September 2009 - mathfun@mailman.xmission.com

[math-fun] Fun Moments in Mathematics II
by Hans Havermann 16 Sep '09

16 Sep '09

This one is about Nicole de Boer (as math student "Leaven") determining primes in the 1997 sci-fi horror-thriller "Cube"... "Uh. OK. Um. Six forty-five." There is a two second pause during which Leaven moves her lips slightly and blinks - apparently calculating. "That's not prime." "Three seventy-two." There is a one-and-a-half second pause. "No." "Six-four-nine." Another two second pause. "Right. Eleven times fifty- nine."

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[math-fun] Is there an iterative way to calculate radii along a scanline?
by Paul Reiners 15 Sep '09

15 Sep '09

This is a programming question I ran across. Does anyone happen to know the answer? I am processing a series of points which all have the same Y value, but different X values. I go through the points by incrementing X by one. For example, I might have Y = 50 and X is the integers from -30 to 30. Part of my algorithm involves finding the distance to the origin from each point and then doing further processing. After profiling, I've found that the sqrt call in the distance calculation is taking a significant amount of my time. Is there an iterative way to calculate the distance? In other words: I want to efficiently calculate: r[n] = sqrt(x[n]*x[n] + y*y)). I can save information from the previous iteration. Each iteration changes by incrementing n, so x[n] = x[n-1] + 1. I can not use sqrt or trig functions because they are too slow except at the beginning of each scanline. I can use approximations as long as they are good enough (less than 0.l% error) and the errors introduced are smooth (I can't bin to a pre-calculated table of approximations). The original question was posted here: http://stackoverflow.com/questions/1423918/is-there-an-iterative-way-to-cal…

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Re: [math-fun] Is there an iterative way to calculate radii along a scanline?
by Henry Baker 14 Sep '09

14 Sep '09

I don't know if it answers your question or not, but you should be aware of Hakmem #149 & following: http://home.pipeline.com/~hbaker1/hakmem/hacks.html#item149 At 02:10 PM 9/14/2009, Paul Reiners wrote: >This is a programming question I ran across. Does anyone happen to know the >answer? > >I am processing a series of points which all have the same Y value, but >different X values. I go through the points by incrementing X by one. For >example, I might have Y = 50 and X is the integers from -30 to 30. Part of >my algorithm involves finding the distance to the origin from each point and >then doing further processing. > >After profiling, I've found that the sqrt call in the distance calculation >is taking a significant amount of my time. Is there an iterative way to >calculate the distance? > >In other words: > >I want to efficiently calculate: r[n] = sqrt(x[n]*x[n] + y*y)). I can save >information from the previous iteration. Each iteration changes by >incrementing n, so x[n] = x[n-1] + 1. I can not use sqrt or trig functions >because they are too slow except at the beginning of each scanline. > >I can use approximations as long as they are good enough (less than 0.l% >error) and the errors introduced are smooth (I can't bin to a pre-calculated >table of approximations). >The original question was posted here: > >http://stackoverflow.com/questions/1423918/is-there-an-iterative-way-to-cal…

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[math-fun] Pentagram snowflakes? In a square grid?
by Bill Gosper 14 Sep '09

14 Sep '09

Due to a 2D pattern-generator discovered by two exceptional brothers I'm tutoring: http://gosper.org/p5d1.png . --rwg STRANGE WOMAN TOWN MANAGERS

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[math-fun] Fwd: Question re two infinite nested radicals expressions
by rcs＠xmission.com 13 Sep '09

13 Sep '09

(from the NMBRTHRY list) This seems to be in our ambit. --Rich ----- Forwarded message from pevnev(a)juno.com ----- Date: Thu, 10 Sep 2009 12:54:47 -0400 From: "Alexander R. Povolotsky" <pevnev(a)juno.com> Subject: Question re two infinite nested radicals expressions To: NMBRTHRY(a)listserv.nodak.edu Hi, Any suggestions on evaluating these particular two (see below) infinite nested radicals expressions ? sqrt(1+sqrt(3+sqrt(6+sqrt(10+sqrt(15+sqrt(21+sqrt(28+ sqrt(36+sqrt(45+sqrt(55+sqrt(66+sqrt(78+sqrt(91+sqrt(.....)))))))))))))) and (1+(7+(20+(44+(...)^(1/3).)^(1/3).)^(1/3).)^(1/3).)^(1/3) Note that for both examples the sum of two adjacent values give the perfect power (particularly square and cube) Thanks, Best Regards, Alexander R. Povolotsky ----- End forwarded message -----

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[math-fun] GR: Solar system's future could be bumpy
by Henry Baker 13 Sep '09

