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[math-fun] paradoxes
by Jon Perry 22 Oct '03

22 Oct '03

Another wow, 'We live for the paradox' 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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Re: [math-fun] greatest integer
by N. J. A. Sloane 22 Oct '03

22 Oct '03

Mike Stay asked: Are there any useful definitions of greatest integer function for algebraic integers? Me: It might be better to look for analogues of the "round to nearest integer" function. Then there's an obvious candidate: tile the space with Voronoi cells centered at the integers, and map any point in the cell to its center. There are fast algorithms too - see Chap 21 of SPLAG (Sphere Packings, Lattices and Groups, by JHC and me). Neil Sloane

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[math-fun] Mr. Perkins Quilt
by Jon Perry 22 Oct '03

22 Oct '03

This MPQ talk is similar to a problem I had considered - what is the smallest square needed to house all 1..n squares? e.g. with 1x1, we need a 1x1 square with 1x1 and 2x2, we need a 3x3 square with 1x1, 2x2 and 3x3, we need a 5x5 square. So a lower bound is 2n-1. However with all the squares from say 1 to 8, this packing fails. 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] plane filling ...
by Joerg Arndt 21 Oct '03

21 Oct '03

Hope it's ok to send a few lines of pari/gp code: /* functions correct to order = 4^N-1 */ fpp(y)= { local( t, Y ); t = 1; Y=y; for (k=1, N, t *= (+1+Y+Y^2-Y^3); Y=Y^4; ); return( t-1 ); } fmm(y)= { local( t, Y ); t = 1; Y=y; for (k=1, N, t*= (-1+Y+Y^2+Y^3); Y=Y^4; ); return( t*y-Y ); } hhpm(y)= { local( tp, tm ); tp = fpp(y); tm = fmm(y); return( ((tp+tm) + I*(tp-tm))/2 ); } start gp, copy & paste above into session ... ? N=4; ? taylor(hhpm(y), y) y + I*y^2 - y^3 + I*y^4 + I*y^5 + y^6 - I*y^7 + y^8 + I*y^9 + y^10 - I*y^11 - I*y^12 - y^13 - I*y^14 + y^15 + O(y^16) Now start at the origin and for each coefficient draw a one-unit line to the right/left/up/down when the coefficient is +1/-1/+I/-I. ... or use: ? taylor(hhpm(y)/(1-y), y) y + (1 + I)*y^2 + I*y^3 + 2*I*y^4 + 3*I*y^5 + (1 + 3*I)*y^6 + (1 + 2*I)*y^7 + (2 + 2*I)*y^8 + (2 + 3*I)*y^9 + (3 + 3*I)*y^10 + (3 + 2*I)*y^11 + (3 + I)*y^12 + (2 + I)*y^13 + 2*y^14 + 3*y^15 + O(y^16) That is connect the points in the complex plane (given as coefficients). The following functions gives the turns of the above curve: hht(y)= { /* correct to order = 4^N-1 */ local( t, Y, F, s, p ); t = 1; Y=y; p = 1; F = y + y^2; for(k=2, N, Y = Y^4; t = -F + Y*F + Y^2*F - Y^3*F; p *= 4; if ( 0==(k%2), t += y^(1*p-1); t += y^(3*p); t -= (y+1)*y^(2*p-1); , /* else */ t += y^(1*p); t += y^(3*p-1); ); F = t; ); return( F-(N%2) ); } If you see how to make the latter algorithm more regular, please let me now. Trivial? Well-known? Nice? Boring? Fun? -- p=2^q-1 prime <== q>2, cosh(2^(q-2)*log(2+sqrt(3)))%p=0 Life is hard and then you die.

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[math-fun] Golden Pearls
by Jon Perry 21 Oct '03

21 Oct '03

Plz see; http://www.users.globalnet.co.uk/~perry/maths/wow/wow.htm for true universal enlightment. 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] 51 stars
by Thane Plambeck 20 Oct '03

20 Oct '03

Did JHC ever reveal his design? If so could someone repost it? Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.qxmail.com/ehome.htm

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[math-fun] PC continued
by Jon Perry 20 Oct '03

20 Oct '03

I found the relation between homeomorphic spaces - the slices have no encapsulated hole. This I believe solves PC, although I will be among the first to admit that the maths needs to be made presentable. Anyway, if anyone would like to publish this (plus other emials) as a proof, please contact me. 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] question about poincare conjecture
by Chris Landauer 19 Oct '03

19 Oct '03

hihi, all - yesterday, i read ``four colors suffice'' by robin wilson, which i found to be a very interesting discussion of the four color theorem, and an entertaining and informative account of the history of its proof, but i also noticed a comment about the poincare conjecture that i'd like to know more about, since it has been so often mentioned here of late Wilson writes (at the bottom of p. 178 of my hardbound edition) about Wolfgang Haken separating and reducing the poincare problem into 200 cases and then successfully addressing 198 of them, but not being able to eliminate the last two cases i'd like to know more about what kinds of features this separation into cases might depend on, whether or not anyone else has taken up this kind of approach, with what (partial) success, and whether it's something like taking out all the trivial and easy cases, so that only the really hard ones remain (as happened frequently for the four color theorem) thanx in advance it has always been interesting to me how hard it is to do any kind of useful geometric reasoning in a computer program (much of what i do is pattern recognition systems), and i've wondered for some time how a computing system might be designed to allow planar or 3d geometric reasoning to be more naturally programmed more soon, cal Chris Landauer Aerospace Integration Science Center The Aerospace Corporation cal(a)aero.org

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[math-fun] Re Mrs P's Quilt
by N. J. A. Sloane 18 Oct '03

18 Oct '03

I've updated A005670 based on the recent discussions, by adding comments, examples and references. But I did not change the list of exact values given there, which are: %S A005670 1,4,6,7,8,9,9,10,10,11,11,11,11,12,12,12,12,13,13,13 or in other words: n a(n) 1 1 2 4 3 6 4 7 5 8 6 9 7 9 8 10 9 10 10 11 11 11 12 11 13 11 14 12 15 12 16 12 17 12 18 13 19 13 20 13 Could someone please clarify just what the known exact values are? Ed? John? Thanks! Neil

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[math-fun] looking for reference to a problem
by Richard Schroeppel 18 Oct '03

18 Oct '03

Nick Baxter sent this a week ago, and I failed to notice he sent it to the old address, math-fun(a)cs.arizona.edu. I switched the list to math-fun(a)mailman.xmission.com last year because of excessive spam (and one virus). I disconnected automatic forwarding months ago, but I try to notice real messages and forward them manually. This one got by me. Sorry, Nick. I think the problem below was on the Math SAT, about a decade ago, but I'm sure someone remembers more specific information. It was written up in newspapers, and maybe Science News. Rich rcs(a)cs.arizona.edu ------------------- Date: Thu, 09 Oct 2003 11:05:00 -0700 From: Nick Baxter <nickb(a)baxterweb.com> To: math-fun <math-fun(a)CS.Arizona.EDU> Subject: looking for reference to a problem Can someone please fill in the details of a story I heard about a high school student who correctly protested the solution to the following problem that appear on a standardized test some years ago. Problem: Given an equilateral pyramid (square base), glue two regular tetrahedrons (same edge length) to two non-adjacent triangular faces. How many faces does the resulting polyhedron have? I'm looking for the name (and any other information) of the student, the year, the test, the given (incorrect) answer, and the original wording of the problem, if possible. Any references to published accounts of the incident are also appreciated. Thanks, Nick

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