November 2019 - mathfun@mailman.xmission.com

[math-fun] Fred Lunnon's Integer sets with distinct subset-sums, Math. Comp. 50 (1988), 297-320.
by Neil Sloane 16 Nov '19

16 Nov '19

I was shocked to see that the sequences in his paper, which are A005318 <https://oeis.org/A005318>, A006754 <https://oeis.org/A006754>, A006755 <https://oeis.org/A006755>, A006756 <https://oeis.org/A006756>, A006757 <https://oeis.org/A006757>, A195683 <https://oeis.org/A195683>, A195684 <https://oeis.org/A195684> have not been extended in 33 years - all we have are the terms Fred computed himself in 1986 (except for the first one, A005318, which is the classic Conway-Guy sequence). Maybe someone would like to extend them!

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[math-fun] Excellent Wikipedia hypocycloid animation
by Bill Gosper 10 Nov '19

10 Nov '19

https://upload.wikimedia.org/wikipedia/commons/d/da/Rolling_Hypocycloids.gif This is the decacuspid (n=10) case. What happens when n➝∞? Why do all the curves coincide only at 3 o'clock and 9 o'clock? Why isn't https://en.wikipedia.org/wiki/Epicycloid nearly as complete as https://en.wikipedia.org/wiki/Hypocycloid? Telling us, e.g., the area of Julian's <http://gosper.org/tripenta.gif> trisectrix (astroid) but not the pentasectrix (quadricuspid epicycloid). Kids: I somehow missed (or completely forgot) the cool integral for the area of the curve enclosed by {x(t),y(t)}: ½∲(x dy - y dx). Supposing a unit radius, what's the area of one of those four "cabochons"? —rwg

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[math-fun] What area is filled by the Dragon joining 0+0i to 1+0i?
by Bill Gosper 10 Nov '19

10 Nov '19

Wikipedia seems not to say. I wonder if expositors of "spacefilling curves" really feel in their gut that the space is filled. Or maybe they give the area, but Wikipedia censors it as "original research". It's probably in Knuth & Davis, Number Representations and Dragon Curves, of which I have at least 2 copies and can find neither. You can guess the answer if you believe the numbers pasted on Heighway Dragon triple point <http://gosper.org/dragtrip!.png>. But there's a direct approach. (Hint: AoCP II.) (Hint <http://gosper.org/basei-1.gif>) The Dragon's image is dense with triple points and has uncountably many double points, but I think they have measure zero, and wouldn't affect the "area" if you counted them thrice and twice. —rwg

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[math-fun] Shoelace formula, twin dragon area, Dekking article
by Joerg Arndt 10 Nov '19

10 Nov '19

* Bill Gosper <billgosper(a)gmail.com> [Nov 08. 2019 15:44]: > [...]] > Kids: I somehow missed (or completely forgot) the cool integral for the > area of the curve enclosed by {x(t),y(t)}: ½∲(x dy - y dx). And for the discrete kids, one also has to mention https://en.wikipedia.org/wiki/Shoelace_formula With this one and an automaton for the border of the twin-dragon ( see Example 4.4 in F.\ M.\ Dekking: {Recurrent Sets}, Advances in Mathematics, vol.~44, no.~1, pp.~78-104, (1982). http://dx.doi.org/10.1016/0001-8708(82)90066-4 (now open access) ) you should be able to compute good approximations of the area of the twin-dragon (you asked about it in another email, I seem to recall). Best regards, jj P.S.: Should I feel stupid because I cannot grasp the commutative diagram on page 79? (I really tried!) > [...]

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[math-fun] Elliptic Curve Animation
by Brad Klee 09 Nov '19

09 Nov '19

Plotting around the poles of Edwards's elliptic function psi it is possible to make a fairly nice picture covering 3/4 of the domain [0,2]x[0,2*tau] : https://youtu.be/m5moOyP0zek ( V 1.0, possibly more improvements upcoming) The animation goes through real "a" from ~0 to ~1. When a~0, the central region approaches to the "harmonic hyperboloid", as do four exterior loops when a~1. Today, I finally straightened out how to reduce the exact solution of pendulum equations of motion, at last using the point doubling addition rule to simplify the argument of theta(t) = 2*arcsin(...). Has anyone else done this derivation? Cheers, Brad

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[math-fun] The Sacred Riemann Sphere, Again.
by Brad Klee 07 Nov '19