13 Sep '09

FYI -- Mercury's orbit was perturbed by just .38 millimeters in the various simulations!! I heard a discussion of this article in which it was stated that general relativity actually provides a bit more stability for the solar system than simulations not including GR. Unfortunately, I don't have access to the article to see if it discusses this GR v. non-GR issue. It makes sense that GR helps to stabilize the solar system, since the ability of GR to "bleed off" information (e.g., perterbations/initial conditions) using gravitational waves provides a mechanism analogous to using heat radiation to stabilize simple mechanical systems. --- http://www.nature.com/nature/journal/v459/n7248/full/nature08096.html Letter Nature 459, 817-819 (11 June 2009) | doi:10.1038/nature08096; Received 17 February 2009; Accepted 22 April 2009 Existence of collisional trajectories of Mercury, Mars and Venus with the Earth J. Laskar1 & M. Gastineau1 Astronomie et SystÃ¨mes Dynamiques, IMCCE-CNRS UMR8028, Observatoire de Paris, UPMC, 77 Avenue Denfert-Rochereau, 75014 Paris, France Correspondence to: J. Laskar1 Correspondence and requests for materials should be addressed to J.L. (Email: laskar(a)imcce.fr) Abstract It has been established that, owing to the proximity of a resonance with Jupiter, Mercury's eccentricity can be pumped to values large enough to allow collision with Venus within 5 Gyr (refs 1Â3). This conclusion, however, was established either with averaged equations 1, 2 that are not appropriate near the collisions or with non-relativistic models in which the resonance effect is greatly enhanced by a decrease of the perihelion velocity of Mercury 2, 3. In these previous studies, the Earth's orbit was essentially unaffected. Here we report numerical simulations of the evolution of the Solar System over 5 Gyr, including contributions from the Moon and general relativity. In a set of 2,501 orbits with initial conditions that are in agreement with our present knowledge of the parameters of the Solar System, we found, as in previous studies 2, that one per cent of the solutions lead to a large increase in Mercury's eccentricityÂan increase large enough to allow collisions with Venus or the Sun. More surprisingly, in one of these high-eccentricity solutions, a subsequent decrease in Mercury's eccentricity induces a transfer of angular momentum from the giant planets that destabilizes all the terrestrial planets 3.34 Gyr from now, with possible collisions of Mercury, Mars or Venus with the Earth. Astronomie et SystÃ¨mes Dynamiques, IMCCE-CNRS UMR8028, Observatoire de Paris, UPMC, 77 Avenue Denfert-Rochereau, 75014 Paris, France Correspondence to: J. Laskar1 Correspondence and requests for materials should be addressed to J.L. (Email: laskar(a)imcce.fr) --- http://www.usnews.com/articles/science/2009/06/11/solar-systems-future-coul… Solar System's Future Could Be Bumpy Study calculates the odds that two planets collide or one crashes into sun in the next 5 billion years Posted June 11, 2009 By Sid Perkins, Science News ItÂs happened before, and it could happen again: Planets in the inner solar system may collide if gravitational interactions substantially disturb now-stable orbits, a new study suggests. Scientists and mathematicians have long known that the equations describing the orbital motions of any group of three or more objects canÂt be solved exactly. Even with the most powerful computers, it is hopeless to try to determine precisely what planetary orbits will look like more than a few million years in the future, says Jacques Laskar of the Paris Observatory. The results of any single simulation offer nothing more than one possible outcome, but running a large number of simulations can provide insights into overall probabilities, he notes. Now, a large-scale study by Laskar and observatory colleague Mickael Gastineau, which appears in the June 11 [2009] Nature, provides a possibly frightening glimpse into the solar systemÂs future. ItÂs a future in which, literally, worlds collide. The computer model used by Laskar and Gastineau accounts for the gravitational interactions among the moon, the eight major planets and Pluto, as well as any effects of general relativity. For the new study, the researchers started with the best known information about the position and orbital velocity of each of the 10 bodies, and marched simulations forward in nine-day steps for the next 5 billion years, the projected life of the sun. To obtain a large sample of possible futures, Laskar and Gastineau ran 2,501 scenarios. The only difference between runs was the size of MercuryÂs orbit. In the second and each subsequent simulation, the size differed by only .38 millimeters. For the first few million years, planetary orbits hardly differed from one run to the next. Eventually, though, variations began to accumulate. Most of the detrimental changes resulted from JupiterÂs gravitational tug on Mercury, the inner planet with the most lopsided orbit, Laskar says. In one simulation, Mercury collides with Venus about 1.76 billion years from now. In