April 2020 - mathfun@mailman.xmission.com

Re: [math-fun] Deep Learning for Symbolic Mathematics
by Steve Witham 01 May '20

01 May '20

From: Marc LeBrun <mlb(a)well.com> > Don't recall this getting discussed here: https://arxiv.org/pdf/1912.01412.pdf Amazing results. My only gripe is that it's not clear how much processing time was used by the neural net, but they gave, e.g., Mathematica 30 seconds. > Interestingly treats symbolic integration as a kind of linguistic translation. Well, they treat it in the same way people treat linguistic translation: with a sequence-in, sequence out interface. Here they encode both as forward polish strings. If you think of a neural net as like a matrix multiplication, you can do a state transition including an input and/or output in a single step. It's actually quite a bit like the different transforms of an IFS fractal. A friend of mine was pushing in and reading out balanced parenthesis strings (known to be not-absolutely-trivial) using a hand-designed 2D IFS around Y2K and I thought it was pointless. https://simondlevy.academic.wlu.edu/files/publications/bics2004.pdf > Leverages the cost asymmetry "trap door" versus differentiation when training. Do you mean the way they generate problem-solution pairs by starting with the solution and differentiating to get the problem? (I don't remember which sage said that intelligence amounts to inverting functions.) --Steve

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[math-fun] Art + Math = Man Ray
by Éric Angelini 29 Apr '20

29 Apr '20

Hello Math-Fun, here is a 70 minutes documentary about "Man Ray and the shakespearian equations", produced in 2019 by... a Fields Medal (Cédric Villani) -- who writes, in the last credit, that "no mathematicians were hurt during the shooting"! 90% of the video is in French without subtitles -- 10% in English. For me the film is a bit "old-fashioned" in its structure and narrative (especially Shakespeare played by a French company) but still interesting due to the wonderful archives material staging Man Ray, the Forties, Fifties and Sixties and the "sculptures" of the Poincaré Institute. Access to this ducumentary is free until May 17th, if I correctly understand the side notes in Le Monde paper. Anyway it is enjoyable! https://www.beauxarts.com/videos/en-exclusivite-le-documentaire-man-ray-et-… Best, É.

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[math-fun] The dodecahedron vertex solid angles
by Bill Gosper 29 Apr '20

29 Apr '20

are, nicely enough, 2 ArcTan[GoldenRatio⁵] = π/2 - ArcTan[11/2]. The good news: Mathematica 12 has polyhedron vertex solid angle. The bad news: In[473]:= ByteCount[PolyhedronAngle[PolyhedronData["Dodecahedron", "Polyhedron"], 12]] // tim During evaluation of In[473]:= 0.689688,0 (* secs *) Out[473]= 44152 (* bytes *) In[474]:= FullSimplify[PolyhedronAngle[PolyhedronData["Dodecahedron", "Polyhedron"], 12]] // tim During evaluation of In[474]:= 14018.13381,2 Out[474]= 2 ArcTan[11/2 + (5 Sqrt[5])/2] In[475]:= 14018./3600 Out[475]= 3.89388888888889 3.9 hours! (Another 12.x produced 180000 bytes and the FullSimplify never answered.) Also nice: Recall that a dodecahedron face subtends about 1 steradian wrt the center, because 4π/12 ~ 1 <http://gosper.org/1steradian.png>. Conwayish question: Given that alternating Dodecahedra and EndoDodecahedra (EndoBob StarPants <http://gosper.org/endogram.png>) "checker" the 3D grid, what are the two vertex solid angles (convex and nonconvex trihedrals) of the latter? (Kids: Remember that 4π steradians is a "full sphere" in the same sense that 2π radians is a "full circle".) Lastly, some numerology with yesterday's DisdyakisTriacontahedron ("d120" die): Its faces are all congruent triangles with the two shortest edges in ratio very nearly π/2: In[471]:= PolyhedronData["DisdyakisTriacontahedron", "EdgeLengths"] Out[471]= {1, 3/10 (3 + √5), 7/5 + 1/√5} In[472]:= 2 N@% Out[472]= {2., 3.14164078649987, 3.69442719099992} In[1718]:= ContinuedFraction[3 (3/5 + 1/√5)] Out[1718]= {3, {7, 16, 1, 1, 1, 2, 1, 6, 1, 2, 1, 1, 1, 16, 7, 2, 1, 1, 5, 1, 1, 66, 1, 1, 5, 1, 1, 2}} (periodic, vs 3 7 15 1 292 1 1 1 ... fer sure not periodic.) —rwg

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[math-fun] Deep Learning for Symbolic Mathematics
by Marc LeBrun 29 Apr '20

29 Apr '20

Don't recall this getting discussed here: https://arxiv.org/pdf/1912.01412.pdf Interestingly treats symbolic integration as a kind of linguistic translation. Leverages the cost asymmetry "trap door" versus differentiation when training. Thoughts? (Might something similar deconvolve a long digit string into two prime factors? Multiplying up a training set of a couple of hundred million products is easy!)

