about same time, who played chess and/or go, and kew the Laskers... but I do not know whether the two Max Langes were related. Although the Laskers thought they were very strong go players, likely among the strongest in the West at the time, they were probably quite weak by Asian standards, plausibly about 1 amateur dan. I unfortunately learned go initially from Lasker's book. I really do not know how best to learn go in English. It is a deep game, much deeper than chess, although the rules are simpler. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Draft of my April 2016 blog post
by James Propp 12 Apr '16

12 Apr '16

I started writing a new draft titled "The Paintball Party Problem, and the Habit of Symmetry" and would appreciate people's feedback. I plan on publishing it on April 17. Here are some issues I'm aware of: (1) What (if anything) should be cut, or at least relegated to the End Notes? Or maybe condensed somehow? (2) Can you think of ways to make the discussion of symmetry arguments clearer? (I think that's the part of the current draft that will cause readers the most trouble.) (3) Do you know of any good images and animations regarding finite geometries that might enhance the essay via embedded links? (4) Do you know of good animations of the symmetries of a cube? (If someone wants to make an animation that specifically shows my blue-and-red-colored cubes rotating, to show how the symmetries of the cube permute the set of colorings, that'd be fantastic!) I found one at https://www.youtube.com/watch?v=TggbcOrALMQ but it's not great. I'll probably make one tomorrow if I can find our family's Rubik's cube. (5) Links to animations of shear transformations would be appreciated. (6) On a more technical note, is there an easy way to see that the order of a Hadamard matrix must be 1, 2, or a multiple of 4? (7) I can't find a good way to make an "|R" (the symbol for the real numbers) in Wordpress; can someone help me find one? And of course I'm sure there are typos! Keep in mind that all feedback will be anonymous unless you include your name. As always, people who help me with the essay will be acknowledged by name unless they prefer not to be. Please leave your feedback here: https://mathenchant.wordpress.com?p=669&shareadraft=570be41c24d66 Title: The Paintball Party Problem, and the Habit of Symmetry Beginning: I was very surprised last fall, when I was running my nine-year-old son’s birthday party, to realize that the geometry of spaces built over finite fields (I’ll explain what that means later) was exactly what I needed to solve the very pressing problem of choosing teams for paintball in a fair way! Thanks, Jim Propp

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[math-fun] Lasker
by Warren D Smith 12 Apr '16

12 Apr '16

Macaulay's book gave a star role to the amazing theorem of Lasker, also mentioned here https://en.wikipedia.org/wiki/Primary_decomposition It is due to Emanuel Lasker, world chess champion 1894-1921, which I think made him the longest-reigning world champ. So, I had not realized this, but Lasker also made a lasting contribution to mathematics.

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Re: [math-fun] Database of math problems
by Keith F. Lynch 12 Apr '16

12 Apr '16

Neil Sloane <njasloane(a)gmail.com> wrote: > I don't have the answer to your question, but I have some suggestions. > . . . > %N A193838 Size k of smallest square of k X k lattice points from > which n points with distinct mutual distances can be chosen. Thanks. That's the inverse function: 1,2,3,4,5,6,7,9,10,12,14,... Since my function is monotonic, it's uniquely specified by yours. Mine is 1,2,3,4,5,6,7,7,8,9,9,10,10,11,... And I've confirmed it through 8, after writing a program that, unlike my first one, won't take until the heat death of the universe to finish. That suggests a new sequence: The number of unique ways N lattice points can be placed on an N by N grid with distinct mutual distances: Not counting rotations or reflections as distinct, it's 1,2,5,23,35,2,1,0,0,0,... I'd add it, except that I seem to have forgotten my OEIS password. So, there's no way to have 8 points in an 8 by 8 grid. How about 7 points in an 8 by 8 grid? There are 11959 ways. Quite a difference. Anyone want to see them all? I didn't think so. Here's the unique solution for 7 on a 7 by 7 grid: * - * - - - - - - * - - - - - - - - - - * * - - - - - - - - - - - - - - - - - - * - - - - - - - *

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[math-fun] universe contains large scale directional alignments?
by Warren D Smith 11 Apr '16

11 Apr '16

https://www.ras.org.uk/images/stories/press/Black_Holes/alignment.pdf "We present a study of the distribution of radio jet position angles of radio galaxies over an area of 1 square degree in the ELAIS N1 field. ELAIS N1 was observed with the Giant Metrewave Radio Telescope at 612 MHz ... The image contains 65 resolved radio galaxy jets. The spatial distribution reveals a prominent alignment of jet [directions] along a 'filament' of about 1 degree. We examine the possibility that the apparent alignment arises from an underlying random distribution and find that the probability of chance alignment is less than 0.1%." These were at a distance corresponding to a redshift of about 1. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Self-column
by Eric Angelini 11 Apr '16

