December 2015 - mathfun@mailman.xmission.com

Re: [math-fun] Raghavendra's algorithm for solving linear systems over finite fields
by Henry Baker 06 Jun '16

06 Jun '16

I can see applications in distributed/parallel implementations. You can do most of the checking & probably a lot of the randomization in parallel. At 09:02 AM 8/15/2012, Warren Smith wrote: >>Preliminary draft: >> http://www.eecs.berkeley.edu/~prasad/linsystems.pdf > >--I don't understand why anybody would want to use this algorithm >for any purpose. It's slower than everything else I can think of and also >randomized i.e. can fail. > >However... perhaps it might be useful if changed to be about finite >RINGS not fields. In rings, you cannot divide, hence gaussian elimination is >not usable. Hence this algorithm then has less competition. >(Even then I'm dubious, but then it'd have at least a hope.)

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[math-fun] 5 and 7 outcome fair dice
by Bill Gosper 06 May '16

06 May '16

Odd barrel dice have the problem that two faces are always uppermost, so how do you label them? Alternative to face-transitivity, there are fair dice which cannot rest on any face, and yet have one face uppermost. Spoiler: gosper.org/5&7dice.png --rwg I assume these are old ideas. Check out Wikipedia if you haven't seen spherical, 6 outcome dice.

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[math-fun] Computational effort
by James Propp 01 Mar '16

01 Mar '16

I don't know what the relevant units of computation are --- floating-point operations are the ones I've heard about most, but what if you're not using floating point? --- but I'm wondering if anyone has tried to quantify "how much computing" took place in (a) code-breaking at Bletchley Park, (b) the Manhattan Project, and (c) the Apollo space program. It seems likely to me that my laptop, running some silly number-crunching problem in Mathematica and returning an answer after six hours that I could probably have figured out in my head if I'd thought a little bit more before jumping in to write some code, has them all beat. What do you think? (And while we're on the subject: What's the canonical example of wasted computational effort? Isn't there some well-known old story of someone who had an early computer calculate the determinant of a singular matrix?) Jim Propp

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[math-fun] San Jose Math Decagon
by Bill Gosper 17 Feb '16

17 Feb '16

A couple of years ago Julian and I showed some animations to the San Jose Math Circle. gosper.org/decazoom.gif suggests why the area of a circle is the radius times half the circumference, and displays the continued fraction of a very lame approximation to π. --rwg

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[math-fun] Poncelet's porism puzzle
by Fred Lunnon 20 Jan '16

20 Jan '16

Recapitulating "Euler's theorem", badly garbled in an earlier post of mine: Relates circumradius R , inradius r , centre displacement d of triangle by R^2 - 2 R r - d^2 = 0 . Actually due to Chapple, according to http://mathworld.wolfram.com/EulerTriangleFormula.html . Synthetic proof at https://en.wikipedia.org/wiki/Euler's_theorem_in_geometry#/media/File:GeometryEulerTheorem.png . Provides foundation of Poncelet's porism for triangles in Euclidean 2-space; see http://mathworld.wolfram.com/PonceletsPorism.html . My Scrooge's Christmas present (to myself, naturally) was to investigate the extent to which Chapple's result generalises to higher dimensions: specifically, *** What can be established about the relation between circumradius, *** *** inradius and displacement of a tetrahedron in Euclidean 3-space? *** I'll post more about this problem after a few days, in case other social misfits out there are motivated to tackle it themselves, or alternatively can disabuse me of any delusional claim to originality. Incidentally, Poncelet's porism now has an entire book devoted to it, Vladimir Dragović, Milena Radnović "Integrable Billiards" Birkhäuser 2011 ; chapter 1 free online at http://www.springer.com/cda/content/document/cda_downloaddocument/978303480… . Fred Lunnon

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[math-fun] GCHQ puzzle
by rcs＠xmission.com 10 Jan '16

10 Jan '16

Has anyone done the GCHQ puzzle yet? The grid & constraints are at http://www.bbc.com/news/uk-35058761 I noticed one row was completely specified by the information supplied. Rich

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[math-fun] 3D version of central angle theorem
by James Propp 05 Jan '16

05 Jan '16

Is there a version of the central angle theorem that applies to spheres and solid angles? The central angle theorem (in one formulation) tells us that if C is a circle containing the point p and C' is a circle centered at p, then projection from C to C' through p halves all angles. if S is a sphere containing the point p and S' is a sphere centered at p, does projection from S to S' through p halve all solid angles? By the way, I spent about three minutes fruitlessly searching the internet for an answer, and I figure that that's enough, since one of you has probably thought about this before and can answer my question in under three minutes. But if some of you think that I (and other posters) should do more googling before we post our questions, please let me know. Jim Propp

