March 2015 - mathfun@mailman.xmission.com

[math-fun] Hurwitz integers --- and it sez 'ere ...
by Fred Lunnon 11 Mar '15

11 Mar '15

https://en.wikipedia.org/wiki/Hurwitz_quaternion << The Lipschitz quaternions L form an index 2 sublattice of H. >> "index 3" , I think ?! See eg. Conway & Smith sect. 5.5 . WFL

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[math-fun] Confessions of a Mathematical Olympian
by Alex Bellos 11 Mar '15

11 Mar '15

Just in case mathfunster Adam P Goucher doesnt post this himself… here’s his funny and insightful take on the new British film X+Y about the International Mathematical Olympiad http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/mar/… <http://www.theguardian.com/science/alexs-adventures-in-numberland/2015/mar/…>

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Re: [math-fun] Numerical nerdism
by Henry Baker 11 Mar '15

11 Mar '15

Every real nerd knows that a number with a zero in the front must be in octal, so your number is bogus. Real nerds will never agree with "Newer languages have been abandoning the prefix 0, as decimal numbers are often represented with leading zeroes" https://en.wikipedia.org/wiki/Octal At 05:12 PM 3/10/2015, James Propp wrote: >Or "My zip code at MIT is two thousand one hundred and twenty nine ... with a zero in the front."

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[math-fun] Structure of the Levy C-Curve
by Allan Wechsler 10 Mar '15

10 Mar '15

This is almost certainly a question for Gosper. Wikipedia claims that the C-curve contains open sets, and therefore has area, and has scaling dimension 2. I had been under the impression that it had a scaling dimension that was just a little less than 2. Is this true? Suppose I have a C-curve in the complex plane with endpoints 0 and 1, with most of the bulk in the first quadrant. Where are some example solid areas? What is the total area of the object? How many differently-shaped "tiles" does it have? On a couple of past occasions I have tried to solve the fairly hairy set of equations arising from the different ways little families of C-curve copies can intersect, in hopes of deriving the scaling dimension. I always failed -- there was too much opportunity for stupid arithmetic errors.

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[math-fun] Numerical nerdism
by James Propp 10 Mar '15

10 Mar '15

I just thought of a new way to be annoying: When communicating to a customer service agent one of those strings of digits that pervade telephone transactions in this century, encode it as a counting number and recite it accordingly, whether or not it's what people do. As in "My Social Security Number, ma'am? It's one hundred twenty-three million, four hundred fifty-six thousand, seven hundred eighty-nine" (or whatever). Or "My credit card number, sir? It's one quadrillion, two hundred and thirty-eight trillion, (etc.)" Or "My zip code at MIT is two thousand one hundred and twenty nine ... with a zero in the front." Though I suppose that with a zip code like that, it may be redundant to say "at MIT". Jim Propp

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[math-fun] (More) area on a sphere.
by Bill Gosper 10 Mar '15

10 Mar '15

I love playing with NeilB's (& Gene's) solid angle integral. In terms of latitude and longitude, the area enclosed by a series of arcs, (lat(t),long(t)), 0<t<1 is just the sum (over the arcs) of spharea[lat_, long_] := long[1] - long[0] - Integrate[D[long[t], t]*Sin[lat[t]], {t, 0, 1}] (times r²). E.g., for the four sides of a "rectangle", (lat(t), long(t) = {{south*(1 - #) + north*# &, east &}, {north &, east*(1 - #) + west*# &}, {north*(1 - #) + south*# &, west &}, {south &, west*(1 - #) + east*# &}}; In[4]:= Factor[Total[spharea @@ # & /@ %]] Out[4]= (east - west) (Sin[north] - Sin[south]) consistent with the "hatbox" (cylindrical projection) equal area theorem. The projection of a (semiaxes) a×b by (height) z ellipse onto the unit sphere has area 2 (π- (2 b z EllipticPi[1 - b^2/a^2, (a^2 - b^2)/(a^2 + z^2)])/(a Sqrt[a^2 + z^2])) (done in two halves), presumably a crypto-symmetric function of a and b. (Does the area formula for the general a,b,c ellipsoid have six-way crypto-symmetry?) Checking, replace z by 1 and a,b by h a, h b, small h, In[37]:= Series[% /. {a -> a*h, b -> b*h}, {h, 0, 2}] Out[37]= SeriesData[h, 0, { 2 Pi - 2 a b^(-1) (a^(-2) b^2)^Rational[1, 2] Pi, 0, a b Pi}, 0, 3, 1] In[38]:= Simplify[Normal[%], a > b > 0] Out[38]= a b h² π I.e., the area of an h a × h b ellipse. I think I once mentioned here NeilB's proud ownership of a plastic-laminated cheat-sheet of handy-dandy geometry formulas, purporting to give the area of an elliptical cone in elementary functions. Probably planted by the KGB or Red Chinese during the Cold War. --rwg

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[math-fun] 24-cell --- it sez 'ere ...
by Fred Lunnon 09 Mar '15

09 Mar '15

At https://en.wikipedia.org/wiki/24-cell we are informed that << The 24-cell is the unique convex regular Euclidean polytope that is neither a polygon nor a simplex. Relaxing the condition of convexity admits two further figures: the great 120-cell and grand stellated 120-cell >> The first sentence is plainly false as it stands (eg. octahedron, tesseract); the second (sic: missing period) yields me no clue as to what might instead have been intended by the first. Any ideas out there? WFL

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[math-fun] proof of work
by Warren D Smith 09 Mar '15

09 Mar '15

we were discussing this topic before... I see John Tromp has a new idea which seems far superior: https://github.com/tromp/cuckoo/blob/master/README.md

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[math-fun] Number of legal 18x18 Go positions
by Warren D Smith 09 Mar '15

09 Mar '15

-- The important thing seems to be the ratio 2.9757341920433572493 rather than the naive estimate 3. Not a huge shrink. Their more precise estimate for mXn grid is L(m,n) =approx= 0.850639925845714538 * 0.96553505933837387^(m+n) * 2.97573419204335724938^(m*n)

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[math-fun] Number of legal 18x18 Go positions
by Henry Baker 09 Mar '15

09 Mar '15

FYI -- https://tromp.github.io/go/legal.html Following 9 months of computation and 4 petabyte of disk IO on a Dell PowerEdge R280 server, generously provided by Piet Hut, and administered by Lee Colbert, at the IAS School of Natural Sciences in Princeton, we determined L(18,18) = 669723114288829212892740188841706543509937780640178732810318337696945624428547218105214326012774371397184848890970111836283470468812827907149926502347633 which factorizes as 7176527950749135946361 * 93321327372370764890001511447003537297637977042784059156126983293481833335226774708913094831974704397065455646983337368444156410553 The computation uses dynamic programming to count the number of paths through a graph consisting of 18^2=324 layers with up to 81 billion nodes each. Each node corresponds to a set of partial board (e.g. the top 7 rows plus 6 points on the 8th row) positions that are equivalent in their set of valid completions. Our paper explains the algorithm in detail, and also explains how the resulting exact counts allow derivation of the approximation formula L(m,n) ~ 0.850639925845714538 * 0.96553505933837387^{m+n} * 2.97573419204335724938^{m*n} which gives us the approximate number of legal positions on a standard board size L(19,19) ~ 2.08168199381982*10^170 https://tromp.github.io/go/gostate.ps

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