February 2019 - mathfun@mailman.xmission.com

[math-fun] I'm so buzzed I can't tell a 6 from a 9.
by Bill Gosper 09 Feb '19

09 Feb '19

In[38]:= ChemicalData["Caffeine", "IUPACName"] Out[38]= "1,3,7-trimethylpurine-2,6-dione" agreeing with https://en.wikipedia.org/wiki/Caffeine#Chemistry in both structural formula (ChemicalData["Caffeine"]) and name. Why isn't it 1,3,7-trimethylpurine-2,9-dione? I.e., Why does the imidazole (pentagonal) ring count for the methyls, but not the -ones? —rwg

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[math-fun] "Smart" soccer balls ?
by Henry Baker 08 Feb '19

08 Feb '19

I've been thinking about what one could do with a sphere whose surface could be deformed slightly -- sort of like those deformable mirrors on modern astronomical telescopes. But I was thinking more about mechanical effects rather than optical effects. So a soccer ball with a somewhat flexible surface material covers a number small actuators which can dimple the surface outwards and inwards by a relatively small amount. Thus, the ball could still be waterproof, for example, and thus protect any electronics inside. Although I initially conceived of this as a *regular Platonic solid*, with modern micro- electronics, the arrangement of actuators doesn't even have to be regular, and indeed could be more-or-less randomly distributed around the surface of the sphere. The soccer ball is relatively well-balanced, so any electronics within would have to be distributed in a mostly symmetrical fashion. So what could such a ball do? If the ball were flying through the air, it could set up asymmetric drag and change its trajectory to some degree. Even if it weren't *guided*, it could still move in a random and impossible-to-predict manner which would thoroughly confuse even the best goalies. If the ball were rolling along a surface, it could change direction and possibly even speed up/slow down if the dimples were large enough and suitably synchronized. I got this idea by watching some of the existing and planned Mars landers, and instead of merely using this idea as part of the landing, I asked myself why the "ball" couldn't simply BE the lander itself, rather than merely a protective shield. One problem might be that the soccer ball finds its way into an appropriate sized hole that makes it impossible for the ball to roll itself out. In this case, it might simply blow up a much larger balloon, climb inside, and by continuing to roll around *inside* the much larger balloon, it might maneuver the larger balloon out of the hole. In order to do this, the soccer ball would have to be able to *climb* up the inside of the larger balloon. However, this is a much easier problem than trying to climb up an arbitrary surface. Of course, the original soccer ball itself could also be an inflated structure, which could be stored in a much smaller volume prior to inflation, and possibly even be used when navigating -- e.g., to get between two trees that might otherwise block the soccer ball's progress. Note that these ideas could also make a "smart" baseball/softball or tennis ball which might be good for "training" (wink, wink). In the case of balls moving through the air, little actuated "hairs" (cilia) might have a greater ability to effect asymmetrical drag. One of the easiest applications would be balls that return on their own -- e.g., during batting or tennis practise or soccer goal-kicking practise. Who knows? With the amount of money involved in international soccer, perhaps such things already exist?

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Re: [math-fun] More Eta nagging
by Bill Gosper 06 Feb '19

