September 2020 - mathfun@mailman.xmission.com

[math-fun] Rolling polyhedra
by James Propp 25 Dec '20

25 Dec '20

I want to think about rolling polyhedra, and the polyhedra I want to think about rolling are polyhedra in which the sum of the angles at each vertex is a submultiple of 360 degrees. For instance, given any triangle T, we can create a tetrahedron whose four faces are all congruent to T, with angle-sum 180 degrees at each vertex (there's a name for such tetrahedra but I forget what it is). Are there other polyhedra in which the angles at each vertex sum to a submultiple of 360 degrees (not necessarily the same one at each vertex)? Jim Propp

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[math-fun] Chess variant
by David Wilson 13 Nov '20

13 Nov '20

In Paul Morphy's "Opera House" chess game, Morphy never moves a piece backward. I propose Opera House Chess, with initial position and moves as in chess, except pieces cannot move backward. Is it playable? With victory conditions modified?

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[math-fun] One parameter quintic solution
by Bill Gosper 31 Oct '20

31 Oct '20

This is probably ho-hum, but while trying to simplify a Chowla-surdberg, I found that 0 = a + 80 x + 20 x^3 + x^5 has solutions {((-a - Sqrt[4096 + a^2])^(1/5) E^((2 I \[Pi])/5))/2^( 1/5) + ((-a + Sqrt[4096 + a^2])^(1/5) E^((2 I \[Pi])/5))/2^( 1/5), ((-a + Sqrt[4096 + a^2])^(1/5) E^(-((2 I \[Pi])/5)))/2^( 1/5) + ((-a - Sqrt[4096 + a^2])^(1/5) E^(-((4 I \[Pi])/5)))/2^( 1/5), ((-a - Sqrt[4096 + a^2])^(1/5) E^(-((2 I \[Pi])/5)))/2^( 1/5) + ((-a + Sqrt[4096 + a^2])^(1/5) E^(-((4 I \[Pi])/5)))/2^( 1/5), (-a + Sqrt[4096 + a^2])^(1/5)/2^( 1/5) + ((-a - Sqrt[4096 + a^2])^(1/5) E^((4 I \[Pi])/5))/2^( 1/5), (-a - Sqrt[4096 + a^2])^(1/5)/2^( 1/5) + ((-a + Sqrt[4096 + a^2])^(1/5) E^((4 I \[Pi])/5))/2^(1/5)} Is it from a two-parameter family? C.f. https://en.wikipedia.org/wiki/Quintic_function#Other_solvable_quintics --rwg

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[math-fun] Three-dimensional flat torus composed of seven cubes each touching the other six?
by Dan Asimov 01 Oct '20

01 Oct '20

Let L be any discrete subgroup of R^3 isomorphic to Z^3. The the quotient R^3/L is a "flat 3-torus", and it carries a well-defined metric. Its metric can be be described as the result of identifying opposite faces of a parallelepiped (the bounded intersection of three slabs* in R^3) in the obvious way. Of course the cubical 3-torus R^3/Z^3 is one example, but a flat 3-torus can have surprising properties. Note that the cubical 3-torus can be tessellated by 8 cubes each touching three of the other seven along two common square faces. Puzzle: ------- Does there exist a flat 3-torus that can be tessellated by seven cubes each touching the other six along a common face? —Dan ————— * A "slab" in R^3 is the closed region between two parallel planes.

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[math-fun] Puzzle: Hilbert space coloring of the cubic tiling
by Dan Asimov 01 Oct '20

01 Oct '20

Let the coordinates of (infinite-dimensional) Hilbert space H be denoted as x_1, x_2, x_3, ..., x_n, .... Note that the integer points of H correspond to arbitrary sequences of integers *only finitely many of which* are non-zero. The cubic tiling of H has one cube for each integer point of H. Definition: ----------- The cube at the integer point x is "adjacent" to the cube at the integer point y exactly if y differs from x in precisely one coordinate, by the amount ±1 in that coordinate, and has all the other coordinates equal to those of x. Puzzle: ------- Prove that given infinitely many colors, indexed by the set of integers, the cubes of the cubic tiling of H can be colored so that each cube is adjacent to exactly one cube of each other color (and not adjacent to any cube of its own color). —Dan

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[math-fun] monkeys are clever than you think
by Simon Plouffe 28 Sep '20

28 Sep '20

Hello, here are 2 films about monkeys. https://www.futura-sciences.com/planete/actualites/zoologie-video-etonnante… https://www.regardecettevideo.fr/video/3966/l-homme-est-plus-intelligent-qu… They can memorize 19 numbers in 210 milliseconds. and more. best regards, Simon Plouffe

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Re: [math-fun] What's New in Timetravel News?
by Éric Angelini 27 Sep '20

