May 2020 - mathfun@mailman.xmission.com

Re: [math-fun] Dodecahedron packing conjecture
by Steve Witham 09 May '20

09 May '20

> From: Bill Gosper <billgosper(a)gmail.com> > Date:Â 5/9/20, 2:46 PM > > Ages ago I mentioned that regular dodex can interlock in an "airtight" sheet > <http://gosper.org/dodex.gif>, > analogous to a sheet of cubes ("Martin's Marbles") > <http://gosper.org/martinsmarbles.png>.Â Isn't this just a plane > section perpendicular to (1,1,1) through the 3D endo-dodec checkerboard... I "see" the cross-section that's a regular hexagonal tiling, I see it as cutting through the "equators" in dodex.gif. I'm pretty convinced the dodecahedrons don't intersect. I see an analogy to martinsmarbles. My brain can't grasp the endo-dodec picture well enough to answer the question. On both the dodex and martinsmarbles, it seems to me that the airtightness is just a 2D slice. There are leaks "of measure zero." Both arrangements have corner points with air on both sides. On dodex it's clearer to me anyway: some of the cutaways show pyramidal air-pockets with their tips touching the "equator". I'm pretty sure the air-pyramids come in pairs with tips (almost?) touching. Is that why you put scare-quotes on "airtight"? On May 9, 2020, at 4:00 PM, James Propp <jamespropp(a)gmail.com> wrote: > I don't understand this [endo-dodec] question. The airtight sheet is a 3-dimensional > structure; a plane section of a 3-d honeycomb is a 2-dimensional structure. From: Veit Elser <ve10(a)cornell.edu> > Jim, look at a the cut-away dodec. on the left of <http://gosper.org/dodex.gif>. > ThereÂ’s a special plane orthogonal to the 3-fold axis where the cross-sections > are regular hexagons. You need to check that when extended above and below > this plane the dodecÂ’s donÂ’t interpenetrate, but it seems to me that part of the > construction also holds water. Heh. --Steve

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[math-fun] Sum of initial nth powers equal to another nth power?
by Dan Asimov 09 May '20

09 May '20

Famously, 1^2 + 2^2 + 3^2 + ... + 24^2 = 70^2 is the only case where the sum of an initial sequence of squares equals another square (ignoring 1^2 = 1^2). Are there any non-trivial examples of this for powers higher than 2 ? (For non-initial cases, 3^3 + 4^3 + 5^3 = 6^3 is the first example, and this website <https://www.mathpages.com/home/kmath147.htm> gives many other examples for cubes.) But I'm interested in when 1^k + 2^k + ... + n^k is an exact kth power for k > 2 and of course n > 1. —Dan

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[math-fun] Playing the numbers: another miraculous hit!
by Brad Klee 09 May '20

09 May '20

During a combinatorial search for more well-integrable geometries (ha ha), the following differential equation turned up: 8*(2 - 3*x)*T - d/dx( x*(1 - x)*(25 - 27*x)*T' ) = 0. As the period constraint of a Hamiltonian: 2H = r^2 - (8/25)*r^4 + (18/625)* r^6 - (2/625)* r^6 *Cos[6*z] . ( in (r,z) polar coordinates ) The integer coefficients of the period expansions are: T@0=1, 16, 330, 7360, 170650, 4050816, 97699056, 2383814400,... T@(25/27)=1, 84, 16830, 3971640, 1030948650, 282553385184, 80264956143456,... T@1=1, 4, 30, 280, 2890, 31584, 358176, 4168560, 49455450, 595480600,... and the integer coefficients of nome expansions are: q@0=0, 1, 40, 1746, 79632, 3730134, 177917808, 8598649428, 419770317216,... q@(25/27)=0, 1, 285, 122346, 59567918, 31213994709, 17129120744817,... q@(1)=0, 1, 65, 5550, 540250, 56586875, 6208576875, 703359012500, ... Only T@1 is in OEIS: https://oeis.org/A274665, and it comes from a list by Bostan et al. see ref. [1,2] entry SS[16]. The authors are interested in connections to physics, so I think it's right in line to mention this alternative interpretation from Hamiltonian mechanics. Entries SS[2] and SS[15] have similar interpretations in terms of Hamiltonian functions with elliptic level curves. I checked the list given on J.A. Weil's website, and didn't find any other matches with my own records. It would be nice to have a more complete list of similar geometries, but so far it has not been too easy to make progress (stronger, faster algorithms are always needed). --Brad [1] https://arxiv.org/pdf/1507.03227.pdf [2] http://www.unilim.fr/pages_perso/jacques-arthur.weil/diagonals/3var/3var_OR…

