November 2006 - mathfun@mailman.xmission.com

[math-fun] Embedding simplicial complexes in Euclidean space
by Daniel Asimov 23 Nov '06

23 Nov '06

Recall what a simplicial complex is: First, an n-simplex delta_n is the convex hull of any n+1 points, in n-space, that lie in no hyperplane. The convex hull of any k+1 of these points is a k-simplex, and is called a k-face of delta_n. The faces of delta_n are all of its k-faces for all k = 0,1,...,n-1. A "simplical complex" is the result of gluing together a disjoint collection of simplices such that if two simplices are glued to each other, they are identified by an affine bijection from a k-face of one to a k-face of another, for some k. Thus in a simplicial complex, any two simplices that intersect do so in solely *one* common face. We shall assume that a simplicial complex K is *connected* -- i.e., for any two simplices s, t of K, there is a sequence of simplices s_0,...,s_L such that s = s_0, s_L = t, and that s_j and s_(j+1) are glued along a (now) common k-face for some k. We also assume a simplicial complex is built from only a *finite* collection of simplices. ---------------------------------------------------------------------------------------------------------- The dimension dim(K) of a simplical complex K is the highest dimension of any simplex of K. QUESTION 1: Is there a least dimension E(n) such that any simplical complex of dimension n can be *topologically* embedded in the Euclidean space of dimension f(n) ? I think this can be done by dipping the complex in goo so that it becomes a subset of a manifold (with boundary) of some dimension d >= n . . . and then use the result that all d-manifolds can be topologically embedded in 2d-dimensional Euclidean space (I think even lower for d-manifolds with nonempty boundary). If a topological embedding of a simplicial complex into Euclidean space is affine on each simplex (and hence must send each simplex to a simplex), we call the embedding *affine*. QUESTION 2: Can every simplicial complex K of dimension n be *affinely* embedded in some Euclidean space? (See Example below.) QUESTION 3: Among those simplicial complexes K (with dim(K) = n) that *do* embed affinely in some Euclidean space, is there a least dimension A(n) such that they can all be affinely embedded in the Euclidean space of dimension A(n) ? Example: Form a strip of three 2-simplices, using equilateral triangles to make an isosceles trapezoid. Identify its left edge with its right edge, abstractly, to form a topological Moebius band. Which can of course be embedded topologically in R^3. But surely this can't be embedded affinely in R^3. It probably can't be embedded affinely in any Euclidean space. (BUT, this is *not* a simplicial complex, since the two outer triangle intersect in both a 1-simplex and a separate 0-simplex.) --Dan

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[math-fun] First known bimagic square of primes
by Christian Boyer 18 Nov '06

18 Nov '06

One century ago, Henry E. Dudeney studied the first magic squares of primes even if, at that time, "1" was considered as a prime number, and was used in his squares. Look for example at http://mathworld.wolfram.com/PrimeMagicSquare.html Hot news: this week, the first BIMAGIC square of primes has been constructed (and not using "1"!). And its order is also prime: 11. Reminder. A bimagic square of order n is a nxn magic square remaining magic after each of its n² numbers have been squared. It is the first solved problem on the 10 open problems published in my Math Intelligencer article, Spring 2005. But it means also that 9 problems are still unsolved. A lot of work remains to be done, including "small" unsolved problems: - 3x3 magic square of squares (only semi-magic are known) - 4x4 magic square of cubes (only semi-magic are known) - 5x5 bimagic square (only semi-bimagic are known) Christian.

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Re: [math-fun] World's hardest sudoku puzzle?
by Daniel Asimov 17 Nov '06

17 Nov '06

Steve Rowley writes: << I was explicit about saying I used only computer time, and that Dan's suggested metric didn't seem to mean much in terms of computer time. >> It's hard to be sure, of course, but the computer programs that solve Sudoku puzzles have algorithms that I suspect aren't all that different from what humans use to solve by hand. So my guess is that Steve's result (and thanks for taking the time to test it!) is indicative that my suggested metric is a dud. And it seemed so promising at first! Oh, well. --Dan

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Re: [math-fun] World's hardest sudoku puzzle?
by Daniel Asimov 17 Nov '06

17 Nov '06

Steve Rowley wrote: << . . . On the other hand, if you mean a "typical" sudoku in the sense that it's randomly selected, then we can MEASURE that! Assuming that "difficulty" translates into computer time, which is of course dubious... . . . >> There ought to be an objective measure of the *paucity* of useful information readily provided by the diagram. IDEA: If (i,j) is empty, let P(i,j) be the subset of {1,2,...,9} that does NOT appear in the union of row i, column j, and the 3x3 box containing (i,j); if (i,j) is filled with the number k, then P(i,j) = {k}. (P stands for "Possible set".) Let s(i,j) := #(P(i,j)). I propose that a good intrinsic measure of the difficulty of a Sudoku is the *geometric* mean of the 81 s(i,j)'s, since the number of possibilities is multiplicative. -------------------------------------------------------------------------------------- It would be interesting to see how this measure compares with the time it takes software to solve a puzzle. --Dan