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[math-fun] Fibonacci/Golden Ratio in Art
by Henry Baker 28 Apr '20

28 Apr '20

A friend of mine who works in an art museum is looking at Fibonacci influences in art. Yes, this has been done many, many times before, and I've seen a number of exhibitions of this type, but I haven't been that impressed by the artistic and mathematical quality or depth of insight provided by most of these previous exhibitions. Furthermore, many (if not most) of the supposed findings of phi (the Golden ratio) in art are spurious coincidences, without any shred of evidence that the artist knew anything about golden ratios. A big problem is finding images/sculptures/animations that are both interesting/fun to look at, but also provide interesting/fun mathematical/geometric insights. I'm hoping that some of you have been equally disappointed in previous exhibitions of this type, and may have some "pet" images/ideas/insights that you'd like to see in such an exhibition. This may be your chance to get artistic attribution! He has not allowed me to go public with his name or museum affiliation, so I can't tell you those things yet -- it may be a number of months. But you may want to be on the lookout for interesting ideas to put into the back of your mind. Some obvious things: Penrose tilings, Archimedes spiral gears, flowers, sea shells, etc. But I'd love to see some 3D-printed sculptures of cool shapes. Animations of mechanisms would also be very cool. I'd love to find any Islamic geometric art that utilizes phi. -------- While Google searching for Fibonacci/phi, I asked myself about the phi number field. Anything interesting there?

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[math-fun] Rational approximate Platonic dodecahedra
by Fred Lunnon 28 Apr '20

28 Apr '20

To construct a family of pentagonal dodecahedra P_k , all their vertices with rational Cartesian components, which successively approximate a regular specimen P . More pedantically, for each real error bound e > 0 , there should exist some h such that dist(P, P_k) < e for all k >= h ; where dist(P, Q) is defined as (say) square root of sum of squared distances between corresponding vertices of (combinatorially isomorphic) polyhedra P, Q . See Adam Goucher, James Buddenhagen https://cp4space.wordpress.com/2013/09/22/rational-approximations-to-platon… (Have I come across those fellows somewhere before?) Günter M. Ziegler (2007) "Non-rational configurations, polytopes, and surfaces" https://arxiv.org/abs/0710.4453 (Undefined terms in the latter may not mean quite what one might expect: like "simple", "polyhedron", "polytope"!) Fred Lunnon

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[math-fun] Game of life L patterns
by Allan Wechsler 27 Apr '20

27 Apr '20

Consider a life pattern that consists of the cell (0,1), plus the straight n-omino (0..n-1,0). This is an L-shaped pattern with one arm of length 2 and the other of length n (counting the corner as belonging to both arms). What is its settle-time as a function of n? Of course, defining the settle time rigorously is philosophically fraught, but all the cases I have looked at resolve eventually, into a simple structure of oscillators and spaceships, so the difficult questions haven't arisen yet. The nth L in this family has n+1 cells. Starting from n = 0, I get the following settle times: 1, 1, 1, 3, 9, 40, 13,122, 30, 786, 342, 186, 2062, 268, 89, 1180, 307, 3526, 49, 79, 875 The last number is the settle time for the case n=20. This sequence is very reminiscent of A152389, which gives the settle time for simple straight n-ominoes. Can anyone confirm the values I've given here, and provide a few more? I'd like to submit the sequence to OEIS.

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[math-fun] "Dedecision" ?
by Steve Witham 27 Apr '20

27 Apr '20

Is "Dedecision" an old pun? Â --Steve

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Re: [math-fun] Fibonacci/Golden Ratio in Art
by Henry Baker 27 Apr '20

27 Apr '20

Interesting! Thanks! Perhaps Q(i*phi) is more visually interesting than Q(phi), since we can get the complex plane ? At 09:53 AM 4/26/2020, Adam P. Goucher wrote: >> While Google searching for Fibonacci/phi, I asked myself about >> the phi number field. Anything interesting there? > >Every regular n-polytope (where n >= 3) is geometrically similar >to a finite subset of the vector space F^(n+1), where F is the >field of characteristic zero generated by phi. > >Best wishes, > >Adam P. Goucher

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[math-fun] i^i^i^i
by James Propp 27 Apr '20

27 Apr '20

It's well known that the expression i^i takes on an infinite set of values if we understand w^z to mean any number of the form exp (z (ln w + 2 pi i n)) where ln is a branch of the natural logarithm function. Since all values of i^i are real, all values of i^i^i (by which I mean i^(i^i))) are on the unit circle, and in fact they form a countable dense subset of the unit circle. I can't figure out what's going on with i^i^i^i, though. I've posted a Mathematica notebook at http://jamespropp.org/iiii.pdf containing some intriguing images starting on page 2. Each image shows the points in the set of values of i^i^i^i lying in an annulus whose inner and outer radii differ by a factor of 10. Can anyone see what's going on? Also, what happens for taller towers of exponentials? Does the set of values of i^i^...^i become dense once the tower is tall enough? Jim Propp

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