Wednesday, July 29, 2020

clumping and the pleasure of recreational computing

Interactive version at
I'm rereading Martin Gardner's book "aha! Gotcha: Paradoxes to puzzle and delight" I found this book in my Aunt and Uncle's library and it left an impact on me, teaching me things like the "liar's paradox" ("this sentence is true!") and the weirdness of the Hotel Infinity, where the seemingly full hotel can make space for any finite number of new guests by having all the current guests move up that many rooms, or even an infinite number of new guests by having everyone take a room that's their current room number x 2.

In the section on Statistics Gardner talks about statistical "clumping". I remember this passage:
A striking experiment in clumping was discovered by A. D. Moore, an engineer at the University of Michigan. Moore calls it the "nonpareil mosaic" because it uses large quantities of nonpareils, a sugar candy manufactured in the shape of tiny colored spheres. Obtain enough red and green nonpareils so that you can fill a glass bottle with equal amounts of each. Shake the bottle until the two colors are thoroughly mixed.

Inspect the sides of the bottle. You would expect to a see a homogenous mix of colors, but instead you see a beautiful mosaic made up of irregular large red clumps interspersed with equally large green clumps. The pattern is so unexpected that even mathematicians, when they first see it, believe that some sort of electrostatic effect is causing spheres of like color to stick to one another. Actually, nothing but chance is operating. The mosaic is the normal result of random clumping.

If this seems hard to believe, try this simple experiment. On a sheet of graph paper, outline a 20-by-20 square. Take each cell in turn and color it red or green, choosing the color by flipping a coin. When the 400-cell square is fully colored, you will see the same kind of mosaic that appeared on the sides of the bottle.
But being in a page of relatively plentiful computer power, I made a digital version of this this morning and saved myself the 400 coin flips... see it yourself at ahaclumps

I made it so if you click higher or lower you get more or fewer colors (up to 8) and if you click left or right the number of squares in the grid changes. With small squares, I feel the clumping effect is a bit less pronounced, so I'm not quite sure I believe the original jar-of-nonpareils effect is well demonstrated here.

It is nice to be able to whip something like this up in an hour or so... two minor bits of cleverness, one is I do love p5's random() function that will return a random member of an array that you pass into (or an ordinary bounded random number  if you pass in numbers). Also if you look in the code there's a good excuse to use a little curry function (so I can put x,y into a single-ly dimensioned array) and then the code to show the current settings and then remove them with a timeout was some decent UX (IMO).

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