December 2019 - mathfun@mailman.xmission.com

[math-fun] A First Look at "Snow Crystals"
by Brad Klee 01 Jan '20

01 Jan '20

1. The substantial work of Kenneth Libbrecht "Snow Crystals" [1] was released in October 2019, and recently popularized by Quanta [2]. Once again, the Quanta article buys in to egocentric optimism, and fails to produce a critical book review (my opinion: what they should be doing here). Yet this article does discuss relatively original and timely research. Also, awesome video header [3]! 2. The book is very long at 500+ pages, but if nothing else at least read the first chapter for history from Kepler to Ukichiro Nakaya. Nakaya, a contemporary of Tomonaga, was the first scientist to chart a temperature-dependent phase diagram for snow crystals, see figures 1.24, 1.25, 1.12 (copied on quanta as "A Wintry Mix"). 3. Figures 3.3 and 3.4 explain discrepancy in adsorption; however, they also contradict the phase diagram figure 1.12. The author even admits that this formation paradox remains incompletely resolved, so still an open question of science. Chapter 3-4 contains in-depth science worth more serious critical reading. 3b. Quanta figure "Growing Snowflakes" is extremely confusing, and excessively over-simplifed. The use of "face" and "edge" is flat out _Wrong_. This is not a good summary of chapters 3-4. 4. Chapter 5 "A Progression of Snow Crystal Models" is an exciting prospect; though, somewhat of a disappointment in execution, especially considering diligent history of Chapter 1. 4b. The "Packard Snowflake" *is not* the earliest relevant C.A. model of dendrite growth. Unfortunately Packard 1986 [4] does not cite Ulam 1962 [5]; though, OEIS does, see [6-8]. Ulam mentions that Holladay already knew of 2^n extinctions as early as 1960 [8]! 4c. A151723 includes a picture [9] that shows the structure of a sectored plate, and features the same 2^n extinctions. This picture is drawn over the half-hexagon tiling. From a mathematical perspective, half-hex is a better analog to 2^n C.A. than the Koch Snowflake. 4d. A similar comparison between Chair tiling and another growth C.A. is available at [10]. After this, it is possible to show that Code 686 C.A. in fact *builds* the chair tiling[11]. It follows in some sense that 686 ~ trilobite and crab. (Did C.G.S. know this?) 4e. We do not know if a variation of the C.A. from A151723 can be used to build the Socolar-Taylor tiling, or any of its variants [12]. This is a worthwhile open question. 5. Even before Holladay & Ulam, L. Pauling began to develop the "ice-type model" in 1935 [13]. Why isn't this cited? 5b. Pursuant to 4b. to 4e. a perspective combining matching-rule tiling theory and cellular automata *is not being presented* in Libbrecht's "Progression of snow crystal models". Does the author have a reason for such an exclusion? 5c. Hypothetical: In a matching-rule tiling theory for describing snow crystals, obviously the binding energies would change with thermodynamic conditions, but also, the matching-rules could change over temperature, pressure, vapour saturation, etc. 6. Chapters 6-9 are out of my price range, but describe incredible experiments. Thankfully lots of pictures and videos are available for purpose of data analysis. 7. Chapter 10, WOO-HOO! Libbrecht speaks: "Keeping an eye out for interesting crystals is a fine pursuit whenever you happen to be outside during a light snowfall. You could be riding the chair lift at your local ski area, taking a stroll through the winter woods, or just waiting in your car somewhere. If the snow is falling all around you, why not have a look from time to time to see what you can find?" 7b. Couldn't agree more, great advice Kenneth! Here are some specimens I collected at about 9800 feet elevation in the Rocky Mountains on Christmas morning: https://0x0.st/zDFW.jpg 7c. The photographs were rushed, but do show plates, sectored plates, stellar dendrites, and simple stars. From the existence of these structures, I think we can infer temperature of formation in the magic range around -15c, and decently high vapour saturation. 7d. Similar structures persisted at least 1000 feet above the height of base camp; though, I did not collect photo evidence. 8. HATE WARNING: The website "snowflakevictory.com" is not actually a website about physical snowflakes. It is a political site for the impeached Donald Trump, and uses the word "snowflake" as derogatory slang [14]. 8b. I supposed that domain name registration is solely a question of money ($$$) at this point. However, if decisions were being made by more fair criteria, I would think that snowflakevictory.com would simply redirect to Kenneth Libbrecht's awesome website: http://www.snowcrystals.com/ Any thoughts on 4e., 5c., or snowflake calculations in general? Anyone else getting decent snowflake pictures this season? Happy new-flakes, --Brad [1] https://arxiv.org/abs/1910.06389 [2] https://www.quantamagazine.org/toward-a-grand-unified-theory-of-snowflakes-… [3] https://d2r55xnwy6nx47.cloudfront.net/uploads/2019/12/SnowStructure_2880x16… [4] http://www.scipress.org/e-library/sof/pdf/0095.PDF [5] https://oeis.org/A002858/a002858.pdf [6] https://oeis.org/A002858 [7] https://oeis.org/A151723 [8] https://oeis.org/A322662 [9] https://oeis.org/A151723/a151723.png [10] https://oeis.org/A147562/a147562_1.png [11] https://demonstrations.wolfram.com/Code686BuildsTheChairTiling/ [12] https://demonstrations.wolfram.com/LimitPeriodicTilings/ [13] https://en.wikipedia.org/wiki/Ice-type_model [14] https://en.wikipedia.org/wiki/Snowflake_(slang)

