September 2003 - mathfun@mailman.xmission.com

[math-fun]
by David Gale 08 Sep '03

08 Sep '03

The solutions of x^2 + y^2 = z^2 the well known "Pythagorean triples" goes back to Euclid. I'd like to know if anyone in the history of Diophantine equations has looked at the same problem with 2 replaced by -2. That is, find all solutions in integers of 1/x^2 + 1/y^2 = 1/z^-2. The solution is easy and rather nice. David

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Re: [math-fun] more 51 star designs
by asimovd＠aol.com 05 Sep '03

05 Sep '03

My ASCII skills fail me, but how about one star in the middle, surrounded by similar ellipses of either 6, 8, 10, 12, and 14 stars, where the ellipses are similar ones, and the stars on each underlying ellipse are equispaced w.r.t. arclength. The arrangement of stars has rectangular symmetry, with the "2 mod 4" ellipses of size 6, 10, 14 each having a pair of stars on the y-axis (to emphasize the horizontal elongation), and the "0 mod 4" ellipses of size 8 and 12 stars each having a pair of stars on the x-axis and (hence) a pair on the y-axis. (The same general idea can be tried using ellipses of 9, 10, 11, 12, and 13 stars, but this has less potential symmetry than the above.) --Dan

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[math-fun] England sees the light.
by Richard Schroeppel 05 Sep '03

05 Sep '03

> This is a problem that's worried me for quite some time : if and when there are 51 United States, then how should the resulting 51 stars be arranged on the "Stars and Stripes" flag? It's all very well to plan ahead, John, but I don't think Tony Blair could take the role reversal of being an American province. California would object to losing its #1 position in population and wackiness. And we're supposed to guarantee our states a Republican form of government -- what would become of the Queen? Besides, we don't even speak the same language. :-) [Or :-o) in the British spelling.] John has * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * for the 49 star flag. The 49 star flag was only current for a year (or two? ~1958). My memory is that the rows were staggered, like this: * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * But I'm not sure if the pattern was this way or left-right reversed -- ie, with the staggering being on odd rows instead of even. Rich rcs(a)cs.arizona.edu

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RE: [math-fun] The problem of the 51st state.
by Meeussen Wouter (bkarnd) 05 Sep '03

05 Sep '03

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O but horizontaly streched W. -----Original Message----- From: Meeussen Wouter (bkarnd) [mailto:wouter.meeussen@vandemoortele.com] Sent: vrijdag 5 september 2003 19:27 To: 'math-fun' Subject: RE: [math-fun] The problem of the 51st state. O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O -----Original Message----- From: Meeussen Wouter (bkarnd) [mailto:wouter.meeussen@vandemoortele.com] Sent: vrijdag 5 september 2003 19:04 To: MathFun (E-mail) Subject: RE: [math-fun] The problem of the 51st state. O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O loosenig at the corners? W. =============================== This email is confidential and intended solely for the use of the individual to whom it is addressed. If you are not the intended recipient, be advised that you have received this email in error and that any use, dissemination, forwarding, printing, or copying of this email is strictly prohibited. You are explicitly requested to notify the sender of this email that the intended recipient was not reached. _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun =============================== This email is confidential and intended solely for the use of the individual to whom it is addressed. If you are not the intended recipient, be advised that you have received this email in error and that any use, dissemination, forwarding, printing, or copying of this email is strictly prohibited. You are explicitly requested to notify the sender of this email that the intended recipient was not reached. _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun =============================== This email is confidential and intended solely for the use of the individual to whom it is addressed. If you are not the intended recipient, be advised that you have received this email in error and that any use, dissemination, forwarding, printing, or copying of this email is strictly prohibited. You are explicitly requested to notify the sender of this email that the intended recipient was not reached.

