September 2010 - mathfun@mailman.xmission.com

[math-fun] new math
by Veit Elser 30 Sep '10

30 Sep '10

Here's a puzzle I made up. In the 1960's it might have been used to get kids interested in "new math". Find two infinite sequences of non-negative integers a0, a1, a2, ... b0, b1, b2, ... such that a0 = b0 = 0, and every non-negative integer can be uniquely expressed as the sum of two integers, one from each sequence. Veit

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[math-fun] Martin Gardner archive
by James Propp 29 Sep '10

29 Sep '10

Does anyone in the math-fun list know what's happening with all of Martin Gardner's files of correspondence? It's obvious to me that they have archival value, but is it obvious to anyone who has the resources to actually make this happen? Jim Propp

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[math-fun] "Area" interpretation of trigonometric angles
by Henry Baker 28 Sep '10

28 Sep '10

One of the usual introductions to hyperbolic trig functions is through the analogy with circular functions. These introductions show that the hyperbolic "angle" can be identified with an _area_ under a hyperbola (y=1/x or one of its rotations), which then leads one to exponentials and logarithms. http://en.wikipedia.org/wiki/Hyperbolic_angle#Imaginary_circular_angle http://myyn.org/m/article/hyperbolic-angle/ BjÃ¸rn Felsager and Christina Sheets have also developed some wonderful teaching materials for hyperbolic trigonometry/geometry (do a Google search). These materials are simple enough to enable a high school student with only a traditional plane geometry background to start appreciating hyperbolic geometry; no advanced trig or even complex numbers are required. Physicists in the early 20th century were uncomfortable enough with these concepts that they stumbled around for several decades without noticing that the usual trig functions with complex arguments work just fine for special relativity. (Google search for "rapidity"; also Scott Walter; John Rhodes and Mark Semon.) Are there any other interpretations of "angle", other than the "area" interpretation and the "rapidity" interpretations?

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[math-fun] Fwd: Mathematical surprises
by Bill Gosper 25 Sep '10

25 Sep '10

George Andrews kindly reminds me that The result: # partitions of n into ordered 4tuples of triangular numbers > = sum of divisors of 2n+1. > is due to Legendre. OK, here's one that isn't Legendre's: The Minsky "circle" recurrence x[0] = 0, y[0]=8, x[n+1] = x[n] - floor(2450365*y[n]/1120052), y[n+1] = y[n] + floor(12497*x[n+1]/8798) has period 51284705405843 (= 7 * 13 * 563568191273). The first multiplier was found by tutor Corey's automatic mapper. I chose the second multiplier in an attempt to *avoid* huge periods. --rwg

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Re: [math-fun] Mathematical surprises
by Bill Gosper 25 Sep '10

25 Sep '10

A few more(?): Somos sequences are integers. And 1,1,-1,x,...,(s[n-1]*s[n-3]+s[n-2]^2)/s[n-4] gives polynomials. Which factor only if n does. # partitions of n into ordered 4tuples of triangular numbers = sum of divisors of 2n+1. # partitions into parts differing by at least 2 = # partitions into parts +-1 mod 5. (Rogers-Ramanujan). The tetrahedral analogy of Koch's Snowflake construction makes a cube. --rwg

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[math-fun] Mathematical surprises
by Ray Tayek 24 Sep '10

24 Sep '10

http://divisbyzero.com/2010/08/18/mathematical-surprises/ Im interested in compiling a list of mathematical surprises. The best possible example would be a mathematical discovery that no mathematician saw coming, but after it was discovered it changed mathematics in some fundamental wayCantors discovery of the nondenumerability of the continuum is such an example. But Ill settle for any surpriseAndrew Wiles surprised everyone with his proof of Fermats Last Theorem, the solution of the Monty Hall problem surprised many capable mathematicians, etc. Ive spent a couple days brainstorming and Ive come up with the following list. Some are better than others, and theyre listed in no particular order. Please add your surprises in the comments below! ... --- co-chair http://ocjug.org/

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Re: [math-fun] Mathematical surprises
by Ray Tayek 24 Sep '10

24 Sep '10

At 01:25 PM 9/21/2010, you wrote: >Could you provide a link for the "Baker & Tayek lists" my attempts to >find it (them?) with google failed. it's not my list. i just posted a link i saw somewhere: http://divisbyzero.com/2010/08/18/mathematical-surprises/ thanks >On Tue, Sep 21, 2010 at 1:03 PM, <rcs(a)xmission.com> wrote: > > I found the mathoverflow list to be disappointing; > > the Baker & Tayek lists are more to my taste. > > > > Rich... --- co-chair http://ocjug.org/

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[math-fun] multi-speed bicycle driven by levers & pulleys
by Henry Baker 23 Sep '10

23 Sep '10

FYI -- Just when you thought that a 150-year-old invention couldn't be improved... http://www.stringbike.com/

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[math-fun] Newton's forward differences : Maclaurin :: Thiele's reciprocal differences : ?
by Bill Gosper 23 Sep '10

23 Sep '10

It seemed like a neat idea, but through 5th order, at least, it coincides with PadeApproximant[f[y],{y,0,{<n or n+1>,n}}]. Coalescing the points of the 5th reciprocal difference took 15 cpu min to get 0/0, which then took 14 l'Hospitalings ("A very large output was generated", 3 min each) to get (3 (5 (f^(4))[0]^2-4 (f^(3))[0] (f^(5))[0]))/(40 (f^(3))[0]^3-60 (f^\[Prime]\[Prime])[0] (f^(3))[0] (f^(4))[0] +15 (f^\[Prime])[0] (f^(4))[0]^2+18 (f^\[Prime]\[Prime])[0]^2 (f^(5))[0]-12 (f^\[Prime])[0] (f^(3))[0] (f^(5))[0]) The required l'Hospitalings (l'Hospitalizations?) went 0,1,2,5,10,14,... --rwg (with Corey and Julian)

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[math-fun] Corn on the cob
by Gareth McCaughan 23 Sep '10

23 Sep '10

Are you an analyst or an algebraist? If you're presented with a delicious butter-dripping corn cob to eat, how do you eat it? According to http://bentilly.blogspot.com/2010/08/analysis-vs-algebra-predicts-eating.ht… your answers to these two questions are strongly correlated. For what it's worth, the alleged correlation works in my case; what about other munsters? -- g

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