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[math-fun] Discrete logs of composite numbers
by Henry Baker 31 Dec '19

31 Dec '19

I've been playing around with arithmetic mod a composite number, and I needed to extend discrete logs to the entire set -- not just those elements representable as g^i for some primitive root g. Consider, for example, Zn, for n = p*q = 47*59 = 2773. We also need phi(n) = lcm(p-1,q-1) = 46*58/2 = 1334. So we have 1334 elements in the multiplicative group. We also have another 1334 *negatives* of this group. We also have 58 powers of 47, and 46 powers of 59. Counting them up, we have 1334 1334 58 46 ---- 2772 non-zero elements of Zn. I.e., p*q = (p-1)*(q-1)+(p-1)+(q-1)+1 We can represent these elements as a quad*: [s,i,j,k] which represents the product (-1)^s * 47^i * (-59)^j * 2^k because "2" happens to be a generator/primitive root for the multiplicative group. (* Different composite n's may require more than 4 elements.) (We use -59 instead of 59 because -59 generates the full subgroup, while +59 only generates half of the subgroup.) Note also that we will never see i>0 and j>0 at the same time, because 47*59 = n = 0 (mod n). So we now define the discrete log of m in Zn to be log(m) = log((-1)^s*47^i*(-59)^j*2^k) = [s,i,j,k] Also, exp([s,i,j,k]) = (-1)^s*47^i*(-59)^j*2^k (mod n) So we can now do multiplications via logs: m1*m2 = exp(log(m1)+log(m2)) where "+" is element-wise addition. I'm certain that others have done this same thing before, but I don't recall seeing this representation.

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Re: [math-fun] Kepler[t]. (Was: Two ellipse circumferences)
by Brad Klee 30 Dec '19

30 Dec '19

This solution is pretty well recognized: Let “t” be the angle around a unit circle, then A=1/2*t gives circular area, and A_t=1/2*e*sin(t) gives the area of a triangle such that the sectorial area can be written: A_s=Sqrt(1-e^2)*(A+A_t). Invert this equation to get the correct time parameter. I doubt that your pals Julian and Neil are the first to have noticed this. Your preferred calculation is very similar to the certificate method I was discussing recently. One area integral is easy to calculate and equals the sectorial area by the addition of a polygon volume. I have already shown *on this list* that the same tactic is capable of defining sectorial area relative to Cartesian area, which is calculated by a relatively simple trigonometric integral. I don’t have my computer with me at the moment, but later I will try to copy your Mathematica code with my alternative approach. Merry Ellipse-Mas, and a Happy New-Circle, —Brad > On Dec 25, 2019, at 6:03 PM, Bill Gosper <billgosper(a)gmail.com> wrote: > > Yes! How could we possibly have gone this long without a > universally recognized Kepler(a,b,t) function? —Bill