another, those planets come within 6,500 kilometers of a collision about 4.9 billion years in the future. In three other simulations, MercuryÂs orbit becomes so lopsided that it falls into the sun. In yet another scenario, Mars swings within 800 kilometers of Earth about 3.34 billion years from now. During such a close approach, tidal flexing within MarsÂ crust could break that planet apart and fragments could strike Earth Â a generally disastrous result, Laskar notes. Using the Earth-Mars close encounter as a starting point, Laskar and Gastineau then ran another 201 simulations to assess the possibility that the near miss could lead to a direct collision. In 196 of those computer runs, one planet in the inner solar system either slammed into another or fell into the sun. Of those scenarios, Earth was struck 48 times: once by Mercury, 18 times by Venus and 29 times by Mars. Despite the dire outcomes of a few of these scenarios, the teamÂs findings hint that the odds are overwhelmingly in EarthÂs favor. ÂI prefer to see the glass as 99 percent full and only 1 percent empty,Â says Gregory Laughlin, an astronomer at the University of California, Santa Cruz. ÂItÂs encouraging to think that the solar system is here for the duration,Â he notes. ---- http://www.sciencenews.org/view/generic/id/44568/title/Solar_systems_future… Solar system's future could be bumpy Study calculates the odds that two planets collide or one crashes into sun in the next 5 billion years By Sid Perkins July 4th, 2009; Vol.176 #1 (p. 9) PLANETARY BUMPER CARS. Chaotic perturbations of orbits, largely due to the gravitational effects of Jupiter, could send planets in the inner solar system crashing into one another (such as Mars and Earth, illustrated here) sometime in the next 5 billion years, a new study suggests. ItÂs happened before, and it could happen again: Planets in the inner solar system may collide if gravitational interactions substantially disturb now-stable orbits, a new study suggests. Scientists and mathematicians have long known that the equations describing the orbital motions of any group of three or more objects canÂt be solved exactly. Even with the most powerful computers, it is hopeless to try to determine precisely what planetary orbits will look like more than a few million years in the future, says Jacques Laskar of the Paris Observatory. The results of any single simulation offer nothing more than one possible outcome, but running a large number of simulations can provide insights into overall probabilities, he notes. Now, a large-scale study by Laskar and observatory colleague Mickael Gastineau, which appears in the June 11 Nature, provides a possibly frightening glimpse into the solar systemÂs future. ItÂs a future in which, literally, worlds collide. The computer model used by Laskar and Gastineau accounts for the gravitational interactions among the moon, the eight major planets and Pluto, as well as any effects of general relativity. For the new study, the researchers started with the best known information about the position and orbital velocity of each of the 10 bodies, and marched simulations forward in nine-day steps for the next 5 billion years, the projected life of the sun. To obtain a large sample of possible futures, Laskar and Gastineau ran 2,501 scenarios. The only difference between runs was the size of MercuryÂs orbit. In the second and each subsequent simulation, the size differed by only .38 millimeters. For the first few million years, planetary orbits hardly differed from one run to the next. Eventually, though, variations began to accumulate. Most of the detrimental changes resulted from JupiterÂs gravitational tug on Mercury, the inner planet with the most lopsided orbit, Laskar says. In one simulation, Mercury collides with Venus about 1.76 billion years from now. In another, those planets come within 6,500 kilometers of a collision about 4.9 billion years in the future. In three other simulations, MercuryÂs orbit becomes so lopsided that it falls into the sun. In yet another scenario, Mars swings within 800 kilometers of Earth about 3.34 billion years from now. During such a close approach, tidal flexing within MarsÂ crust could break that planet apart and fragments could strike Earth Â a generally disastrous result, Laskar notes. Using the Earth-Mars close encounter as a starting point, Laskar and Gastineau then ran another 201 simulations to assess the possibility that the near miss could lead to a direct collision. In 196 of those computer runs, one planet in the inner solar system either slammed into another or fell into the sun. Of those scenarios, Earth was struck 48 times: once by Mercury, 18 times by Venus and 29 times by Mars. Despite the dire outcomes of a few of these scenarios, the teamÂs findings hint that the odds are overwhelmingly in EarthÂs favor. ÂI prefer to see the glass as 99 percent full and only 1 percent empty,Â says Gregory Laughlin, an astronomer at the University of California, Santa Cruz. ÂItÂs encouraging to think that the solar system is here for the duration,Â he notes.

