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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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RE: [math-fun] The problem of the 51st state.
by Schroeppel, Richard 05 Sep '03

05 Sep '03

The 46-star field is like the 48 field, with rows 2 & 5 having only 7 stars and being centered. * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * I recall seeing a different design somewhere: * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Sort of "no nonsense". This must have been inspired by the recent science articles about the 3-D version of the problem for small clusters of atoms. Rich -----Original Message----- From: John Conway To: math-fun Sent: 9/5/2003 9:37 AM Subject: Re: [math-fun] The problem of the 51st state. On Fri, 5 Sep 2003, Thane Plambeck wrote: [clip] > I've thought the 46 star flag shown at the URL below must not be the best possible. I don't know how to read that - can you roughly draw or describe it for me? JHC _______________________________________________ math-fun mailing list math-fun(a)mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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[math-fun] 12 exponentials conjecture
by Jon Perry 04 Sep '03

04 Sep '03

wrt. http://mathworld.wolfram.com/SixExponentialsTheorem.html and http://mathworld.wolfram.com/FourExponentialsConjecture.html what is known if x1,x2 are linearly independent complex numbers, as are y1,y2 and z1,z2. Then must one of; e^x1y1, e^x1y2, e^x2y1, e^x2y2, e^x1z1, e^x1z2, e^x2z1, e^x2z2, e^y1z1, e^y1z2, e^y2z1, e^y2z2 be transcendental? Jon Perry perry(a)globalnet.co.uk http://www.users.globalnet.co.uk/~perry/maths/ http://www.users.globalnet.co.uk/~perry/DIVMenu/ BrainBench MVP for HTML and JavaScript http://www.brainbench.com

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[math-fun] Perry's Observation
by Jon Perry 02 Sep '03

02 Sep '03

According to; http://www.users.globalnet.co.uk/~perry/maths/fourcolour/fourcolour.htm the 3-sphere can be constructed in plastic but not in wood. What other mediums render the 3-sphere impossible, and what other PO objects can not be constructed in wood? Jon Perry perry(a)globalnet.co.uk http://www.users.globalnet.co.uk/~perry/maths/ http://www.users.globalnet.co.uk/~perry/DIVMenu/ BrainBench MVP for HTML and JavaScript http://www.brainbench.com

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[math-fun] Re: inversion in finite field ~= fourier transform?
by Mike Stay 01 Sep '03

01 Sep '03

I wrote: >In GF(2^n), inversion behaves very much like a >change between conjugate observables and made a vague comparison to a Fourier transform; after a little more thought, I guess it's more like convolution, since it's the sum of products going in opposite directions. So (x+h)^-1 = x * h = sum over even k ( x^{N-k} h^k ) where N=2^n-2. I guess I need to figure out whether this definition of convolution gives anything useful... -- Mike Stay staym(a)clear.net.nz http://www.xaim.com/staym

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[math-fun] astroid: 3 recipes and a question
by R. William Gosper 01 Sep '03

01 Sep '03

For a circle of radius r rolling inside the unit circle: 1) r=1/4: trajectory of point on circumference. 2) r=3/4: trajectory of point on circumference. (opposite dir) 3) r=1/2: boundary of area swept by diameter. All draw x^(2/3) + y^(2/3) = 1. What density function rho(r) must a planet have so that an astroid is a brachistochrone? --rwg

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