27 Sep '20

... Yes, the 5 entries of the Sator square are quoted in the movie: https://www.vox.com/culture/21419050/tenet-explained-sator-square-nolan > travel in time — Please come in! — Knock, knock, knock Best, É. Catapulté de mon aPhone > Le 26 sept. 2020 à 20:43, Brad Klee <bradklee(a)gmail.com> a écrit : > > Hi Éric, > > Thanks for this piece of information, > I assume you are referring to this > magic square: > > https://en.m.wikipedia.org/wiki/Sator_Square > > My feeling is that a true protagonist is > more likely to be working outdoors, > on behalf of the “cherry tomatoes”. > > In a worst case scenario, I doubt that > we are going to reach circumstances > where we could travel back in time for > a harvest season. > > —Brad > >>> On Sep 26, 2020, at 12:17 PM, Éric Angelini <eric.angelini(a)skynet.be> wrote: >> >>> COVID danger in closed rooms >> ... watch out, Brad, as it reads backwards >> DIVO COVID (God) if we follow the Sator >> Arepo TENET path! >> à+ >> É. >> Catapulté de mon aPhone >>>> Le 26 sept. 2020 à 18:48, Brad Klee <bradklee(a)gmail.com> a écrit : >>> COVID danger in closed rooms >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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[math-fun] What's New in Timetravel News?
by Brad Klee 27 Sep '20

27 Sep '20

According to my newsfeed, a mathematical proof of a grandfather-nonparadox theorem has been achieved in almost perfect timing with the release of Christopher Nolan's new movie "Tenet". Briefly perusing the paper, it seemed to be above my paygrade. Their idea about time traveling "agents" confused me--are the "agents" supposed to be subatomic particles or can they be equipped with extra, dangerous technology? Seeking an answer to this question, I decided to travel back in spacetime (Elsewhere in the Buddhaverse), Cordoba, Argentina, 1992, where A Plenary Lecture on closed timelike curves (CTCs) was given by Nobel Prize winner Kip S. Thorne: https://s3.cern.ch/inspire-prod-files-9/9e243ddb6420681ac2362781c9754c04 This article I could almost understand, and the conclusion in section four seems to me, most reasonable. A feature of the Feynman diagrams is that they contain CTC's at a subatomic scale. Since a theory of quantum gravity is lacking, we don't know if quantum CTC's could ever extrapolate to macroscopic scales. Later in the future, why should we be surprised if known quantum behaviors reach a semiclassical or classical cut-off, beyond which extrapolation is nonsense? The other totally stupid idea I've heard is that, due to tunneling solutions of Schroedinger's equation, people might someday be able to walk through walls. I guess I'll change my position if such a measurement is ever made during my lifetime. As for "Tenet", I risked a large theatre with ten or so other people, and liked the movie much more than interstellar (a movie that I couldn't understand in the least). A sequel could turn out even better. Terms are already defined, and we have a few questions left to wonder about. Who is this woman being compared to Oppenhiemer? What type of mechanisms exist for enacting an uncertainty principle between the orthogonal dimensions of protagonist and antagonist? All the facts of the movie point strongly to the interpretation that "the protagonist" should not be trusted! When venturing to public spaces, be advised of inherent COVID danger in closed rooms. --Brad

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Re: [math-fun] What's New in Timetravel News?
by Éric Angelini 26 Sep '20

26 Sep '20

Thanks for yr great post, Brad! I didn’t know that Sylphium was ... laser! A last link: I guess Christopher Nolan knows this 24-year old clip by Michel Gondry (BTW co-inventor of the « bullet-time Matrix effect »): https://m.youtube.com/watch?v=EN9auBn6Jys Best, É. > Le 26 sept. 2020 à 22:25, Brad Klee <bradklee(a)gmail.com> a écrit : > > Silphium!

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Re: [math-fun] Scrambled necklaces
by Keith F. Lynch 26 Sep '20

26 Sep '20

Dan Asimov <dasimov(a)earthlink.net> wrote: > Does the circular character string have marked "ends", or merely a > circular ordering? That depends on what you mean by ends. I consider it to have a specific beginning and end. I regard a rotation of a string to be different from the original. But the scrambling process ignores the ends, i.e. the scramble of a rotation will be the rotation of the scramble: reverse -> erersev everser -> rerseve versere -> ersever > Is it assigned a direction (orientation)? Yes. Otherwise the approved term would be bracelet, not necklace. > It seems that the reversals are performed consecutively. Does it > matter what order they're performed in? If so, what order should > they be performed in? In the lexical order of delimiters, i.e. ASCII order if we're talking about ASCII strings. Or are you asking about the multiple reversals with each specific delimiter? The order there is irrelevant, since no reversal bridges a delimiter character. For instance when the delimiter is the space bar, this line turns into: tni ecnatsni nehw eht retimiled si eht ecaps ,rab siht enil snrut roF:o regardless of the order of the reversals. (Note that the delimiters don't move.) The next delimiter is the comma, and you get he when instance into:For turns line this bar, space the is delimiter t James Propp <jamespropp(a)gmail.com> wrote: > 14 is divisible by 7 and 16320 is divisible by 40. > Could it be that the size of each orbit is always divisible by the > length of the string? No. For instance if every character is the same, it repeats after 1. If every character is different, it repeats after 1 or 2 depending on whether the number of characters is even or odd. If there are only two distinct characters (e.g. the string is all ones and zeros), it repeats after 1. My main question is really whether this is a known transform, and if so what is it called and where can I learn more about it? Thanks.

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