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Re: [math-fun] Looking for Feller Probability Theory" volume 2
by Axel Vogt 09 May '20

09 May '20

You may try abebooks, https://www.abebooks.de/servlet/SearchResults?an=feller&bi=0&bsi=30&bx=off&… > Message: 2 > Date: Sat, 9 May 2020 12:32:21 -0400 (GMT-04:00) > From: Dan Asimov <dasimov(a)earthlink.net> > To: math-fun(a)mailman.xmission.com > Subject: [math-fun] Looking for Feller Probability Theory" volume 2 > Message-ID: > <1644941167.2652.1589041941376(a)wamui-agami.atl.sa.earthlink.net> > Content-Type: text/plain; charset=UTF-8 > > Does anyone have a copy of > > *[volume 2]* of "Introduction to Probability Theory and Its Applications" ? 2nd edition, by William Feller ? *[copyright 1971]* > > in *very good condition* that they'd be willing to sell (or donate) to me? > > I recently bought a copy online that turned out to be in much worse physical condition than advertised plus strongly smelling of mildew, so I need another copy to replace it. > > If you do, please write me directly at <asimov(a)msri.org>. > > Many thanks, > > Dan

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[math-fun] Moebius-like torus name?
by Steve Witham 09 May '20

09 May '20

What does one call a tube looped around as if to make a torus, but connected clockwise-to-counterclockwise, as in this sewing pattern? z y>--C-->z y +---------+ x|w>--A-->x|w v|^ v|^ ||| ||| B|D B|D ||| ||| v|^ v|^ y|z<--C--<y|z +---------+ w x<--A--<w x Unless I'm fooling myself, this makes a way to wrap the sides of the board around for a game like Life or Langton's Ant, in such a way that the space is locally flat everywhere--without any local kinks or anomalies that would snarl the game rules. I have tried to think of such a thing so many times in the past and failed... Is this consistent? What's it called? --Steve

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[math-fun] evil hangman
by Thane Plambeck 08 May '20

08 May '20

evil hangman is the usual word-guessing game with a puzzle-setter who reserves the right to secretly change the sought word mid-game to another one that's still consistent with all your guesses so far. we put up a version you can try here (as the guesser) http://labs.counterwave.com/evilhangman/ there's a recent paper about evil hangman in the arxiv, too (not our work), but we simply wanted to play it, so we created this thing. -- Thane Plambeck tplambeck(a)gmail.com http://counterwave.com/

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Re: [math-fun] [EXT]Re: Non-standard "polyhedra" ...
by Dan Asimov 07 May '20

