April 2013 - mathfun@mailman.xmission.com

[math-fun] Fwd: harder 7x7x7?
by Bill Gosper 03 Jun '15

03 Jun '15

I sent this to the kids last week. Neil 3D printed two 2x3x1s and searched manually, with Burrtools, and with a too-slow Mathematica program, and was unable to find a solution in which the new pieces relaxed the constraints on the other key pieces. (Not firm) conclusion: Replacing the three 2x2x1s by two 2x3x1s makes the puzzle harder (for anyone who doesn't already know how to solve it). --rwg But replacing the the 2x2x2 and 2x2x1 by two 2x3x1s makes Conway's 5x5x5 *much* easier.http://gosper.org/7%5E3leg.pdf ---------- Forwarded message ---------- From: Bill Gosper <billgosper(a)gmail.com> Date: Wed, Oct 24, 2012 at 12:00 PM To: Neil Bickford <techie314(a)gmail.com> Cc: Julian Ziegler Hunts <julianj.zh(a)gmail.com>, Jon Ziegler < jonathan.zh(a)gmail.com>, Corey Ziegler Hunts <corwin.zh(a)gmail.com>, Michael Beeler <mikebeeler(a)verizon.net>, rcs(a)xmission.com, stan(a)isaacs.com Working Stan [Isaac]'s copy at the Gardner thing, I was reminded that it always seems easy to permute a solution so that the three 2x2x1s form a 2x2x3. The puzzle might be harder if we force this configuration by replacing the 2x2x1s with two 2x3x1s (e.g. by truncating two spare 2x4x1s), assuming that the temptation to use a 2x3x1 singly in the solution proves both frequent and fatal. If nonfatal, it would probably make the puzzle way too easy. --Bill

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[math-fun] What is elliptic K and E closed form?
by Bill Gosper 25 May '15

25 May '15

I think the answer has changed in the last 50 yrs. A query from constantologist Steve Finch revived my interest in whether special value identities like EllipticK[(1 + Sqrt[2])^4] == ((1/4 - I/2)π^(3/2))/ ((2 + Sqrt[2]) Gamma[3/4]^2) and EllipticK[8 2^(1/4) (-1 + Sqrt[2])^3* E^(I ArcTan[(3 (3 + Sqrt[2]))/(4 2^(1/4))])] == (I (1 + Sqrt[2]) E^(-2 I ArcCot[2^(1/4)]) π^(3/2))/ (4 Sqrt[2] Gamma[3/4]^2) are already known, and where one might find tables much more extensive than, e.g., http://functions.wolfram.com/EllipticIntegrals/EllipticK/03/01/ or http://dlmf.nist.gov/19.6 . DLMF says "Exact values of K(k) and E(k) for various special values of k are given in Byrd and Friedman (1971, 111.10 and 111.11)" (1st English edition, 1954) Neil and his Mom were kind enough to fetch me this comprehensive (*Handbook* of *Elliptic* Integrals for Engineers and Physicists<http://www.amazon.com/Handbook-Elliptic-Integrals-Engineers-Physicists/dp/B…>), and to my surprise and disappointment, there was not a single identity of the above form, i.e. algebraics times Gammas(rationals). How could this be? My guess: In the 1950s K and E were considered closed forms while Gamma was not! It makes sense: K and E are of lower computational complexity. We Funsters once concluded that Gamma[k/24] was rapidly computable thanks to elliptic K identities. But now, in papers like "EVALUATION OF COMPLETE ELLIPTIC INTEGRALS OF THE FIRST KIND AT SINGULAR MODULI" (people.math.carleton.ca/~*williams*/papers/*pdf*/*299*.*pdf*) "evaluation" means "determin[ation] explicitly in terms of Gamma values". Which is what I always thought. See, e.g., http://en.wikipedia.org/wiki/Elliptic_integral#Special_values --rwg

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Re: [math-fun] parametrizing the Koch snowflake curve
by Bill Gosper 24 Apr '15