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[math-fun] Kepler[t]. (Was: Two ellipse circumferences)
by Bill Gosper 30 Dec '19

30 Dec '19

> > HGB> What about time around an elliptical orbit? Any closed form solutions > > for -- e.g., approximations to #days from solstice? > > Yes! How could we possibly have gone this long without a universally > recognized Kepler(a,b,t) function? —Bill > > The function is ridiculously simple, but I *never* could have found it without crucial help from Julian and NeilB. Let lips[t_, e_] := {e + Cos[t], Sqrt[1 - e^2] Sin[t]} be an ellipse of eccentricity e with unit semimajor axis, one focus at the origin, center at e, and other focus at 2e. If t is proportional to time, this is *not* Keplerian (= constant swept area per unit time = constant angular momentum) motion. The parameter t needs to be fudged by a pacing function, which is merely fuj[t_,e_] := InverseFunction[e Sin[#] + # &]@t Then kepler[t_,e_]:={e + Cos[#], Sqrt[1 - e^2] Sin[#]}&@fuj[t,e] . Proof: In[40]:= Area[r kepler[t, eps], {r, 0, 1}, {t, t0, t0 + duration}] Out[40]= 1/2 duration Sqrt[1 - eps^2] I.e., the area of the swept elliptical sector is independent of t0. (The actual area needs to be scaled by the square of the actual semimajor axis, which we called "1". One "year" is a duration of 2π.) Area is a fancy Mathematica function that, in this case, correctly guesses that its first argument is a pair parametric rectangular coordinates. Here are the 52 weeks of a 364 day year <http://gosper.org/keplerwks.png> (with eccentricity exaggerated to √½) (This discussion applies to hyperbolic and parabolic trajectories as well.) http://www.tweedledum.com/rwg/pizza.htm makes a fuss over numerical methods for InverseFunction[e Sin[#] + # &], but Mathematica is *so* seductive. Now I can easily realistically animate orbiting planets. Modulo Einstein. —rwg

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Re: [math-fun] (Corollary of) Kepler's 3rd Law:
by Keith F. Lynch 28 Dec '19

28 Dec '19

William R Somsky <wrsomsky(a)gmail.com> wrote: > Well, your "bouncing" planet only occurrs w/ point-like planets and > sun which can pass through each other, so it's understandable that > that is not normally considered. Also, assuming it's a true radial orbit rather than an ellipse too narrow to see with my screen resolution, if the planet could pass through the sun it wouldn't bounce or whip around, it would leave the far side with the same speed it entered the near side, and its orbit would be bilaterally symmetrical. Of course I'm assuming that neither the sun's mass nor the planet's mass are concentrated at a point. If they are concentrated at a point, and the points collide, the program gets a division by zero error and terminates. Or the Runge-Kutta numerical simulation is swamped by noise and gives unpredictable results. > Any eccentricity zero orbit w/ non zero perihelion we'll be an > escaping parabolic path. No, an eccentricity zero orbit is a circle. I'm still curious as to how far out Earth's orbit can be gradually moved (to save us from the sun's red giant stage) before resonances with other planets increase eccentricities and cause planetary collisions or expulsions. We may need to move all the planets simultaneously.

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Re: [math-fun] Discrete logs of composite numbers
by Henry Baker 28 Dec '19