11 Apr '16

Hello Math-Fun, We want to build a column of true statements; every step makes a true statement about the count of the letters involved so far in the column. Example of such a column: TWO T THREE E THREE H FOUR R ... Explanation: At stage 1 the column is made of this true statement about the column itself: "There are 2 T's so far in C" ("C" is short for "column") At stage 2 the column is made of this true statement about the column itself: "There are 3 E's so far in C" At stage 3 the column is made of this true statement about the column itself: "There are 3 H's so far in C" At stage 4 the column is made of this true statement about the column itself: "There are 4 R's so far in C" etc. There are lots of such "self-columns", but we want here the alphabetically first one. It will start, I guess, like this: ONE A ONE B FOUR O FIVE O FIVE E EIGHT E etc. Question: when will this column end? Best, É. à+ É. Catapulté de mon aPhone

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[math-fun] General formulation for Rupertability, & linear/not
by Warren D Smith 10 Apr '16

10 Apr '16

A convex polytope (or other shape, but I'll concentrate on convex polytopes) P can be "Ruperted" if it is possible to drill a hole in it, then pass a copy of P thru the hole. Actually I'll only consider "line Rupertability" where the motion has to be along a line by pure translation; no twisting and bending motions allowed. SIDE QUESTION: Can anybody devise some examples of shapes P which can be Ruperted, but only in a non-line manner? ANSWER: I suspect such shapes do exist! Consider a "thickened helix", e.g wind some wire, of thickness 1 cm, round a helix. I think this shape can be Ruperted in a limit sense where the "drilled hole" is actually the entire shape shaved by epsilon, with the motion necessarily being a helical slido-rotation -- but any straight-drilling attempt disconnects? Also, instead of a helix, another possibility is a "thickened circular arc." You may object that construction was silly and degenerate. However, it may be possible to avoid that objection either by going to higher dimension or in 3D by making the "wire" have noncircular cross section. The point I am trying to drive at, is that for some shapes, it is very natiural to try to worm them thru twisting and/or curving drilled holes, and you pay a major penty if you try to drive them stright thru holes via pure translation. Specific 3D try: consider a thickened unit-radius semicircular arc in the XY plane, cut out of a piece of sheet metal so it has 3D thickness. Choose the sheet-thickness and the radial thickness right, that's 2 parameters. Well, I think this shape is (if parameters chosen right) just barely Rupertable, but NOT by any pure-translation motion, it needs to be a nonlinear motion. Crude Sketch: https://dl.dropboxusercontent.com/u/3507527/NonlinRupert.png Note, this object is nonconvex. I have no idea whether convex P could require nonlinear Rupertizing. RETURN TO MAIN TRACK (Linear Rupertability only): Greg Huber got in touch with me & Gosper, and anyhow in emails with him I outlined proofs the n-octahedron and n-simplex both can be linearly Ruperted for each n>=3; and the n-cube also is Rupertable as was already known. He sent a proof, which was much like a post by me but done better 23 years earlier that indeed the n-cube is Rupertable with shrink factor 1+const/9^n. The n-sphere of course is NOT Rupertable, and the status of the sporadic regular polytopes (icosahedron, dodecahedron, 600-cell, 120-cell, 24-cell) is presently unknown but could be solved by computer by the method I will now describe. Here is how to decide if convex polytope P is Rupertable. Choose a direction d. Project the vertices of P along d onto a plane perp to d, call that operation Proj[d](P). Also choose a rotation (or orthogonal, if we allow mirror reflect...) matrix Q. P is Rupertable in direction d using Q if and only if the vertex set of Proj[d]( Q P ) is translatable to lie inside ConvexHull( Proj[d](P) ) -- which question is a linear programming problem once d and Q are given. Now perform that test for lots of d and Q, covering the space of rotation matrices and sphere of directions with a dense set and trying them all until you find a successful test. Then you have proven it Rupertable. This will always eventually succeed (with probability 1 using random points...) in proving Rupertability if it is Rupertable, because the spaces to be searched are compact manifolds and the success set is algebraic and hence has nonzero measure, so there is a positive probability of success each try. Now if P is not Rupertable, then this kind of crude approach, rather less obviously, can eventually prove that too, in finite time. Because: we can produce, a priori, some rather weak but valid degree and "height" bounds on the algebraic set. We then try as our set of trials, all algebraic numbers of low degree and height... or actually any dense-enough finite set of points (using known "separation" bounds for algebraic numbers of given degree and height arising from discriminants...) if they all fail, that meant there were no solutions. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] the incredible mathematical mind of the pufferfish, geometric patterns
by Simon Plouffe 10 Apr '16

10 Apr '16

Hello, I saw a documentary about the wonders of nature and one topic left me with my jaws that dropped on the floor (les bras m'en tombent), about the pufferfish. This incredible little fish is very talented in patterns and geometric shapes. http://www.livescience.com/40132-underwater-mystery-circles.html That fish is not even 4 inches long and shows definite intelligence in making a nest for the female, the patterns are wonderful. see other photos here : https://www.google.fr/search?q=fugu+fish+geometric&biw=1590&bih=853&source=… best regards, Simon Plouffe

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Re: [math-fun] Lasker
by Guy Haworth 10 Apr '16

10 Apr '16

John Nunn has an excellent book out which teaches some principles of chess through Lasker's games. John regards Lasker as the first World Champion to play the opponent as well as 'the board', i.e., to make a possiblyl debatable move on the grounds that it was more likely to confound the opponent. The anecdote is, sort of, a case in point. Guy

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