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[math-fun] How do you solve this kind of problem?
by Daniel Asimov 15 Nov '06

15 Nov '06

Question -- I've seen many problems of the kind << Express in closed form the number x represented by x = sqrt(1 + sqrt(2 + sqrt(3 + . . . + sqrt(n + . . . . . . ) . . . ))) >> And when I numerically take the limit of sqrt(1 + sqrt(2 + sqrt(3 + . . . + sqrt(n) . . . ))) as n -> oo (using 16-digit precision) the answer converges to 1.7579327566180045 by the time n = 20. But how would one identify this number? And what about the many closely related problems? (For example, if in the quoted problem the integers 1,2,3,... are replaced by their squares, the answer numerically (again to 16 places) converges to 1.9426554227639874 even sooner -- by the time n = 15. In fact, I wonder if there's a way to express in closed form the limit when the integers 1,2,3,... are replaced by their pth powers (p an arbitrary real) ? Call this limit rad(p). Numerically, I was quite surprised to see what limit as p -> 0+ of rad(p) is (though I shouldn't have been.) --Dan --Dan

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[math-fun] (off topic) Geographical data base error
by Daniel Asimov 14 Nov '06

14 Nov '06

In looking up the telephone number of a store I shop at, I typed its name & approx. location into Google and got the info I sought . . . plus a map that showed the wrong location (on the right street), off by about a mile. Since then I've typed the address into MapQuest, and separately into the Microsoft map software on the store's own website -- and ALL of them show the wrong location on the right street. (I called the store just to confirm that they were exactly where I thought they were, in case they had just moved or something. They were indeed where I thought. So here's the problem -- How do I figure out who provides the evidently common database that is being used by Microsoft, Google and MapQuest so I can try to get them to correct the error? (Contacting the three mapping software companies mentioned is, I quickly found, not going to help at all -- it's seemingly impossible to reach anyone with any chance of knowing how to reach the source of the problem.) --Dan

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Re: [math-fun] (off topic) Geographical data base error
by Henry Baker 13 Nov '06

13 Nov '06

Dan: There are all sorts of errors in the maps used by the various GPS and database vendors. Navteq is one of the largest vendors of the data itself, while ESRI provides a lot of the geographical information system software. While following the I-5 pavement, I drive off the roadway of the I-5 in central California all the time -- if I am to believe the GPS system in my car. I understand that Navteq hires college students & others to exhaustively drive the entire roadway graph (this is the Euler tour problem, not the Hamiltonian tour problem) in order to update the GPS data & flag strange turn situations, etc. Even when the GPS has the roads correct, they often don't get the addresses along the roads correct -- sometimes they rely on an interpolation algorithm which can be wildly wrong. The store should be happy that the GPS data is wrong -- they don't have to worry about "smart bombs" finding them. :-) At 01:52 PM 11/13/2006, Daniel Asimov wrote: >In looking up the telephone number of a store I shop at, >I typed its name & approx. location into Google and >got the info I sought . . . plus a map that showed the >wrong location (on the right street), off by about a mile. > >Since then I've typed the address into MapQuest, and >separately into the Microsoft map software on the >store's own website -- and ALL of them show the >wrong location on the right street. > >(I called the store just to confirm that they were exactly >where I thought they were, in case they had just moved >or something. They were indeed where I thought. > >So here's the problem -- How do I figure out who provides >the evidently common database that is being used by >Microsoft, Google and MapQuest so I can try to get them >to correct the error? > >(Contacting the three mapping software companies mentioned >is, I quickly found, not going to help at all -- it's seemingly >impossible to reach anyone with any chance of knowing >how to reach the source of the problem.) > >--Dan

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[math-fun] Why study math in high school?
by Henry Baker 13 Nov '06

13 Nov '06

Metrics are important if you're going to become a drug dealer. You can't use the "Manhattan metric" in determining whether you are selling drugs within 1000' of a school: http://www.law.cornell.edu/nyctap/I05_0145.htm

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[math-fun] Packing squares in circles
by Jason Holt 11 Nov '06

11 Nov '06

I've been playing around with packing things, and ran across Erich Friedman's site: http://www.stetson.edu/~efriedma/packing.html I wrote a program to pack unit squares into circles: http://lunkwill.org/src/square-in-circle/ It even generates SVG images of the smallest circles it can find that support a given number of squares (svg is neat!). They're in the svg/ directory, but my browser doesn't like to display them, so I converted them into .PNGs as well. An interesting integer sequence might be the number of unit squares that pack into circles of increasing integer radii. My program gives that sequence as the following for circles of r=1..19: 0,8,21,40,65,97,135,180,229,286,350,419,495,575,664,761,860,966,1079 (but it might be off by a few). -J

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[math-fun] Fib-haikus
by Olivier Gerard 10 Nov '06

10 Nov '06

(hope it was not already posted here) FIBONACCI POEMS By Shelley Batts http://www.scq.ubc.ca/?p=599 (the number of syllabes of each verse is 1,1,2,3,5,8, ...)

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