06 Feb '19

The Theta constants in terms of eta: {Equal[EllipticThetaPrime[1, 0, q], 2*(DedekindEta[-((Log[q^2]*I)/(2*Pi))])^3], Equal[EllipticTheta[2, 0, q], ((2*(DedekindEta[-((Log[q^4]*I)/(2* Pi))])^2)/(DedekindEta[-((Log[q^2]*I)/(2*Pi))]))], Equal[EllipticTheta[3, 0, q], (((DedekindEta[-((Log[q^2]*I)/(2* Pi))])^5)/((DedekindEta[-((Log[q]*I)/(2* Pi))])^2*(DedekindEta[-((Log[q^4]*I)/(2*Pi))])^2))], Equal[EllipticTheta[4, 0, q], (((DedekindEta[-((Log[q]*I)/(2*Pi))])^2)/(DedekindEta[-((Log[ q^2]*I)/(2*Pi))]))]} Actually, there are many other Theta constants, like ...Theta[...,π/3,q], Theta[...,π/4,q],... that can be done with etas. I didn't mention π/2 because things like In[522]:= FullSimplify@%519 Out[522]= EllipticTheta[1, \[Pi]/2, q] == EllipticTheta[2, 0, q] should autosimplify! In[523]:= Series[%, {q, 0, 69}] Out[523]= True Using the modular property of eta, plus the above, provides such futuristic benchmarks as EllipticThetaPrime[1, 0, E^(-(1/13) (I + 2 Sqrt[3]) \[Pi])] == ((-1)^(1/4) 3^( 3/8) (((11 - 4 I Sqrt[3]) (Sqrt[2] + Sqrt[3]))/((1 + Sqrt[3]) (2^( 3/4) + 2^(1/4) Sqrt[3])))^(3/4) Gamma[1/3]^(9/2))/( 8 2^(11/16) \[Pi]^3), EllipticTheta[2, 0, E^(-(1/13) (I + 2 Sqrt[3]) \[Pi])] == ( 3^(1/8) Sqrt[1 + Sqrt[2]] ((11 - 4 I Sqrt[3]) (Sqrt[2] + Sqrt[3]))^( 1/4) (98 + 40 Sqrt[6])^(1/16) Gamma[1/3]^(3/2))/( 2 2^(23/48) Sqrt[1 + Sqrt[3]] \[Pi]), EllipticThetaPrime[1, 0, E^(-(2/13) (I + 2 Sqrt[3]) \[Pi])] == ((-1)^(1/8) 3^( 3/8) ((1 + Sqrt[2]) (11 - 4 I Sqrt[3]) (Sqrt[2] + Sqrt[3]))^(3/4) Gamma[1/3]^(9/2))/(32 2^(1/16) (1 + Sqrt[3])^(9/8) \[Pi]^3), EllipticTheta[4, 0, E^(-(2/13) (I + 2 Sqrt[3]) \[Pi])] == ((-3)^( 1/8) ((11 - 4 I Sqrt[3]) (Sqrt[2] + Sqrt[3]))^(1/4) Gamma[1/3]^( 3/2))/(2 2^(5/48) (1 + Sqrt[2])^(1/4) (1 + Sqrt[3])^(1/8) Sqrt[ 2^(3/4) + 2^(1/4) Sqrt[3]] \[Pi]) —Bill On Sat, Jan 26, 2019 at 12:36 AM Bill Gosper <billgosper(a)gmail.com> wrote: > (Probably repeating myself, but) > In[86]:= eta124 > > Out[86]= 0 == -DedekindEta[2 \[Tau]]^24 + > DedekindEta[\[Tau]]^16 DedekindEta[4 \[Tau]]^8 + > 16 DedekindEta[\[Tau]]^8 DedekindEta[4 \[Tau]]^16 > > E.g., > In[81]:= eta124 /. \[Tau] -> I/2/\[Sqrt]3 > > Out[81]= 0 == -DedekindEta[I/Sqrt[3]]^24 + > DedekindEta[I/(2 Sqrt[3])]^16 DedekindEta[(2 I)/Sqrt[3]]^8 + > 16 DedekindEta[I/(2 Sqrt[3])]^8 DedekindEta[(2 I)/Sqrt[3]]^16 > > In[84]:= FullSimplify@%81 > > Out[84]= DedekindEta[I/Sqrt[3]]^24 == > DedekindEta[I/(2 Sqrt[3])]^8 DedekindEta[(2 I)/Sqrt[ > 3]]^8 (DedekindEta[I/(2 Sqrt[3])]^8 + > 16 DedekindEta[(2 I)/Sqrt[3]]^8) > > 12.0 can't even do this case, let alone symbolic 1-2-4. > > In[85]:= N@% > > Out[85]= True > In[91]:= %84 /. etavals > > Out[91]= (19683 Gamma[1/3]^36)/(4294967296 \[Pi]^24) == ( > 729 Gamma[1/ > 3]^24 ((27 Gamma[1/3]^12)/(512 (1 + Sqrt[3])^2 \[Pi]^8) + ( > 27 (1 + Sqrt[3])^2 Gamma[1/3]^12)/(2048 \[Pi]^8)))/( > 16777216 \[Pi]^16) > > In[92]:= FullSimplify@% > > Out[92]= True > > Written as infinite products, 1-2-4 is Jacobi's Æquatio Identica Satis > Abstrusa. > > Besides 1-2-4, there are relations for 1-3-9, 1-4-16, 1-5-25, ... > There is even 1-2-3: > DedekindEta[\[Tau]]^60 DedekindEta[2 \[Tau]]^24 - > DedekindEta[\[Tau]]^72 DedekindEta[3 \[Tau]]^12 - > 12 DedekindEta[\[Tau]]^48 DedekindEta[2 \[Tau]]^24 DedekindEta[ > 3 \[Tau]]^12 - > 119168 DedekindEta[\[Tau]]^24 DedekindEta[2 \[Tau]]^48 DedekindEta[ > 3 \[Tau]]^12 - > 16777216 DedekindEta[2 \[Tau]]^72 DedekindEta[3 \[Tau]]^12 + > 196830 DedekindEta[\[Tau]]^36 DedekindEta[2 \[Tau]]^24 DedekindEta[ > 3 \[Tau]]^24 + > 19131876 DedekindEta[\[Tau]]^24 DedekindEta[2 \[Tau]]^24 DedekindEta[ > 3 \[Tau]]^36 + > 387420489 DedekindEta[\[Tau]]^12 DedekindEta[ > 2 \[Tau]]^24 DedekindEta[3 \[Tau]]^48 > > and presumably a-b-c for arbitrary distinct positive rationals. With > Macsyma I once > computed a big list of these, which seems now to break off in the middle > of 1-4-5. > Anyway, this claims that, e.g., DedekindEta[22I/7] > is some monstrous algebraic number times DedekindEta[I] = Gamma[1/4]/(2 > \[Pi]^(3/4)). > —Bill > The reason I nag is that nearly everything is made out of Etas—Theta > constants, elliptic Integrals, > moduli, lambda*, Klein's invariant, . . ., and then there is Eta' (another > case where a function's > derivative is hairier than the function). >