07 May '20

There are many "regular maps on surfaces" — an old-fashioned term for a tiling of a surface by p-sided polygons, q per vertex, with a regularity condition: Any regular map (or chiral map as below) can be made of regular polygons as long as the metric on the surface is adjusted accordingly. * If any polygon is mapped onto any polygon in any of the 2p dihedral ways, that map extends to an isomorphism of the whole structure. If this seems unnecessary, consider a tiling of a torus by just 2 squares next to each other — a 2 x 1 rectangle with opposite sides identified: _________________ | | | | | | | | | | | | ————————————————— (There does, however, exist a regular map of the torus made of just 2 squares.) There is also a weaker condition: * If any polygon is mapped onto any polygon in any of the p rotational ways, that map extends to an isomorphism of the whole structure. If a map on a surface is not a regular but satisfies this condition, it's called a "chiral map". The first and extraordinarily lovely examples of chiral maps occur on tori: There is one made of 5 squares and one made of 7 hexagons. The one of 7 hexagons realizes the 7-coloring of a torus. Its automorphism group is size = 42. A theorem of Hurwitz puts a bound on the size of the symmetry group G of a regular or chiral map. For an orientable surface of genus g: |G| <= 168(g-1). For a chiral map, |G| <= 84(g-1). My favorite regular map is the first one where this bound is reached. It consists of 24 heptagons, 3 per vertex, tiling the 3-hole torus. Its symmetry group is size 336 and its subgroup of orientation-preserving symmetries is the (unique) simple group of size 168, the second-smallest nonabelian simple group. (The smallest one is the alternating group A_5, the group of rotations of the dodecahedron — which is of course an example of a regular map.) I was very surprised when first hearing of this regular map: The 3-holed torus seems to have nothing to do with the number 7, and yet this regular map has a great deal to do with 7. In case you know about complex projective varieties, a 3-holed torus can be expressed as a Riemann surface with exactly this symmetry by the equation X Y^3 + Y Z^3 + Z X^3 = 0 for [X : Y : Z] a point of the complex projective plane CP^2. This is the famous "Klein quartic", about which a whole book was written: <http://library.msri.org/books/Book35/contents.html>. —Dan

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Re: [math-fun] Non-standard "polyhedra" ...
by Dan Asimov 06 May '20

06 May '20

Combinatorily a polyhedron can be just thought of as a decomposition of the sphere into geodesic polygons. If you expand the definition to include other compact surfaces besides the sphere, there are many lovely examples of regular polyhedra. For instance, the 3-holed torus can be tiled in a regular way with 8 hexagon, with each vertex surrounded by 4 of them. Resulting in a symmetry group of size 96. (Determined by: one hexagon going to any of the 8, in any of 12 ways.) But if you try to tile the sphere with hexagons, you find it won't work. Suppose there are N hexagons averaging q >= 3 around each vertex. The number of vertices must be V = 6N/q. Then the number of edges must be E = 6N/2 = 3N. The number of faces is F = N. Since the Euler characteristic X(S^2) = 2, this means 2 = X(S^2) = V - E + F = 6N/q - 3N + N = N(6/q - 2). But since q >= 3, this is impossible. —Dan Colin writ: ----- Forgive me if this is obvious and well-known. Because of Euler we know that for polyhedra without holes we have V+F=E+2. As a result we can't have a polyhedron made entirely of hexagons, and need at least 12 pentagons or an equivalent set of shapes. Etc. But things change if we allow intersecting faces. Clearly we have a lot of scope for variations, but we can always retain limits to make things reasonable. But ... Can we make a "polyhedron" consisting only of hexagonal "faces"? -----

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Re: [math-fun] math-fun Digest, Vol 207, Issue 16
by Colin Wright 06 May '20

06 May '20

>> Can we make a "polyhedron" consisting >> only of hexagonal "faces"? > From: Mike Stay <metaweta(a)gmail.com> > https://en.wikipedia.org/wiki/Szilassi_polyhedron Fabulous ... thank you. Colin -- I love to go sit in the woods with a favorite book. That way the trees know what happens to them if they try anything. -- twitter.com/TheAndrewNadeau/status/1156712445903290368

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[math-fun] Non-standard "polyhedra" ...
by Colin Wright 06 May '20

06 May '20

Forgive me if this is obvious and well-known. Because of Euler we know that for polyhedra without holes we have V+F=E+2. As a result we can't have a polyhedron made entirely of hexagons, and need at least 12 pentagons or an equivalent set of shapes. Etc. But things change if we allow intersecting faces. Clearly we have a lot of scope for variations, but we can always retain limits to make things reasonable. But ... Can we make a "polyhedron" consisting only of hexagonal "faces"? Thanks Colin -- Every mathematical proof is a one liner if you start far enough on the left -- Anon

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