24 Apr '15

Fourier epicycles thru nth: (d16) sum(((product((1 - (( - 1)^j * tan(((%pi * (2 * k + (1/3)))/(2^j)))/(sqrt(3)))),j,1,inf)) * (%e)^(%i * (6 * k + 1) * t)/((2 * k + (1/3))^2)),k, - n,n) See http://www.tweedledum.com/rwg/cog.htm Also http://gosper.org/fst.dvi Pix: http://gosper.org/fst1.png ,http://gosper.org/fst2.png , http://gosper.org/fst3.png ,http://gosper.org/fst4.png . (For parametric, take Re and Im.) Rich's base 4 recursion loops for rational arg. Find the fixed point of the pending stack ops to simulate infinite recursion, and get an exact val on the curve. Equate to this Fourier series for a *non-obvious* identity. --rwg > If you view the starting triangle as running from 0 to 3, > and write the fraction base 4, you can use the > fraction digits to select which part of the (sub)edge > you are on. The orientation of the edge can be calculated > from the 1s and 2s of the high order portion of the fraction, > and that should give a somewhat kludgy sum for the (x,y) > coordinates of the image point. > RWG likely has some nice FFT clock sum expression. > > Rich > > > -----Original Message----- > From: math-fun-bounces(a)mailman.xmission.com [mailto: math-fun-bounces(a)mailman.xmission.com] On > Behalf Of N. J. A. Sloane > Sent: Thursday, February 25, 2010 12:43 PM > To: math-fun(a)mailman.xmission.com > Cc: njas(a)research.att.com > Subject: Re: [math-fun] parametrizing the Koch snowflake curve > > Jim Propp asked: > >> Does anyone (Bill Gosper, maybe?) know an explicit parametrization > of the Koch snowflake curve? > > The Cantor set consists of the points 0 <= x <= 1 which have a base 3 "decimal" expansion that > does not contain a 1. Is there a similar characterization of the Koch snowflake? > > Neil Sloane > >

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[math-fun] That fractal pentagram again
by Bill Gosper 09 Apr '15

09 Apr '15

For some pictorial. mildly trick questions: http://gosper.org/foupent.pdf (1.6M) --rwg

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[math-fun] why can a line of slope 10^n determine n decimal digits of π?
by Gary Antonick 19 Jun '14

19 Jun '14

Just noticed this yesterday and can't figure out why it works. Draw a circle with radius 1 around 0,0. Draw a line with slope -10 that hits the circle at 1,0. This line will also hit the circle again and make a little arc. The length of the arc will be about 1/31 the perimeter of the circle. Now try again with a line of slope -100. The line will create an arc 1/314 of the circle. Starting to look a bit like pi. Am wondering why this is. In case I made a mistake here's what I'm trying to describe.. if the circle were a clock the first line would cut an arc from 15 minutes (3:00) to almost the 13 minute mark.

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Re: [math-fun] Iridium & Indium (inside future analogue of light bulbs?)
by Henry Baker 13 May '14

13 May '14

The good news is that "quantum dots" don't depend upon normal valence electron energy levels in order to achieve their properties. Their properties are primarily a result of their size & configuration, which is in the 5-20 nm range -- almost achievable in the next 1-2 generations of lithography. So many of the needs for exotic materials with very specific bandgaps will be replaced by these engineered quantum dots. --- It was my impression that most, if not all, iridium comes from outer space; there is very little indigenous iridium on the Earth's surface. There may be a lot in the Earth's core, but it's going to be a little difficult to mine it from there. At 04:42 PM 12/4/2012, Warren Smith wrote: >Iridium is 40x rarer than gold and total world production is only >about 10 tons annually. > >Indium is also pretty rare, but not as rare: 1000 ton annual production. >So I think if iridium is a substantial component of solid plastic >lighting source then that will not be a useful technology for Average >Joe Consumer. I would think >you'd want ingredients common enough for million-ton production. >If they can remove the need for Iridium and other rare elements, the >general concept sounds excellent as a future lighting source. > >-- >Warren D. Smith >http://RangeVoting.org <-- add your endorsement (by clicking >"endorse" as 1st step)

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[math-fun] Size of the Cherbakul Lake Meteorite Fragment
by Hans Havermann 16 Oct '13

16 Oct '13

From a National Post (newspaper) article today: "Viktor Grokhovsky, who led the expedition from Urals Federal University, said Monday the meteorites plucked from the ice-covered Chebarkul Lake so far are less than a centimeter and had an iron content of about 10 percent. Locals saw a big meteorite fall into the lake on Friday, leaving a six-meter-wide hole in the ice. Grokhovsky said a meteorite up to 50-60 centimeters could eventually be found in the lake." When the initial reports of the 6 m hole surfaced, I thought (looking at a picture of the hole in the ice): That must be a 4 m (or more) chunk of rock. But .5 m? An impact in a surface ice layer surely is not like an impact on ground where a wider hole might be expected.

