January 2020 - mathfun@mailman.xmission.com

[math-fun] Rough Ellipso-Spectrometry of a "Gruppenpest"
by Brad Klee 19 Jan '20

19 Jan '20

Subtitle: Stranger Than Fiction. Fayetteville, AR, Jan. 19. A green June bug was found dead and dismembered near a sewer at the University of Arkansas on the afternoon of Friday January 17. The untimeliness of the occurrence raised enough suspicions for students and faculty to initially avoid any dealings with the mess of ill-strewn body parts. Later on Sunday evening, an intrepid reporter decided to return to the scene of the crime, where he then proceeded to discretely collect remnant fragments of the beetle's shell--the pronotum and a few abdominal sections. The beetle (possibly beetles, plural) appears to have been of the local variety Cotinis Nitida. However, if the remains were, in fact, dumped inappropriately by a university researcher, they might have come from Cotinis Mutabilis. No missing beetle reports have been filed recently, and the relevant authorities do not care to help with the investigation. The plight of the insects has largely been ignored by humans, and many species of Coleoptera have recently been derided for their role in ruining crops. Despite growing inter-species tensions, the unexpected, serendipitous appearance of a chiral-reflecting corpse can do no less than to inspire! Opportunist scientists, anyways, are usually not patient enough to just wait until the June emergence. Recent efforts have reconstructed a 100-200Ma family tree fo the Scarabaeidae, which also correlates development of anthophagy following the angiosperm takeover [1]. Other scientists have taken a narrower view of the Pleurosticti radiation, especially around the sub-family Cetoniinae [2]. As first noticed by Michelson in 1911, many of these beetles have iridescent shells which selectively reflect circular polarized light. One hypothesis is that this adaptation evolved concurrently with the rise of predatory animal species, as a new method of crypsis. In Scarab-shell ellipsometry, researchers today measure reflective change of polarization in terms of Mueller matrix elements, as functions of wavelength and scattering angle. Emphasis continues to fall on the element M_{1,4}, which determines conversion of unpolarized light into left or right circular polarization. For qualitative measurement, a full scientific setup is not necessary. Two opposed CPL camera filters or a pair of 3D glasses (see also [3]) are sufficient to verify chiral selectivity in the visible spectrum. For example, on the recently collected "specimen", we find the following: https://0x0.st/zF0V.jpg with row 1: ambient, row 2: CPL, row 3 opposite CPL. One of the circular polarizing filters eliminates almost all of the green reflection, while the other does not. This indicates that the species--probably a Cotinis Scarab--does have chiral-layered shell. A quick search of the literature did not reveal if full ellipsometery of Cotinis Nitida is known. Such an experiment may be of some interest during the summer months when specimens are readily available. Ultimately scientists would like to have a complete picture as to when and why chiral selectivity evolved, and this effort needs to include all relevant species, even the green June bugs from Arkansas. --Brad [1] https://royalsocietypublishing.org/doi/pdf/10.1098/rspb.2014.1470 [2] https://www.diva-portal.org/smash/get/diva2:800022/FULLTEXT01.pdf [3] https://www.itp.uni-hannover.de/fileadmin/arbeitsgruppen/zawischa/static_ht…

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[math-fun] Would someone with a clearer head please explain why
by Bill Gosper 18 Jan '20

18 Jan '20

these are unequal: {Sin[InverseFunction[(π + #1) + Sin[π + #1] &][t]], Sin[InverseFunction[#1 + Sin[#1] &][π + t]]} (real t) while these aren't? {Sin[InverseFunction[(π + #1) + Sin[π + #1] &][π - t]], Sin[InverseFunction[#1 + Sin[#1] &][π + t]]} —rwg

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[math-fun] Mathematicians Catch a Pattern By Figuring Out How To Avoid It
by Ray Tayek 16 Jan '20

16 Jan '20

https://science.slashdot.org/story/19/11/28/180213/mathematicians-catch-a-p… We finally know how big a set of numbers can get before it has to contain a pattern known as a "polynomial progression." <https://www.quantamagazine.org/mathematicians-catch-a-pattern-by-figuring-o…> From a report:/A new proof <https://arxiv.org/abs/1909.00309> by Sarah Peluse of the University of Oxford establishes that one particularly important type of numerical sequence is, ultimately, unavoidable: It's guaranteed to show up in every single sufficiently large collection of numbers, regardless of how the numbers are chosen. "There's a sort of indestructibility to these patterns," said Terence Tao of the University of California, Los Angeles. Peluse's proof concerns sequences of numbers called "polynomial progressions." They are easy to generate -- you could create one yourself in short order -- and they touch on the interplay between addition and multiplication among the numbers. For several decades, mathematicians have known that when a collection, or set, of numbers is small (meaning it contains relatively few numbers), the set might not contain any polynomial progressions. They also knew that as a set grows it eventually crosses a threshold, after which it has so many numbers that one of these patterns has to be there, somewhere. It's like a bowl of alphabet soup -- the more letters you have, the more likely it is that the bowl will contain words. But prior to Peluse's work, mathematicians didn't know what that critical threshold was. Her proof provides an answer -- a precise formula for determining how big a set needs to be in order to guarantee that it contains certain polynomial progressions. Previously, mathematicians had only a vague understanding that polynomial progressions are embedded among the whole numbers (1, 2, 3 and so on). Now they know exactly how to find them./ -- Honesty is a very expensive gift. So, don't expect it from cheap people - Warren Buffett http://tayek.com/

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[math-fun] Need help identifying a fractal
by Neil Sloane 14 Jan '20

14 Jan '20

I've been trying to understand the Conant Gasket https://oeis.org/A328078 but without much success. The pictures are gorgeous. It is a kind of fractal. (The definition is clear, but what are the numbers?) By accident I found a similar-looking genuine fractal in an old pile of papers - see https://oeis.org/A328078/a328078_2.pdf - does anyone recognize it?