28 Dec '19

Damn! "phi" below should be "lambda": At 04:46 AM 12/28/2019, Henry Baker wrote: >I've been playing around with arithmetic mod a composite number, >and I needed to extend discrete logs to the entire set -- not >just those elements representable as g^i for some primitive >root g. > >Consider, for example, Zn, for n = p*q = 47*59 = 2773. We also >need phi(n) = lcm(p-1,q-1) = 46*58/2 = 1334. Sould be: lambda(n) = lcm(p-1,q-1) = 46*58/2 = 1334. >So we have 1334 elements in the multiplicative group. > >We also have another 1334 *negatives* of this group. > >We also have 58 powers of 47, and 46 powers of 59. > >Counting them up, we have > >1334 >1334 > 58 > 46 >---- >2772 > >non-zero elements of Zn. > >I.e., >p*q = (p-1)*(q-1)+(p-1)+(q-1)+1 > >We can represent these elements as a quad*: > >[s,i,j,k] > >which represents the product > >(-1)^s * 47^i * (-59)^j * 2^k > >because "2" happens to be a generator/primitive >root for the multiplicative group. > >(* Different composite n's may require more than >4 elements.) > >(We use -59 instead of 59 because -59 generates >the full subgroup, while +59 only generates half >of the subgroup.) > >Note also that we will never see i>0 and j>0 at >the same time, because 47*59 = n = 0 (mod n). > >So we now define the discrete log of m in Zn to be > >log(m) = >log((-1)^s*47^i*(-59)^j*2^k) = [s,i,j,k] > >Also, > >exp([s,i,j,k]) = (-1)^s*47^i*(-59)^j*2^k (mod n) > >So we can now do multiplications via logs: > >m1*m2 = exp(log(m1)+log(m2)) > >where "+" is element-wise addition. > >I'm certain that others have done this same thing before, >but I don't recall seeing this representation.

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Re: [math-fun] stupid number theory problem
by Henry Baker 26 Dec '19

26 Dec '19

Bingo! Thanks! So my F(p) is actually "Sophie Germain degree" of p. It appears that my F(p)>6 may be exceedingly rare. I wonder how fast F^(-1) grows... At 08:40 AM 12/26/2019, Fred Lunnon wrote: > See OEIS A063377 : Sophie Germain sequence, >tabulated up to p = 10^5 by Antti Karttunen (though >omitting p = 0 !) > >WFL > >On 12/26/19, Henry Baker <hbaker1(a)pipeline.com> wrote: >> Consider the sequence of integers generated by: >> >> Given a prime p, >> >> Ap_0 = p >> Ap_(n+1) = 2*Ap_n + 1 >> >> This sequence *terminates* when Ap_n *isn't* prime. >> >> Finally, define the function F(p), p prime, as the >> *length* of this sequence Ap_n. >> >> Informally, F(p) is the length of an all-prime sequence >> consisting of p, 2p+1, 2(2p+1)+1, ... >> >> These sequences of primes seem to be common enough, that >> I'm guessing that for *every* n, there exists a prime p >> such that F(p)=n. >> >> Here's the real kicker: there probably isn't a >> prime p s.t. F(p)=oo, is there ?

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[math-fun] stupid number theory problem
by Henry Baker 26 Dec '19

26 Dec '19

Consider the sequence of integers generated by: Given a prime p, Ap_0 = p Ap_(n+1) = 2*Ap_n + 1 This sequence *terminates* when Ap_n *isn't* prime. Finally, define the function F(p), p prime, as the *length* of this sequence Ap_n. Informally, F(p) is the length of an all-prime sequence consisting of p, 2p+1, 2(2p+1)+1, ... These sequences of primes seem to be common enough, that I'm guessing that for *every* n, there exists a prime p such that F(p)=n. Here's the real kicker: there probably isn't a prime p s.t. F(p)=oo, is there ?

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[math-fun] Boustrophedon primes
by Walter Trump 24 Dec '19

24 Dec '19

The next Boustrophedon primes are 3821 and 3989. Here is a continuation of Eric's arrangement of numbers (you cannot read the digits, they are too small): https//www.trump.de/Boustrophedon-Primes.png Walter

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[math-fun] Hacking RSA/DH in Lisp
by Henry Baker 23 Dec '19