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[math-fun] Wieferich numbers?
by Bill Gosper 30 Apr '13

30 Apr '13

Is it obvious that if n∈N and 2==Mod[2^n,n^2] then n is (a Wieferich) prime? Why don't we just call A001220 Wieferich numbers? --rwg

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Re: [math-fun] Kenneth I. Appel, Mathematician Who Harnessed Computer Power, Dies at 80
by Bill Gosper 30 Apr '13

30 Apr '13

Dave Makin>I assume these "countries" didn't include any "holey" ones i.e. rings etc. ? Connectedness is obviously required. I don't see simple-connectedness mattering. DM> Also given the equivalent assumptions is the answer for 3D just 5 colours ? Infinity, obviously. Scott Kim once showed me that the answer for *convex* 3D countries is "arbitrarily large". On 29 Apr 2013, at 13:48, Henry Baker wrote: FYI -- How is it possible that I never knew that (computer scientist) Andrew Appel was Kenneth Appel's son? Obviously, Appel's don't fall far from the tree... http://www.nytimes.com/2013/04/29/technology/kenneth-i-appel-mathematician-… Kenneth I. Appel, Mathematician Who Harnessed Computer Power, Dies at 80 By DENNIS OVERBYE Published: April 28, 2013 Kenneth I. Appel, who helped usher the venerable mathematical proof into the computer age, solving a longstanding problem concerning colors on a map with the help of an I.B.M. computer making billions of decisions, died on April 19 in Dover, N.H. He was 80. <snip> DM>The meaning and purpose of life is to give life purpose and meaning. The instigation of violence indicates a lack of spirituality. And spirituality instigates jihad.

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[math-fun] Kenneth I. Appel, Mathematician Who Harnessed Computer Power, Dies at 80
by Henry Baker 30 Apr '13