07 Nov '19

These notes from IPhT are starting to look nice, especially with color figures: https://arxiv.org/pdf/1805.06405.pdf , and middle of page 16 mentions the realization that everyone needs to have about genus zero: "(it has the topology of a cylinder)". Shouldn't this be more than an after-thought? It is another way of saying "both sinh(t) and cosh(t) are singly-periodic functions over the complex plane". This analogy between surface topology and periodicity of parametric functions extends to higher genus, and begets the grand "Riemann Uniformization Theorem". My personal issue, today's gripe, is with Theorem 3.5 "Genus zero = Riemann sphere". This theorem is copied from place to place in just about every reference on algebraic topology. Such practice excludes that "Genus zero = Harmonic Hyperboloid", arguably a more fundamentally correct statement. Students continue to go unaware, while the same mantra gets over-used, again and again. I feel like I have already complained about this, repeatedly. This whole thing about the Riemann Sphere being the one True Form of Genus Zero (TFGZ) has become too dogmatic for me. I'm planning to release more detailed Heresy in the near future, so watch out! Cheers --Brad PS. For more info about "Harmonic Hyperboloid", see also: https://demonstrations.wolfram.com/SemiclassicalApproximationForQuantumHarm…

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[math-fun] Cumulative sums and hits
by Éric Angelini 05 Nov '19

05 Nov '19

Hello Math-Fun, We write, after each digit D of K, the cumulative sum of the digits so far. When this is done, we concatenate the results and iterate the process. If, at some stage, the concatenation starts with the string K, we have a hit. K = 344567 Cumulative sums of K's digits: 3, 7, 11, 16, 22, 29 Concatenation and iteration: 3711162229 3, 10, 11, 12, 13, 19, 21, 23, 25, 34 Concatenation and iteration: 3101112131921232534 3, 4, 4, 5, 6, 7, 9,... HIT! Are all integers potential hits? Best É.

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[math-fun] How can Mathematica not have Mediant?
by Bill Gosper 04 Nov '19

04 Nov '19

To get all "best" rational approximations to a real, instead of continued fractions, use Mediant: Mediant[a_] := a; Mediant[a__] := Divide @@ (Total /@ {Numerator[{a}], Denominator[{a}]}) (Just sum of numerators over sum of denominators.) Then we can, e.g., In[62]:= mediate[π, 10] Out[62]= {0, 1, 2, 3} | {22/7, 19/6, 16/5, 13/4, 10/3, 7/2, 4, 1/0} where the fractions are in numerical order and improve toward the "|". In[60]:= mediate[√2, 6] Out[60]= {0, 1, 4/3, 7/5} | {10/7, 3/2, 2, 1/0} In[41]:= mediate[√2, 12] Out[41]= {0, 1, 4/3, 7/5, 24/17, 41/29, 140/99} | {99/70, 58/41, 17/12, 10/7, 3/2, 2, 1/0} (I confess to being unable to convince Mathematica to say 1/0 vs ComplexInfinity.) Here's mediate: Unprotect[Numerator, Denominator]; Numerator[ComplexInfinity] = 1; Denominator[ComplexInfinity] = 0; Off[Power::infy] (* 1/0 is an important rational number! *) mediate[x_?NumericQ, ct_?NumericQ /; ct > 0, lo_ : {0}, hi_ : {ComplexInfinity}] := Piecewise[{{mediate [x, ct - 1, Append[lo, #], hi], # <= x}}, mediate[x, ct - 1, lo, Prepend[hi, #]]] &@ Mediant[lo[[-1]], hi[[1]]] /; ct >= 1 mediate[_, _, lo_, hi_] := lo | hi (Julian, is there a cool way to avoid repeating "mediate [x, ct - 1" with Sequence?) Pushing harder on π, In[39]:= mediate[π, 26] Out[39]= {0, 1, 2, 3, 25/8, 47/15, 69/22, 91/29, 113/36, 135/43, 157/50, 179/57, 201/64, 223/71, 245/78, 267/85, 289/92, 311/99, 333/106} | {355/113, 22/7, 19/6, 16/5, 13/4, 10/3, 7/2, 4, 1/0} so we see that 22/7 and 355/113 are both overestimates. And we see the drawback of not using continued fractions. The "15" in π = {3,7,15,1,292,...} caused 15 consecutive slowly converging underestimates. Soon, we shall face 292 more! To see how easy it is to perform mediate by hand, arithmetic with boxes <http://gosper.org/rectarith12.pdf> . —rwg

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[math-fun] Fwd: Unlikely looking Sin[π/22]
by Bill Gosper 01 Nov '19

01 Nov '19

More puzzles for FullSimplify: Sin[π/22] == 1/10 + √11 (Cos[1/5 ArcTan[5/Sqrt[877 + 1958/√5]]] - Cos[π/5 - 1/5 ArcTan[5/109 Sqrt[4385 + 1958 √5]]])/5 == 1/10 + 2/5 √11 Sin[1/10 (π- ArcTan[5/199 Sqrt[71045 - 26378 √5]])] Sin[ 1/10 (π - ArcTan[5/199 Sqrt[71045 + 26378√5]])] "Explanation": Sin[π/22] satisfies a (solvable) quintic. But isn't π/22 preferable to ArcTan[...]/5? Not if you want the radicals. Whatever happened to Mathematica's quintic solver? I have something that seems to solve (the solvable) sextics and septics as well. —rwg

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