February 2004 - mathfun@mailman.xmission.com

[math-fun] Help with a sequence needed
by Jon Perry 09 Feb '04

09 Feb '04

Consider a tower of height n on the left hand side of a square, and we wish to create a horizontal line along the bottom of the square. We can push any block at the top of a column 1 space to the right, and then gravity takes over. How many ways can we do this? So far I have for n=1, 0,1,2,5, and I guess n=5 is 15 (Catalan numbers? A000108) If we label the blocks for convience, then the 5 options for n=4 are: 444332 444322 443222 433222 433322 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] Apollonian gaskets and Dionysian fillings
by Daniel Asimov 06 Feb '04

06 Feb '04

Jim asks: << ... I've thought about switching to even more confusing families of circles, which (if they haven't been named yet) OUGHT to be called "Dionysian space-fillers". To get one of this hairy objects, start with an Apollonian gasket and begin promiscuously inverting circles in the gasket with respect to other circles in the gasket.... >> This sounds like a case of sexual inversion.... Jim: Since there are various versions of Apollonian gaskets, could you tell us exactly the one you are using to get started here? Dan <asimov(a)msri.org>

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[math-fun] Apollonian gaskets and Dionysian fillings
by James Propp 05 Feb '04

05 Feb '04

I've lately been trying (without much success) to understand the combinatorics behind the integers that turn up as the reciprocals of the radii of circles in Apollonian gaskets. (See Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks, "Beyond the Descartes Circle Theorem", American Mathematical Monthly, vol. 109, pp. 338-361, available on the web as http://arxiv.org/abs/math.MG/0101066 .) With the somewhat perverse post-Polya attitude "if you can't solve a problem, replace it by a harder problem you also can't solve", I've thought about switching to even more confusing families of circles, which (if they haven't been named yet) OUGHT to be called "Dionysian space-fillers". To get one of this hairy objects, start with an Apollonian gasket and begin promiscuously inverting circles in the gasket with respect to other circles in the gasket. (The inverse of a point P relative to a circle with radius r centered at O is the point Q on the ray OP satisfying (OP)(OQ)= r^2.) Now invert new and old circles with respect to new and old circles. Etc. Taking this to the limit, we get a collection of circles with the property that for every point P in the plane, and any positive epsilon, there is a circle in the collection that is contained within an epsilon-neighborhood of P. Also, all the circles have radii that are reciprocals of integers (I think I know how roughly a proof of this last assertion might go, but I have not worked out the necessary details). Has anyone looked at collections of circles like this? A simplified version of this construction starts with the partial gasket consisting of just the circles of radius 1 centered on all the even integers (viewed as complex numbers). We're not using the "gasket swap" ("Given four mutually tangent circles, replace one of them by the other circle that's tangent to the other three"); we're just using inversion of circles with respect to circles (as in the book "Indra's Pearls", which I still mean to look at one of these days!). Are things known about this? It ought to be possible to give a complete description of the combinatorics of the set of circles, at least in this simplified case. To see one step in the construction of a Dionysian space-filler of a more genuinely two-dimensional kind, see Hal Canary's nice picture http://ups.physics.wisc.edu/~hal/SSL/apollonian.pdf Jim Propp

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[math-fun] limit question
by Mike Stay 04 Feb '04

04 Feb '04

Does anyone happen to know what the limit of the sum of the nth catalan number over 4^n is? inf ___ \ C(n) > ---- /__ 4^n n=0 Thanks! -- Mike Stay staym(a)clear.net.nz http://www.cs.auckland.ac.nz/~msta039

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[math-fun] 24-sided dice
by Michael Kleber 04 Feb '04

04 Feb '04

Hello funsters, I just learned that Kaplow Games, one of the main manufacturers of polyhedral dice, started making a 24-sided die! It's the most mundane of the four d24's -- the one you get by building a square pyramid on each face of a cube (the "tetrakis hexahedron"). But it's still pretty cool. I clearly need to buy a bunch of these. In fact, I'm giving my "What are all the fair dice?" talk at Wellesley on Monday, and I'd love to have some of these to show off then. Kaplow is a wholesaler only, located in Boston. They are happy to sell to educators and the like, but only if you meet their usual minimum order of $150. A box of 50 24-sided dice is $30 wholesale, and probably twice that retail. So... is there anyone else out there interested in a bulk order of 50 geometrically interesting (but all identical) dice? (I could imagine buying two boxes myself, at a stretch, but surely not five.) If not, I can get a local game store to order them for me and pay retail, of course. The picture of the box of d24's is on their web site, at http://koplowgames.com/page54.html It's in the upper-right corner. Next to it, by the way, are boxes of 30-sided dice... I already own a large bag of those, but if someone wants both 24s and 30s, we'd reach the threshold that much sooner, I suppose. If you are interested, please answer quickly, since I'd like to have these in hand on Monday, and I'll want to order through my local gaming store soon if there's not enough interest to buy wholesale. --Michael Kleber kleber(a)brandeis.edu