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[math-fun] RandomPartition[n]
by James Propp 14 Jan '20

14 Jan '20

Has RandomPartition[n] been dropped from Mathematica? The documentation says that it's part of the Combinatorica package, which has now been folded into the Wolfram Language, so in theory RandomPartition[n] should be part of Mathematica. I figured out on my own (no thanks to the documentation) that Partitions[n], which appeared to have been dropped, is still available as IntegerPartitions[n], so when RandomPartition[n] didn't work I tried RandomIntegerPartition[n], but that doesn't work either. Jim Propp

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[math-fun] I just wrote out a "contingency pdf" of my G14 talk
by Bill Gosper 14 Jan '20

14 Jan '20

g14 talk <http://gosper.org/gosper-bill-talkproposal-dragon%20function,vs%20dragon%20…> . I hope I can get a big enough time slot to throw in some animations, or have time to remedy a technical catastrophe such as befell me at G13. —Bill

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[math-fun] 99% accurate Kepler solution, e < 0.5
by Brad Klee 13 Jan '20

13 Jan '20

When e< 0.5, M = E - e*sin(E) is approximately solved by: E = M + e/(1-e)*sin(M)-(2*e^2)/(1-e^2)/Pi^2*M*(2*Pi-M)*sin(M), This seems to be the most simple three-term solution, which matches E(M) and E'(M) at integer multiples of Pi. The error increases with e, so the approximation becomes unusable after e = 0.5, where the error is still less than 1% of 2*Pi. All planets are well within this bound, but some bound comets are not. Check with the wiki-tabulation at: https://en.wikipedia.org/wiki/Orbital_eccentricity For worst-case planet Mercury with e=0.21, accuracy is better than 99.7%. The approximate E(M) is so simple, I think it would have been discovered hundreds of years ago, but this is the first time I've seen it. Anyone with a Reference? Or how about a simple 99% approximation for e > 0.5? --Brad (* Mma Accuracy Check *) Manipulate[Plot[Times[Subtract[ M + e/(1 - e)*Sin[M] - (2*e^2)/(1 - e^2)/Pi^2*M*(2*Pi - M)* Sin[M], InverseFunction[# - e Sin[#] &][M]], 100/2/Pi], {M, 0, 2 Pi}, PlotRange -> {-1, 1}], {e, 0.001, 0.5}]

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[math-fun] When FindIntegerNullVector is too fussy
by Bill Gosper 13 Jan '20

13 Jan '20

For some stuffy reason, UnitConvert refuses to convert square degrees to steradians. An ill-chosen web page said there were 41253 square degrees in a sphere. (It didn't even say "approximately".) What is this number, exactly? It's pretty trivial to Google, but it's also easy just to guess. It's presumably some product of small rationals and π. FindIntegerNullVector[Log[{41253, 2, π, 3, 5, 7}]] FindIntegerNullVector::rnfu: FindIntegerNullVector has not found an integer null vector for {Log[41253],Log[2],Log[\[Pi]],Log[3],Log[5],Log[7]}. Don't underrate LatticeReduce! For N digit accuracy, you might want to keep this two-liner handy: LR[L_, N_] := Block[{$MaxExtraPrecision = N}, LatticeReduce[Transpose[Prepend[IdentityMatrix[Length[L]], Round[10^N*L]]]]] For six digits, In[114]:= LR[Log[{41253, 2, π, 3, 5, 7}], 6] Out[114]= {{3, 1, -6, 1, -4, -2, 0}, {10, 1, 3, -12, -7, 3, 2}, {9, 1, 12, 2, -4, -2, -7}, {-2, 1, -1, -4, 0, -13, 8}, {6, -1, -9, 1, 17, 3, -4}, {2, -5, 3, 4, -3, 14, 14}} The first vector is always shortest. Its first component, 3, is the error. This looks like the answer, since the exponents of 41253 and 7 are respectively 1 and 0: In[115]:= {2, π, 3, 5, 7}^{-6, 1, -4, -2, 0} Out[115]= {1/64, \[Pi], 1/81, 1/25, 1} In[117]:= Times @@ %% Out[117]= \[Pi]/129600 I.e., 1 ~ 41253 ⨉ π/360², a satisfying mixture of π, "degrees", and "square". —rwg

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[math-fun] AnAgraM + aNAGram = AnagRAm
by Éric Angelini 12 Jan '20

12 Jan '20

Hello Math-Fun (Gilles E.-F. and Alexandre W. in BC copy), Let a + b = c and a < b < c and a, b, c = anagrams of each other. What is the smallest possible "a" with this property? Have all a's under one million been computed already? Two examples: a = 41904 b = 49140 c = 91044 and a = 164907 b = 604197 c = 769104 Sorry of this is old hat. Best, É.

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[math-fun] Coordination sequences for polyhedra
by Tom Karzes 07 Jan '20

07 Jan '20

Neil, do you have coordination sequences for all of the Platonic/Archimedean/Catalan solids? Those seem like the most fundamental ones for polyhedra. I could probably generate them without too much trouble if needed. Tom

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