Universal laws behind growth patterns, from Tetris to coffee stains | Hacker News Reader

Universal laws behind growth patterns, from Tetris to coffee stains

empiricalzeal.com

56 pointsaatish12 comments

What I find cool is that it seems that all sorts of growth rates in nature (and even computer science) can be expressed in terms of elementary mathematical functions. Things can grow at exponential rates, logarithmic rates, hyperbolic rates, etc., and we actually have the terminology to express all these types of growth curves, whether they apply to bacterial colonies, or a nautilus shell, or a tree-trunk's diameter. Growth is described by log x, or a^x, or x^n, or sin x, or other functions we've already catalogued, and there's nothing "in-between" these curves that isn't already covered.

Are there any natural generative processes that produce/grow at a rate that isn't modellable by a linear combination of basic functions? Have we essentially discovered all the possible functions that describe how things grow or accumulate? Do more complicated functions sometimes describe changes in nature? I'm no mathematician, but I'd enjoy learning about this, because it puzzled me back in school.

I've wondered this, too!

It helps that elementary functions are in some sense dense in the set of all other functions. Results like Stone-Weierstrass show that on a compact interval, any function can be arbitrarily-well approximated by a polynomial.

Other things that might interest you-- how one might know if a function even has an antiderivative that can be expressed as an elementary function: http://en.wikipedia.org/wiki/Liouville%27s_theorem_(differen...

> Are there any natural generative processes that produce/grow at a rate that isn't modellable by a linear combination of basic functions? Have we essentially discovered all the possible functions that describe how things grow or accumulate? Do more complicated functions sometimes describe changes in nature?

Global weather patterns come to mind. E.g. how hurricanes, tornadoes, etc. form and change. As I understand it (and I'm no expert) there is some funky math involved in modelling those, and we're not really there yet. I'm not sure if those fall under "generative processes" though.

This is because we try to fit these patterns into elementary functions, and we do that because it is most useful.

Natural generative processes come down to growth force. When you write them down as the results of your most basic observations of them, like "Growth is exactly proportional to the population size" you end up putting everything in terms of these "forces" which yields a differential equation p'(t) = k p(t) which has a solution of p(t) = c e^(kt).

There are a number of basic tweaks that reflect real life situations that can quickly make these functions unsolvable by elementary functions. One example are time-varying carrying capacity. If you're on an island that generates a limited amount of resources each year you get the basic carrying capicity logistic function p'(t)=p(t)(1-p(t)) which is solved by p(t) = 1/(1+c e^t). But if that your population has a delayed affect on the available resources you get p'(t)=p(t)(1-p(t-t1)) which can't be expressed as elemantary equations. http://en.wikipedia.org/wiki/Logistic_function#Time-varying_...

Also, preditor-prey models which occur naturally where you have two populations that depend on each other which can't be reduced to elementary functions either:

http://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equation

This reminds me a lot of Stephen Wolfram's A New Kind of Science. A great read!

www.wolframscience.com/nksonline/toc.html

Chapter 11, section 9 (on universality) is probably why! www.wolframscience.com/nksonline/section-11.9

I was about to zone out, and then I got to: “Alexei Borodin, a mathematician from MIT, contacted us after we published a paper on how particle shape affects particle deposition regarding the coffee-ring effect. He saw our experimental videos online and was reminded of simulations that he has performed. I think this is a great example of the value of reaching out across disciplines – we never would have studied this topic without Alexei bringing it to our attention.”

Wow, small world. Alexei Borodin taught my freshman math class when he was still at Caltech.

You might want to play around with https://github.com/matthiasgoergens/Coral, a toy to explore the second pattern. Needs PyGame and Python2.

Science discovers there are two general categories of things: sticky things that don't tile, and slippery things that do tile.

Not to be a smart ass, but this seems like common sense, and it's not remotely surprising that things within each category behave similarly.

I think for me the sense of wonder comes from the fact that from very simple rules, complex behaviour can emerge.

Yep, it's about Poisson distributions.

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