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Re: [math-fun] Pentagram snowflakes? In a square grid?
by Dan Asimov 12 Sep '09

12 Sep '09

<< On Fri, Sep 11, 2009 at 2:18 AM, Fred lunnon <fred.lunnon(a)gmail.com> wrote: > This might make an interesting wall design for a public space ... > > Any comments on how it's done? WFL I'd guess it's something like the dodecahedra in pyrite crystals: http://math.ucr.edu/home/baez/dodecahedron/14.html >> That dodecahedron is described as non-regular on its web page. But (as is well-known) a perfectly regular dodecahedron can be inscribed in a perfect cube with 3 pairs of opposite edge passing through the face centers of the 3 pairs of opposite faces, resp., of the cube. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] Schmitter words, sequences and numbers
by Eric Angelini 10 Sep '09

10 Sep '09

Hello Math-Fun [and Jean Fontaine], I've coined "Schmitter sequences" yesterday because I had to write a little something for a friend's birthday (don't spread elsewhere msg and links until sept. 29th please) A Schmitter seq is a seq of positive integers, none of which is re- peated, built on a name, a word, a sentence, etc. Example (in English) with "Fred Schmitter" (paper in French there http://www.cetteadressecomportecinquantesignes.com/Schmitter.htm ): Two simple rules : - reproduce ad infinitum the block FREDSCHMITTER - replace one by one each letter with the smallest integer not already used in the sequence and containing (in english) the said letter. So, the first FREDSCHMITTER block produces: Four 4 zeRo 0 onE 1 one hunDred 100 Six 6 one oCtillion 10^27 tHree 3 one Million 10^6 fIve 5 Two 2 eighT 8 ninE 9 thiRteen 13 See: "Spelling n in English requires a new letter of the alphabet." http://www.research.att.com/~njas/sequences/A050933 0, 1, 2, 3, 4, 5, 6, 8, 11, 20, 100, 101, 1000000, 1000000000, 1000000000000000, 1000000000000000000000000, 1000000000000000000000000000 The large numbers are (new letter in uppercase): "one Billion" (10^9), "one Quadrillon" (10^15), "one sePtillion" (10^24) and "one oCtillion" (10^27). Their names and definitions are based on the American system. and "Number of distinct letters needed to spell English names of numbers 1 through n" http://www.research.att.com/~njas/sequences/A082655 3, 5, 7, 9, 11, 13, 13, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 15, 16 -- Late increases are at twentY, hunDred, thousAnd, Million, Billion, Quadrillion, sePtillion, oCtillion. Only J and K are never used for English number names. Z is used only for zero. The second FREDSCHMITTER block will be: Fourteen 14 twenty-thRee 23 sEven 7 one hunDred one 101 Sixteen 16 one oCtillion one 10^27 + 1 eigHteen 18 one Million one 10^6 + 1 fIfteen 15 Ten 10 Twelve 12 Eleven 11 twenty-fouR 24 etc. The seq arising from "Fred Schmitter" is the concatenation of all blocks -- and starts like this (if I didn't make too many mistakes -- but you have the general idea now): S = 4,0,1,100,6,10^27,3,10^6,5,2,8,9,13,14,23,7,101,16,... Question: What is the smallest (english spelled) number yielding to a "complete Schmitter sequence" -- a sequence which is the re- arrangement of the complete set of integers? In French it is 15 (QUINZE). And what are the next such "Schmitter numbers"? I've more questions (see the French link above), but my wife desperately asks me to step out for a movie... Best, É.

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Re: [math-fun] Legitimate word with a triple letter
by James Propp 05 Sep '09

05 Sep '09

Mike Stay writes: >For information on the other meaning of "triply repeated letter", >contact your bookkeeper. And if your bookkeeper's gone out to lunch, you can ask the subbookkeeper. Jim Propp

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Re: [math-fun] induction from completeness?
by James Propp 04 Sep '09

04 Sep '09

Joshua Zucker writes: >Is this based on the assumption that 1 is the smallest positive >integer, which is what you said you needed as an additional >assumption? This step seems like the key to the proof. If there were an integer y strictly between x-1 and x, where x is an integer, then x-y would be a positive integer less than 1. >But x-1 might not be positive? Hm, some details to worry about. Oops. This uncovers another error in my write-up. Every induction needs a base case! Putting it differently: My earlier write-up never used the assumption that S contains 1, so you can tell it isn't quite right. But I'm fairly sure all the bugs have been fixed now. >Induction is equivalent >to the statement that any set of positive integers has a least >element. So all you need is the part that establishes that the >greatest lower bound of E is itself in E, right? You don't even need >the set S at all. Good point! >But wait a minute. How do you know that for any x that is not an >integer, you can find x' > x with no integers between x and x'? This >seems also equivalent to what you're trying to prove. (I'm imagining >counterexamples of the form "all fractions with denominator a power of >2".) This isn't hard. There is at most one integer between x and x+1 because if there were two of them, say y and z with y < z, then z-y would be positive integer less than 1. (In fact there's exactly one integer between x and x+1, but say we don't know this yet.) If there are no integers between x and x+1, take x' = x+1 (or take x+1/2 if you prefer). On the other hand, if there is an integer n between x and x+1, take x' = n (or (x+n)/2 if you prefer). Either way, there are no integers between x and x'. I guess what surprises me is that you don't need any second-order assumptions to get from the completeness of the reals to the least element principle --- just the first-order assumption that every positive integer is greater than or equal to 1. Jim Propp

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