23 Dec '19

The Linux bash script at the end of this message might be useful for anyone who wants to hack RSA or DH crypto in Lisp. The script converts an OpenSSL .pem file into a .lsp (Lisp) file. YMMV... cat key512.pem -----BEGIN RSA PRIVATE KEY----- MIIBOwIBAAJBALgG3dbTsWaMMd6l5KYLCVpP4O0xZ9OcMqWJHvPQ050BaFdgLRsj DOECyWb1HzWJ25E5EmwVV7xv6RTe/JzVcscCAwEAAQJBAKpFQDErQfMeOG5HlF3F zGuRiQ4uLmDvHzVmwArb51BWkQK/rYs2FRNdHwFVbECzn2FmakjokX17qJ4NiFFO QZECIQDjAnODewWyLr77b8QIFLz+Apo0tPcxWe9AW2frlsEp/QIhAM+HMce/j13H 1DANadMFnNAYAaYEHGRaDd5b722AnjkTAiAYQPjKgWhYYzowbv6Ixn5yvSBwpS0L 7snrHFj8CL1WmQIhALTx1e+qEKDeOR5ks55wyFeQdu9/qhm9g1Ybptk9SbfNAiBK 1COirtJRWGD4cWY4doI+qpR1/kGZ/w6mo6GTaQc4Vg== -----END RSA PRIVATE KEY----- cat key512.pem | ./rsa2lsp.sh ;;; RSA Private-Key (512 bit, 2 primes) (setq modulus #x00b806ddd6d3b1668c31dea5e4a60b095a4fe0ed3167d39c32a5891ef3d0d39d016857602d1b230ce102c966f51f3589db9139126c1557bc6fe914defc9cd572c7) (setq publicExponent 65537) (setq privateExponent #x00aa4540312b41f31e386e47945dc5cc6b91890e2e2e60ef1f3566c00adbe750569102bfad8b3615135d1f01556c40b39f61666a48e8917d7ba89e0d88514e4191) (setq prime1 #x00e30273837b05b22ebefb6fc40814bcfe029a34b4f73159ef405b67eb96c129fd) (setq prime2 #x00cf8731c7bf8f5dc7d4300d69d3059cd01801a6041c645a0dde5bef6d809e3913) (setq exponent1 #x1840f8ca816858633a306efe88c67e72bd2070a52d0beec9eb1c58fc08bd5699) (setq exponent2 #x00b4f1d5efaa10a0de391e64b39e70c8579076ef7faa19bd83561ba6d93d49b7cd) (setq coefficient #x4ad423a2aed2515860f871663876823eaa9475fe4199ff0ea6a3a19369073856) cat dhkey1.pem -----BEGIN PRIVATE KEY----- MFwCAQAwMwYJKoZIhvcNAQMBMCYCIQDOxpswAa6BOww02q1Y3OdzFblZJkIiQoxF FbUZgNHliwIBAgQiAiBCXWiQd0cpVXcNKj6j+knJWnLX6mbx/9o8oa99XPr4fA== -----END PRIVATE KEY----- cat dhkey1.pem | ./rsa2lsp.sh ;;; DH Private-Key (256 bit) (setq private-key #x425d689077472955770d2a3ea3fa49c95a72d7ea66f1ffda3ca1af7d5cfaf87c) (setq public-key #x4ac901eff14eb826b1c01d17408913f73fd55a30eb6d98d110f05ec07424a28a) (setq prime #x00cec69b3001ae813b0c34daad58dce77315b959264222428c4515b51980d1e58b) (setq generator 2) cat rsa2lsp.sh # Use "sed" to transform .pem keys into something readable by Common Lisp. openssl pkey -text -noout | \ tr '\n' '|' | \ sed 's/$/)\n/g' | \ sed 's/[[:space:]]*(0x10001)[[:space:]]*//g' | \ sed 's/[[:space:]]*(0x2)[[:space:]]*//g' | \ sed 's/|[[:space:]]*/|/g' | \ sed 's/RSA/\;\;\; RSA/g' | \ sed 's/DH/\;\;\; DH/g' | \ sed 's/|modulus:|/\n(setq modulus #x/g' | \ sed 's/|private-key:|/\n(setq private-key #x/g' | \ sed 's/|public-key:|/)\n(setq public-key #x/g' | \ sed 's/|prime:|/)\n(setq prime #x/g' | \ sed 's/|publicExponent: /)\n(setq publicExponent /g' | \ sed 's/|generator: /)\n(setq generator /g' | \ sed 's/|privateExponent:|/)\n(setq privateExponent #x/g' | \ sed 's/|prime1:|/)\n(setq prime1 #x/g' | \ sed 's/|prime2:|/)\n(setq prime2 #x/g' | \ sed 's/|exponent1:|/)\n(setq exponent1 #x/g' | \ sed 's/|exponent2:|/)\n(setq exponent2 #x/g' | \ sed 's/|coefficient:|/)\n(setq coefficient #x/g' | \ sed 's/|//g' | \ sed 's/://g'

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