February 2014 - mathfun@mailman.xmission.com

[math-fun] time dilation
by Wouter Meeussen 11 Feb '14

11 Feb '14

in http://www.nist.gov/pml/div688/2013_1_17_newera_atomicclocks_3.cfm a atomic clock, using Y-Sr ‘lattice’, is said to achieve 3E-18 accuracy & stability. This would allow to detect time dilation caused by a mere 4 cm altitude difference. If guestimating correctly, this would be equivalent to the gravitational effect of a 19 kg mass at a horizontal distance of 10 cm. Question: would a second 19 kg mass on the opposite side anihilate the effect (null net acceleration) or double it ? Is gravity a normal ‘garden&kitchen’ vectorial field ? Wouter.

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Re: [math-fun] More 4th grade math
by Henry Baker 10 Feb '14

10 Feb '14

At 06:26 AM 2/10/2014, Andy Latto wrote: >Do the people who say that 1 is a prime also say that 0 is a composite? Yes, because gcd(0,n)=|n| for all rational integers n. http://en.wikipedia.org/wiki/Greatest_common_divisor

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Re: [math-fun] Triangle puzzle for näifs
by Dan Asimov 10 Feb '14

10 Feb '14

Yes, Andy missed something because I did. Find all pairs (T,N) such that T is a right triangle and there exists a tiling of T by N congruent right triangles similar to itself. --Dan P.S. Er, multitasking made me do it. On 2014-02-10, at 1:46 PM, Andy Latto wrote: > I must be missing something; can't you dissect *any* triangle into 4 > congruent triangles, each similar to the original, by joining the > midpoints of the sides?

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[math-fun] New party trick
by Henry Baker 10 Feb '14

10 Feb '14

Math homework in Afghanistan: what is the probability that each (deranged) terrorist _doesn't_ get his own SIM card back ? https://firstlook.org/theintercept/article/2014/02/10/the-nsas-secret-role/ ... Some top Taliban leaders, knowing of the NSAÂs targeting method, have purposely and randomly distributed SIM cards among their units in order to elude their trackers. ÂThey would do things like go to meetings, take all their SIM cards out, put them in a bag, mix them up, and everybody gets a different SIM card when they leave,Â the former drone operator says. ÂThatÂs how they confuse us.Â As a result, even when the agency correctly identifies and targets a SIM card belonging to a terror suspect, the phone may actually be carried by someone else, who is then killed in a strike. According to the former drone operator, the geolocation cells at the NSA that run the tracking program Â known as Geo Cell Âsometimes facilitate strikes without knowing whether the individual in possession of a tracked cell phone or SIM card is in fact the intended target of the strike. ÂOnce the bomb lands or a night raid happens, you know that phone is there,Â he says. ÂBut we donÂt know whoÂs behind it, whoÂs holding it. ItÂs of course assumed that the phone belongs to a human being who is nefarious and considered an Âunlawful enemy combatant.Â This is where it gets very shady.Â ...

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[math-fun] Ingredients: ...,water,...
by Bill Gosper 10 Feb '14

10 Feb '14

rwg> Puzzle: Why does this absolutely dry and crispy snack I am munching list water as an ingredient? APG>Being edible, I imagine that it is composed of biological cells (plant? animal? fungus?), and the cytoplasm is full of water.<Adam No, these things are seriously sere, unpleasantly chewy a few hours after unsealing the package. --rwg I've probably said this before, but in the 60s, Egan's Market in Cambridge, MA carried a line of Indonesian condiments--tiny bottles in little pastel boxes, identical but for color and unfamiliar name featuring lots of "ng"s, each bearing the helpful legend "Contents: ingredients."

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[math-fun] name for golden ratio triangle tiling?
by Steve Witham 10 Feb '14