30 Apr '13

FYI -- How is it possible that I never knew that (computer scientist) Andrew Appel was Kenneth Appel's son? Obviously, Appel's don't fall far from the tree... http://www.nytimes.com/2013/04/29/technology/kenneth-i-appel-mathematician-… Kenneth I. Appel, Mathematician Who Harnessed Computer Power, Dies at 80 By DENNIS OVERBYE Published: April 28, 2013 Kenneth I. Appel, who helped usher the venerable mathematical proof into the computer age, solving a longstanding problem concerning colors on a map with the help of an I.B.M. computer making billions of decisions, died on April 19 in Dover, N.H. He was 80. The cause was esophageal cancer, which was diagnosed in October, his son Andrew said. Since the time of Euclid and Pythagoras, proofs of mathematical theorems had consisted of long strings of equations or geometric notations that any mathematician could read and quibble with, all marching logically, step by step, toward a conclusion. But the proof that Dr. Appel and a colleague, Wolfgang Haken, established in 1976 was of a different order. Their conclusion, that four colors would suffice for any map, depended on 1,200 hours of computer time Â the equivalent of 50 days Â and 10 billion logical decisions all made automatically and out of sight by the innards of an I.B.M. computer at the University of Illinois in Urbana. Hailed in some circles, including this newspaper, as Âa major intellectual feat,Â the proof shepherded computers toward a greater role in higher math. But it made many mathematicians uneasy; they worried about computer bugs and wondered how they could check or understand a ÂproofÂ they could not see. And it ignited a long-running debate about what constitutes a mathematical proof. ÂLike a landmark Supreme Court case, the proofÂs legacy is still felt and hotly debated,Â said Edward Frenkel, a mathematician at the University of California, Berkeley. Kevin Short, a mathematician at the University of New Hampshire, where Dr. Appel spent his later years, called the feat Âa watershed for modern mathematics.Â ÂIt has spawned whole fields of study,Â he said. Kenneth Ira Appel (pronounced ah-PEL) was born on Oct. 8, 1932, in Brooklyn and grew up in Queens, where he graduated from Queens College with a degree in mathematics in 1953. His father, Irwin, was an electrical engineer, and his mother, the former Lillian Sender, had been an office worker. After a short stint as an actuary and two years in the Army, Kenneth Appel enrolled at the University of Michigan, where he earned a Ph.D. in math in 1959. During the summers, he programmed computers for Douglas Aircraft. Dr. Appel soon went to work for the Institute for Defense Analysis in Princeton, N.J., doing research in cryptography and number theory for the federal government. He joined the University of Illinois as a professor in 1961. Long interested in Democratic politics, he also served a term on the Urbana City Council. Some of the thorniest problems in math are simple to state but hideously complex under the surface. Such is the case with the four-color theorem, first enunciated by an English mapmaker, Francis Guthrie, in 1852. He asserted that to create a map in which no adjacent countries are the same color, only four colors are needed. Although everyone believed it was true, proof had eluded a century of mathematicians until Dr. Appel attended a lecture in 1972 by Dr. Haken. Because of the bewildering variety of map configurations, Dr. Haken was contemplating using computers to solve the problem, but as he related in his lecture that evening, experts had convinced him that it was not possible. Dr. Appel, familiar with computers from his defense and government work, was more optimistic. ÂI donÂt know of anything involving computers that canÂt be done; some things just take longer than others,Â he said to Dr. Haken afterward, according to an account in the journal Social Studies of Science by Douglas MacKenzie of the University of Edinburgh. ÂWhy donÂt we take a shot at it?Â The two started off by showing that the universe of all possible maps must contain what mathematicians call an Âunavoidable setÂ of 1,936 different configurations. One configuration might be a country surrounded by four neighbors, for example. Their task, then, was to prove that each of these configurations could be rendered on a map using only four colors in such a way that no two adjacent land areas were of the same color. That was where the heaviest computation would come in. To help, they recruited a computer science graduate student, John Koch, and Dr. Appel persuaded the university to let them use its I.B.M. 370-168 computer, newly acquired for administrative services. Those were the days when computers filled an entire room, although their memory capacities were minuscule compared with a modern smartphone. Dr. Short recounted an occasion, as described by Dr. Appel, when the computer gave an unexpected answer. ÂOh, that wire must have fallen out again,Â Dr. Appel said. Dr. Appel began to think of the computer as a partner, though with a different kind of brain, with almost Âan artificial intelligence,Â he told Dr. MacKenzie. ÂThe computer was, to the best of my feeling about the subject, not thinking like a mathematician,Â he said. ÂAnd it was much more successful, because it was thinking not like a mathematician.Â In the summer of 1976, Dr. Appel and Dr. Haken announced their result to their colleagues by leaving a note on the department blackboard: ÂFour colors suffice.Â Their work was published in 1977 in the Illinois Journal of Mathematics. Their four-color proof earned newspaper headlines and a prestigious award in mathematics, the Delbert Ray Fulkerson Prize. But the notion of computer proofs drew skepticism in some academic circles. In a visit to one university, Dr. Appel and Dr. Haken said, professors barred them from meeting graduate students lest the studentsÂ minds become contaminated. Dr. Appel became the chairman of the mathematics department at the University of New Hampshire in 1993. He retired in 2003. He also served on the Dover School Board and for a time was the treasurer of the Strafford County Democratic Party. Before their revolutionary work was published, Dr. Appel and Dr. Haken enlisted their entire families to check hundreds of pages of calculations, making sure that diagrams of map configurations matched the computer printouts and did not have typos. Andrew Appel said his sister, Laurel, found some 800 mistakes, most of which she could fix herself. Laurel F. Appel, a biology professor at Wesleyan University, died this year. Besides his son Andrew, a computer science professor at Princeton, Dr. Appel is survived by his wife, Carole S. Stein; another son, Peter; a sister, Lois Green; and five grandchildren. Despite the criticism in more traditionalist quarters, Dr. Appel never agonized about his reliance on a computer to arrive at the four-color theorem, his son Andrew said. The mathematician Alan Turing, he noted, had shown long ago that even very short theorems could have very long proofs, running hundreds of pages. As his son recalled, Dr. Appel used to say, ÂWithout computers, we would be stuck only proving theorems that have short proofs.Â A version of this article appeared in print on April 29, 2013, on page A19 of the New York edition with the headline: Kenneth I. Appel, 80, Dies; Computerized Higher Math.

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