meet the clumps

July 29, 2020
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.

UPDATE: I made a variant clumpswap

It starts with equal and divided red green populations, and then starts letting things mix by swapping adjacent squares.

Of course at first the action seems slow, because most of the swaps are of a color with itself.

I didn't make it interactive but I find the end result kind more fun watch than the earlier program... instead of swapping, it feels more like individual cells are moving ... but of course, the "color" of the mover is determined by which color is more in contrast with its neighbors.


A major difference between dogs and cats is that when you take care of a dog, the dog assumes you must be some sort of king, while when you take care of a cat, the cat assumes that it must be some sort of king
Col0nelFlanders on /r/Showerthoughts

Gizmodo's I Miss series is pretty fun.