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[math-fun] q^9 in a theta constant
by Bill Gosper 03 Jan '16

03 Jan '16

https://en.wikipedia.org/wiki/Dedekind_eta_function gives a peculiar theta expression for Dedekind eta. Combining with the more obvious DedekindEta[\[Tau]] -> EllipticTheta[1, Pi/3, E^((I Pi \[Tau])/3)]/Sqrt[3] gives the weird-looking EllipticTheta[1,Pi/3, q] == -Sqrt[3] q^(25/4) EllipticTheta[4, 15/2 I Log[q], q^9] == (Sqrt[Pi/3] EllipticTheta[1, Pi/3, E^(Pi^2/(9 Log[q]))])/Sqrt[-Log[q]] (The latter via Jacobi's imaginary transformation.) --rwg

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[math-fun] Not quite squares
by Bill Gosper 30 Dec '15

30 Dec '15

I only now notice that https://en.wikipedia.org/wiki/Gateway_Arch#/media/File:Gateway_Arch_windows… is clad in (many hundreds of) quadrilaterals, presumably all slightly different! This reminds me of the elliptical staircase and undulating bookshelf construction nightmares in Nathan Myrhvold's house|science museum, where every piece was individually water- jetted and mechanically keyed with a binary code. Except the Gateway Arch-itects had puny computers and no Computer Algebra! Another goodie from the wiki: On October 28, the arch was topped out <https://en.wikipedia.org/wiki/Topping_out> as then Vice President <https://en.wikipedia.org/wiki/Vice_President_of_the_United_States> Hubert Humphrey <https://en.wikipedia.org/wiki/Hubert_Humphrey> observed from a helicopter.[58] <https://en.wikipedia.org/wiki/Gateway_Arch#cite_note-Bryant-63> A Catholic priest and a rabbi prayed over the keystone,[32] <https://en.wikipedia.org/wiki/Gateway_Arch#cite_note-Duffy-35> a 10-short-ton (9.1 t), eight-foot-long (2.4 m) triangular section.[59] <https://en.wikipedia.org/wiki/Gateway_Arch#cite_note-Courant-64> It was slated to be inserted at 10:00 a.m. local time <https://en.wikipedia.org/wiki/Central_Time_Zone_(North_America)> but was done 30 minutes early[32] <https://en.wikipedia.org/wiki/Gateway_Arch#cite_note-Duffy-35> because thermal expansion <https://en.wikipedia.org/wiki/Thermal_expansion> had constricted the 8.5-foot (2.6 m) gap at the top[59] <https://en.wikipedia.org/wiki/Gateway_Arch#cite_note-Courant-64> by 5 inches (13 cm).[58] <https://en.wikipedia.org/wiki/Gateway_Arch#cite_note-Bryant-63> To mitigate this, workers used fire hoses to spray water on the surface of the south leg to cool it down[49] <https://en.wikipedia.org/wiki/Gateway_Arch#cite_note-Tribune-54>and make it contract.[58] <https://en.wikipedia.org/wiki/Gateway_Arch#cite_note-Bryant-63> The keystone was inserted in 13 minutes,[32] <https://en.wikipedia.org/wiki/Gateway_Arch#cite_note-Duffy-35> only 6 inches (15 cm) remained. For the next section, a hydraulic jack <https://en.wikipedia.org/wiki/Hydraulic_jack> had to pry apart the legs six feet (1.8 m). --rwg Also: Since 1965 (my graduation), US currency has inflated fourteenfold. "Seventeen Fifty Is Too Damned Much!" (MIT tuition "riot" chant.)

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[math-fun] Dodgson and non-euclidean geometry
by Bernie Cosell 30 Dec '15

30 Dec '15

In today's times, in a completely random place [in the *sports* section, I think] I ran into this statement: ... Lewis Carroll was a pseudonym for Charles Dodgson, a mathematics prof at Oxford, and Alice in Wonderland was born out of his antipathy for new mathematical concepts in the late 1800s. Math had been pretty straight-forward up until then, so Dodgson created a world based on abstract math to show that being on drugs made more sense than embracing new, ridiculous concepts. Wonderland was born out of spite! Alice was a representation of a Euclidian mathematician while Wonderland stood for Christ Church College at Oxford, where Dodgson worked. I've *NEVER* heard this "analysis" of Alice before [and the "new mathematical concepts" date to Gauss's, et alia, work in the early-to-mid 1800s, don't they?]. Have any of you ever heard anything like this? I think it is pretty unlikely.... /Bernie\ -- Bernie Cosell Fantasy Farm Fibers mailto:bernie@fantasyfarm.com Pearisburg, VA --> Too many people, too few sheep <--

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