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Re: [math-fun] limit question
by jdb＠math.arizona.edu 04 Feb '04

04 Feb '04

Dear Bill, I thought I might mention a paper by D. H. Lehmer on results concerning the central binomial coefficent which may be of interest to you in case you don't already know it: Interesting series involving the central binomial coefficient, Amer. Math. Monthly, 92(1985), 449-457. Best, John

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Re: [math-fun] twenty questions
by James Propp 03 Feb '04

03 Feb '04

I wrote: > The discussion of Twenty Questions reminded me of a game I learned > from Alan Frank (who I believe also invented it), called "Plenty > Questions". In this game, all queries must take the form "Is it > more like X or Y?" In the warm-up version of the game, that's > the only constraint. But in the real version of the game, there's > a further catch: each question must re-use the better of the two > matches from the previous question. That is, if the answer to > "Is it more like X or Y?" is "X", then the next question must > be of the form "Is it more like X or Z?", whereas if the answer to > "Is it more like X or Y?" is "Y", then the next question must > be of the form "Is it more like Y or Z?". Andy Latto tells me that the game I called "Plenty Questions" is not original with Alan Frank; in fact, Alan learned it from Andy, who learned it from Diana Steifbold. That's as far back as I've been able to trace things with any measure of certainty. Jim Propp

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[math-fun] Two questions in geometric combinatorics
by asimovd＠aol.com 01 Feb '04

01 Feb '04

(I don't know whether either of these problems has been solved. Here are the 3D versions of what could asked of any dimension.) 1. Analogous to a recent question posed here about triangles: Let A_n denote a regular tetrahedral arrangement of points in 3-space, n to an edge. (The layers successively contain 1, 3, 6,... points.) Define a "tetrad" to be any four mutually adjacent points of A_n. QUESTION: For which n is it possible to partition A_n into disjoint tetrads ??? Call such an n "tetrizable". Clearly, any such n = 0 (mod 4). It's not hard to show that the number of points #(A_n) is divisible by 4 exactly when n = 0,2,4,6, or 7 (mod 8). QUESTION: More concretely, n = 2 is the first tetrizable number. What is the next one (if any) ??? --------------------------------------------------------------- 2. Given any arrangement of congruent spheres in 3-space where any two are either disjoint or tangent, consider the graph formed by using each sphere as a node and each tangency as an edge. Call such a graph a "packing graph". QUESTION: What is the maximum chromatic number of a packing graph (if the maximum exists) ??? ------------------------------------------------------------- --Dan Asimov -- NOTE: Please direct any responses to <asimov(a)msri.org>, since I may not keep this AOL e-address much longer -- thanks.

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[math-fun] Taking on Mars
by Thane Plambeck 01 Feb '04

01 Feb '04

Since it's in the news Here's a passage from Arthur Koestler's "The Watershed," (a biography of Kepler). Kepler is trying to work out the orbit of Mars (pg 132, "The Giving of the Laws") * * * ** He chose out of Tycho's treasure four observe positions of Mars at the convenient dates when the planet was in opposition to the sun. The geometrical problem which he had to solve was, as we saw, to determine, out of these four positions, the radius of the orbit, the direction of the axis, and the position of three central points on it.... ...at times he was despairing, he felt, like Rheticus, that a demon was knocking his head against the ceiling, with the shout, "These are the motions of Mars!" At the other times, he appealed for help to Maestlin (who turned a deaf ear), to the Italian astronomer, Magini (who did the same), and thought of sending an SOS to Francois Vieta, the father of modern algebra: "Come, O Gallic Apollonius, bring your cylinders and spheres and what other geometer's houseware you have..." But in the end he had to slog it out alone, and to invent his mathematical tools as he went along. Halfway through [his] dramatic sixteenth chapter, he bursts out: "If thou [dear reader] art bored with this wearisome method of calculation, take pity on me who to go through with at least seventy repetitions of it, at a very great loss of time; nor wilst thou be surprised that by now the fifth year is nearly past since I took on Mars..." Thane Plambeck 650 321 4884 office 650 323 4928 fax http://www.plambeck.org

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