Mathematicians have proved that a random process applied to a random surface will yield consistent patterns.

They establish that thereâ€™s a certain intermediate size cluster that occurs most often, and that the frequency with which larger or smaller clusters appear decreases exponentially as you move away from that intermediate size.

The Laplace or Double Exponential distribution!

The title of the article is yet another reminder that Christians should realize that Godâ€™s sovereignty is not somehow challenged by â€śrandomness.â€ť And randomness and order are not some sort of radical dichotomy in contradiction of one another.

Sometimes randomness and fascinating patterns can be just different ways of looking at the same thing.

This is only roughly related to this but I can still remember my delight as a child in learning of Buffonâ€™s Needle Problem. It had never occurred to me that something as random as scattering toothpicks on a mosaic title floor could be used to estimate the value of Pi. Randomness was actually â€śpredictableâ€ť and could not only generate order but an order related to my favorite irrational number! (OK, when I got older, Napierâ€™s constant e replaced Pi as my favorite irrational number but at the time Pi struck me as quite fun.)

Yet more years later I would assign my Analysis of Algorithms students the implementation of a Monte Carlo simulation based on Buffonâ€™s work. For extra credit they could further triple-check their results using the integral calculus formulas Buffon published some two hundred years ago.

Good times.

Amazing

Eulerâ€™s identity blows me away.

e^{iĎ€} + 1 = 0

And then there are fractals.

Me too! Itâ€™s a spectacular equation. Who would have thought that the most â€śfundamentalâ€ť constants (0,1, pi, i, and e) could be related so beautifully? (No doubt some mathematical geniuses see the interrelationships intuitivelyâ€”but I needed to work at it.)

Was it Bertrand Russell who said of this equation, â€śIf God actually exists, this equation would be the proof of himâ€ť? Or something like that. (Please correct me if my memory failed me on the attribution. I think it was some British mathematician from that era.)

Agreed. Fractals are amazing.

(I think it is very cool that in a forum like Peaceful Science we get to talk about the grandeur of the universe in every facetâ€”especially those topics which donâ€™t really come up in our daily interactions with most people. Of course, for those who are still actively working in university environments, these kinds of conversations may not be so atypical. In retirement I miss that atmosphere.)