10 Feb '14

If you can divide a triangle into two similar triangles, then the small triangles are right triangles. Which means the small triangles are similar to the original triangle. If you want the small triangles' areas to be 1 : phi, then the sides of the triangles are 1 : r : r**2, where r = sqrt(1/phi) ~= .7861513777574233. r = sqrt( (sqrt(5) - 1) / 2 ) I don't see a simpler expression for that. As far as I can tell, this triangle is not related to Penrose tiles. But, googling for "golden ratio tiling" finds only mentions of Penrose tiles. -> Is this a familiar construction with a name? <- It seems about tied for simplicity with these other ways of recursively dividing the plane into similar shapes: square -> four squares 1 : sqrt(2) rectangle -> two of them (European paper sizes). 90-45-45 triangle -> two of them. From this you can get "gradual" tiling (see below) "gradual" Hilbert-like curve "gradual" sampling of a 2D space, "peppering." By gradual tiling, I mean that you can divide the original triangle into N smaller triangles by splitting in a systematic order. If N is a Fibonacci number, you only need two sizes, else three. Similarly you can evolve a path by putting a dot in the center of the original triangle, and when you split a triangle, replace the dot by a line segment between the two new centers (rubber-banding any segments that were already attached). By peppering I mean, as you split a triangle, put a dot at the middle of the split line. Gradually, the accumulating dots sample the triangle at finer and finer resolutions. Something like the way you (or a sunflower) can sample the unit interval using phi. Conway's tessellation / Charles Radin's "pinwheel tiling" divides a sqrt(5) : 2 : 1 triangle into five similar triangles. With the original triangle oriented correctly, all the vertices of the little triangles have rational coordinates! With this 1 : r : r**2 tiling, all of the coordinates are irrational. --Steve P.S. Strange, I had to add Penrose, Conway and Fibonacci to my spelling dictionary for the first time, all in the same email. And sqrt.

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Re: [math-fun] More 4th grade math
by Henry Baker 10 Feb '14

10 Feb '14

Yes, 0 is the ultimate composite number. The use of the phrase "characteristic zero" is merely a hack, and nothing should be read into this usage. Once again, the primary reason for considering these 'corner cases' is to reduce the number of 'special cases' in definitions and proofs. In many cases, a formula can be extended to include the case n=0 or p=1 if the appropriate extended definition is used. It is easy to handle some of these definitions with "for all x in A, E(x)", so long as you realize that this works even when A is the empty set. Unfortunately, many speakers of English assume that when you talk about such an x, you seem to imply that such an x actually exists. At 06:26 AM 2/10/2014, Andy Latto wrote: >Do the people who say that 1 is a prime also say that 0 is a composite?

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Re: [math-fun] More 4th grade math
by Adam P. Goucher 10 Feb '14

10 Feb '14

R is a ring (specifically an integral domain), not \mathbb{R}. Sorry! Sincerely, Adam P. Goucher > ----- Original Message ----- > From: Andy Latto > Sent: 02/10/14 02:26 PM > To: math-fun > Subject: Re: [math-fun] More 4th grade math > > On Mon, Feb 10, 2014 at 5:45 AM, Adam P. Goucher <apgoucher(a)gmx.com> wrote: > > > > > Well, the official definition of `prime' is `for all a,b in R, we > > have p|ab <==> (p|a or p|b)'. > > What do you mean in R by x|y? > > If you mean that there's a z in R such that xz = y, then x|y as long > as x is nonzero, and every nonzero number is prime. > > If you mean there's a z in X such that xz = y, then 2 is not prime > (and neither is anything else), because 2 | 2pi * 1/pi, > but does not divide either 2pi or 1/pi. > > If you meant Z when you said R, then 1 is prime by your definition, > since for any a and b, both sides of your equivalence are true. > > If you want people to use a sensible definition of prime, you have to > have them deal with the more complicated world where there aren't just > primes and composites, but 4 types of numbers: 0, units, primes and > composites. > > Do the people who say that 1 is a prime also say that 0 is a composite? > > Andy > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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Re: [math-fun] More 4th grade math
by Adam P. Goucher 10 Feb '14