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RE: [math-fun] The problem of the 51st state.
by Meeussen Wouter (bkarnd) 05 Sep '03

05 Sep '03

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O -----Original Message----- From: Meeussen Wouter (bkarnd) [mailto:wouter.meeussen@vandemoortele.com] Sent: vrijdag 5 september 2003 19:04 To: MathFun (E-mail) Subject: RE: [math-fun] The problem of the 51st state. O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O loosenig at the corners? W. =============================== This email is confidential and intended solely for the use of the individual to whom it is addressed. If you are not the intended recipient, be advised that you have received this email in error and that any use, dissemination, forwarding, printing, or copying of this email is strictly prohibited. You are explicitly requested to notify the sender of this email that the intended recipient was not reached. _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun =============================== This email is confidential and intended solely for the use of the individual to whom it is addressed. If you are not the intended recipient, be advised that you have received this email in error and that any use, dissemination, forwarding, printing, or copying of this email is strictly prohibited. You are explicitly requested to notify the sender of this email that the intended recipient was not reached.

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RE: [math-fun] The problem of the 51st state.
by reid＠math.arizona.edu 05 Sep '03

05 Sep '03

i propose a pentagonal pattern with D_5 symmetry: a central star, surrounded by a ring of 5 more stars, which itself is surrounded by another ring of 10 stars, then two more, of 15 and 20 stars. it's not so easy to draw it in ascii. start with * . * * . * * * * . * * * * * * . * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * and stretch it around the central star, so that the dots are covered by the stars on their left. i hope this gives the idea! mike

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RE: [math-fun] The problem of the 51st state.
by Meeussen Wouter (bkarnd) 05 Sep '03

05 Sep '03

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O loosenig at the corners? W. =============================== This email is confidential and intended solely for the use of the individual to whom it is addressed. If you are not the intended recipient, be advised that you have received this email in error and that any use, dissemination, forwarding, printing, or copying of this email is strictly prohibited. You are explicitly requested to notify the sender of this email that the intended recipient was not reached.

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[math-fun] seven of three
by wouter meeussen 05 Sep '03

05 Sep '03

yet another pretty matrix: k = 4; vec = IntegerDigits[Range[0, 3^k - 1], 3, k] - 1; or, all (-1,0,1) vectors of length k, sev = Outer[Plus, vec, vec, 1] /. (q : {__Integer}) /; (Min[q] < -1 || Max[q] > 1) :> 0 /. (q : {__Integer}) :> 1 + FromDigits[q + 1, 3]; .. added in all possible ways, and replacing (-1,0,1) vecs with 1, the others with 0 (those containing -2 or 2). ListDensityPlot[sev /. i_?Positive -> 1, Mesh -> False] .. results in a fractal plot of 7 hexagons, made of 7 hexagons, made of ... and its determinant is 1. The '1''s are of course counted as 7^k. Beauty is in the eye of the beholder. W.

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Re: [math-fun] The problem of the 51st state.
by Marc LeBrun 05 Sep '03

05 Sep '03

>=John Conway > How should the resulting 51 stars be arranged on the "Stars and Stripes" flag? OK, I'll bite: it doesn't seem particularly clever, but how does * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * (51 = 7x9 - 3x4) strike you? >=Dan Asimov > Further, can you define explicitly what "nicest" means? I doubt it, but I could hazard a few obvious criteria that probably apply: 1. stars are subset of square or triangular lattice 2. bounding rectangle is "nearly square" 3. at least bilaterally symmetric Others? I'm curious to see how John's design pushes the envelope!

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Re: [math-fun] The problem of the 51st state.
by asimovd＠aol.com 05 Sep '03

05 Sep '03

John Conway wrote: << This is a problem that's worried me for quite some time : if and when there are 51 United States, then how should the resulting 51 stars be arranged on the "Stars and Stripes" flag? There are nice arrangements for 48, 49, 50 stars: * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * >> Good question. When I was in 4th grade my teacher asked the class to come up with arrangements for 49 and 50 stars, and I'm proud of coming up with the above patterns, especially the one for 50 (rotated 90 degrees). (Though as it happens the gov't chose for 49 a pattern of 7 staggered rows of 7 stars, possibly because it looked better filling the rectangular field available for it.) But (ignoring the rectangular field condition) this is actually a fascinating problem for all lowish numbers: What is the nicest arrangement of n points in the plane, for each n up to 100? Further, can you define explicitly what "nicest" means? --Dan

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