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[math-fun] Number-of-pieces question (in the limit)
by Dan Asimov 06 Feb '19

06 Feb '19

Imagine a large unit circle C centered at the origin of the plane. Also assume the plane is endowed with the usual hexagonal tessellation T but after the tessellation has been uniformly shrunk toward the origin by a factor of f with 0 < f < 1, to obtain the tessellation we call fT. Given f, define H(f) = the union of the hexagons of fT lying entirely inside or on C. Now suppose that 50% of the hexagons of fT are selected independently at random. Call the *union* of these hexagons X. Questions: ---------- What is the expected number N(f) of connected components of X as a function of f ??? As f —> 0+, is N(f) asymptotic to (a constant times) some power f^p of f? —Dan

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[math-fun] Number of 15x15 crossword puzzle grids
by Michael Kleber 04 Feb '19

04 Feb '19

The 538 Riddler from two and a half weeks ago asked for the number of 15x15 black-and-white grids that are legal for crossword puzzles. https://fivethirtyeight.com/features/how-many-crossword-puzzles-can-you-mak… Rules: Every white square must be "checked" (in both a horizontal and vertical word), words must be at least 3 letters long, the whole grid must be symmetric under 180-degree rotation, and the set of white squares must be connected. Evidently nobody solved it, so I decided to :-) Answer: there are 347804238364806 legal 15x15 grids according to those rules (347 trillion). And if you don't like the fact that those rules allow some all-black rows around the edges of the puzzle, and want them to be "really" 15x15, then that cuts down the count to 342935406100702. To convey the information to the outside world (including the 538 Riddler guy), I've posted some more about this on Twitter. Please forgive me. https://twitter.com/Log3overLog2/status/1092472516571000839 NJAS, the sequence of counts of nxn grids for n=3,5,7,9,11,13,15 is: All grids: 1, 15, 397, 35184, 17431781, 37147554097, 347804238364806 No black edges: 1, 12, 312, 31047, 16459558, 36076951460, 342935406100702 --Michael -- Forewarned is worth an octopus in the bush.

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Re: [math-fun] disyllabic panvowel
by Dan Asimov 02 Feb '19

02 Feb '19

Not just the first but the second word below was also coined in the 20th century. In the second case, it was a successful attempt to beat the longest-word record in a certain dictionary. —Dan ----- supercalifragilisticexpialidocious pneumonoultramicroscopicsilicovolcanoconiosis -----

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[math-fun] disyllabic panvowel
by Bill Gosper 01 Feb '19

01 Feb '19

Everybody knows FACETIOUS and ABSTEMIOUS. (Which also have the vowels in order, and have adverbs if you demand y.) But can you find a disyllabic less QUESTIONABLE than HOUSEPAINT? I was shocked when Select[WordData[], ContainsAll[Characters@#, Characters@"aeiou"] &] returned 6899 hits. But most were multiword phrases like AMMONIUM NITRATE and VIRTUAL MEMORY. Precluding " " still left 1093(!), including some cheats like ACRYLONITRILE-BUTADIENE-STYRENE ("The three major food groups!" —Beldar Conehead) There is at least one aeiou single word nation, chemical element, common vegetable, hideous parasite, medical condition, impossibly common word (automobile, inoculate) . . . I wonder if it's a quiz show category. —rwg

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Re: [math-fun] disyllabic panvowel
by Dan Asimov 01 Feb '19

01 Feb '19

Let us not forget euoia. —Dan

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[math-fun] disyllabic panvowel
by Bill Gosper 01 Feb '19

01 Feb '19

(Impressive SPOILERs from Nick Baxter) In[111]:= DictionaryLookup[] // Length Out[111]= 92518 isn't as large as In[112]:= WordData[] // Length Out[112]= 149191 Yet it has In[132]:= Position[DictionaryLookup[], #] & /@ {"Cointreau", "cautioned", "auctioned"} Out[132]= {{{15133}}, {{12388}}, {{4816}}} In[134]:= WordDefinition /@ {"Cointreau", "cautioned", "auctioned"} Out[134]= {Missing[UnknownWord], {"warn strongly; put on guard"}, {"sell at an auction"}} Why doesn't WordData? In[133]:= Position[WordData[], #] & /@ {"Cointreau", "cautioned", "auctioned"} Out[133]= {{}, {}, {}} WordDefinition does not necessarily exclude say, proper names, even French disyllabic panvowels: In[130]:= WordDefinition@"Beauvoir" Out[130]= {"French feminist and existentialist and novelist (1908-1986)"} —rwg

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