10 Feb '14

> Bill Gosper wrote: > > ES> > > According to a friend who volunteers in the Santa Cruz CA Public > Schools, the official view is that 1 is a prime, because its only > divisors are 1 and itself. However, a teacher did mention that in > more advanced mathematics, 1 is not a prime. > > -- Gene > > We need to displace that definition! Well, the official definition of `prime' is `for all a,b in R, we have p|ab <==> (p|a or p|b)'. I think that could possibly be de- algebraicised to make it compatible with 4th-graders (whoever they are; I don't know how American grades translate into actual ages). Of course, you probably want to use the definition of `irreducible' instead (they coincide in the integers, or more generally any PID), in which case it is `we cannot decompose p into a product of two factors, neither of which is a unit'. So: `A positive integer is prime if it cannot be decomposed into a product of two factors, both of which are greater than 1.' Or equivalently: `A positive integer is prime if it cannot be decomposed into a product of two strictly smaller factors.' > "An integer is prime iff it appears > exactly twice in the infinite times table." Twice twice in the "± times > table": It might be difficult to make the conceptual leap from that into a general intdom, though, since there is no nice generalisation. (I suppose if there are finitely many units, as is usually the case, you can say `appears exactly 2n times in the times table'.) Anyway, considering how many times things appear in the Cayley table of the monoid of positive integers under multiplication is quite nice. A _Klein number_ (square of a prime) is something that appears exactly thrice, for example. (The reason for the name is because Felix Klein was born on 1849-04-25, and 4, 25 and 1849 are all squares of primes.) > Puzzle: Why does this absolutely dry and crispy snack I am munching list > water as an ingredient? Being edible, I imagine that it is composed of biological cells (plant? animal? fungus?), and the cytoplasm is full of water. As Martin Chan astutely remarked, `Oh, so ROM bars are 40% rum in the same way that a tomato is 90% water?'. (The rum is absorbed into a brown paste permeating the chocolate bar.) http://lurnq.com/lesson/ROM-The-American-Takeover-Case-Study/ They're also tasty and conducive to solving mathematical problems. Indeed, I can't think of another chocolate bar to which I am equally indebted. I suppose I should acquire one next time I contemplate this question: "Let alpha be an ordinal. Does there exist a function from the set of limit ordinals below alpha to the power-set of alpha such that the following properties hold? 1. f(beta) is a subset of beta for all beta; 2. If beta and gamma are distinct, f(beta) is not a subset of f(gamma)." (The second condition is essentially saying `the image of f forms an antichain under inclusion'.) The answer is certainly `yes' up to the first weakly-inaccessible cardinal, so there are models of ZFC in which the answer is `yes' for all ordinals. However, I don't know whether it is provably true for all ordinals in ZFC alone. Sincerely, Adam P. Goucher http://cp4space.wordpress.com

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Re: [math-fun] More 4th grade math
by Bill Gosper 10 Feb '14

10 Feb '14

ES> According to a friend who volunteers in the Santa Cruz CA Public Schools, the official view is that 1 is a prime, because its only divisors are 1 and itself. However, a teacher did mention that in more advanced mathematics, 1 is not a prime. -- Gene We need to displace that definition! "An integer is prime iff it appears exactly twice in the infinite times table." Twice twice in the "± times table": Back in 2012-8-30, Subject: zooming in on the reals, I interrupted our regularly scheduled program to bring you the following commercial message: >Apropos nonelectronic calculation, if you have a youngster, hardcopy it this >extended times table <http://gosper.org/kellytable.png>. Filling this >in should cement intuitions about minus >times minus, even times odd, primes, and progressions, e.g.. New! Improved! http://gosper.org/kellytbl.pdf (This message will end in 1.5 KB) When your kid can add and subtract two-digit numbers, and understands the idea of multiplication, s|he might enjoy filling in the rest of this unusual times table. It's easy: just repeatedly add going down or right, and repeatedly subtract going left or up. But don't stop subtracting when you get to zero! What's unusual is that this table contains both negative and positive numbers. You find the multiplier and multiplicand not along the edges, but down in the green rows and columns. E.g., you find (blue) 15 at the intersection of either row containing a green 3 and either column containing a green 5. Likewise at the intersection of a 5 row and 3 column. (For future purposes, all the square numbers are red.) Note that primes can only be green. (1 is NOT a prime!) Filling this in is a useful drill, and it self-checks: You'd better get the same answer whether you reach a box along a column or a row. Farther down the road, the big deal is that this exercise convinces you that negative times negative equals positive, a 7th grade idea. So 15 is also -3 times -5 and -5 times -3. The table also shows how odd times odd = odd (pale pink), and even times anything = even (yellow). And, of course 0 times anything is 0. Exercise: Where is 0 times 0? And when the kid gets it all filled correctly, s|he can show it off at school and maybe drum up additional customers. --rwg Apologies to recipients of a recent version with only a single green row and column. I wrote a pdf in the vain hope that you could paste the InString into Mathematica, but Adobe has other ideas. Also, reading the pdf with Firefox fails intermittently, at least for me. Puzzle: Why does this absolutely dry and crispy snack I am munching list water as an ingredient?

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