Monday, June 28, 2010

football fallacy

Arnold King's short post on Soccer and the Law of Large Numbers made me wonder - when does the law large numbers 'kick in?' I also had to wonder whether King's analysis was an appropriate application of the law of large numbers, or is the expectation that in a high scoring game the lucky scores will even out an example of the law of averages fallacy?

Points in a game of skill are not random, so the law of large numbers applies only to what could be thought of as the occasional 'lucky' or 'unlucky' (i.e. somewhat random) points that disrupt the overall score. To have the law of large numbers apply to these, the amount of random points scored in a game would have to increase dramatically, I would think.

As pointed out by John Allen Paulos in his book Innumeracy, the law of large numbers won't help make a game fair - it is likely that one team will be ahead in 'luck' for the entire course of the game, no matter how long it is played. The unlucky gambler does not get luckier by playing longer.

In a higher scoring game, it may be argued (and perhaps this is what King is really saying) that the points accrued through skill will outweigh those acquired by luck. This reasoning does not suffer from the 'law of averages' fallacy, rather its correctness may depend on how you model the role of luck within a game of skill. Does the number of lucky points stay fixed, or is it proportional to the overall score? If we assume something closer to the former, then skill will eventually dominate luck; if the later is closer to the truth, then bigger scores won't help.

Exploring the law of large numbers and the fallacy of the law of averages is easier with coin flips than with soccer balls (consider the game played by Rozencrantz and Guidenstern).

(BTW, we all know that you can turn a doughnut into a coffee cup, but did you know you can also turn it into a doubly covered soccer-ball?)

Monday, June 21, 2010

colored phizz

In "Transitive Decompositions of Graphs and their Links with Geometry and Origami" which appeared in April's American Mathematical Monthly, Geoffrey Pearce described how to edge-color a unit-origami dodecahedron symmetrically using five colors.

Constructing a model based on Pearce's instructions is pretty straight forward if you use Tom Hull's Phizz Unit. Pearce's instructions make use of the usual planar projection of the dodecahedral graph - you need to color your model based on the diagram below (the dark lines represent a single color, you need to repeat the pattern five times with five distinct colors).

If you do this (hopefully with less garish colors), you end up with a model that looks like the one below. In a single picture, you can't really get an appreciation of the symmetry of the coloring, but if you make it yourself and handle it, you'll see that it is indeed a symmetrical, transitive, coloring.

If you want a minimal coloring, rather than a transitive one, you'll only need three colors. A neat way to generate a three-coloring of the dodecahedron is to first find a Hamiltonian circuit for it. To do this, start with two colors - one color to mark an edge-path that touches each vertex only once, and the other color to mark the edges that are not part of the path. Below is a phizz unit dodecahedron with a Hamiltonian circut in yellow (other edges blue). A question before you begin: how many blue edge units and how many yellow edge units should you fold?

Once you have a Hamiltonian circut on your dodecahedron, replace alternate edges on your circut with your third color. Doing this will give you a dodecahedron whose edges are colored such that every vertex has three distinctly colored edges intersecting at it (can you see why?).

Tom Hull describes how to use this method to color phizz buckyballs in his book Project Origami.

If instead of using Phizz units you use Sonobe units, you will obtain models that have the same characteristics, but look more like icosahedrons than dodecahedrons due to the nice duality between these two unit types.

Thursday, June 17, 2010

Apollonius, Descartes, Ford, and Farey

A while ago I posted briefly on Ford Circles. I wanted to point out an interesting short post on the Wolfram Blog by Ed Pegg Jr. that shows how to draw these in Mathematica.

If you can, you should also check out the article by Dana Mackenzie on Apollonian gasket circle packing. Both Pegg and Mackenzie connect these packings to Descartes circle theorem.