June 22, 2004

Design Patterns of Nature

Mathematics in Nature: Modelling Patterns in the Natural World by John A. Adam, Princeton University Press, 2003.

As a kid I used my Sinclair ZX81 computer to simulate things: overlapping cratering on the moon, planetary orbits, population ecology and plant growth. Simulation was just a natural outgrowth of my interest in creating worlds. As I grew up I learned about other ways of understanding reality and creating worlds: mathematical models and physical theories.

Why does the refrigerator door resist opening? Because the lower temperature inside causes the air molecules to move more slowly, reducing the pressure and producing a force inbalance. How large is it? Can you get a pressure difference that makes it impossible for even a strong human to open the door?

Adam's book is all about exploring reality through mathematical models. It is about the enjoyment of thinking about the small observations we make - that waves in puddles behave differently from waves in lakes, that trees change shape as they grow and clouds seem to like particular shapes. We see these things all the time, but actually trying to understand them brings new wonder. The book shows why both ducks and boats leave wakes that are angled at 33 degrees composed of waves angled 35 degrees regardless of the speed of the duck or boat. And why river meander lengths are close to 4.7 times their radius of curvature, why halos appear att 22 and 46 degrees from the sun and why pavement cracks into polygons. A world of elegant interactions between opposing forces and mathematical facts opens up.

The book starts out by discussing the ideas of mathematical modeling and then moves on to making estimation (so-called Fermi problems) and dimensional analysis. Much of mathematical modeling is about braving the problem. Just because we don't know all details, all physical laws and how to solve it from the start doesn't mean it is intractable. We can start out with a rough approximation and refine it when we see discrepances; seeking perfection from the start will just prevent us from ever starting.

The book later examines various phenomena. The selection has mainly been about things we can experience ourselves, like the weather, waves, geography, biological pattern formation and minimal surfaces. This helps to anchor the mathematics in everyday reality.

My main complaint is just that the narrative jumps between so many different things that it never has the time to get into them deeply. Often a model is introduced, applied to the original problem, and then applied to other interesting problems. This helps to show how the world hangs together mathematically, but it leaves the analysis often a bit shallow. Of course, going into the detailled electromagnetism of wave refraction in droplets might be too much to ask for, but the book leaves you hungry for more.

The text was apparently the result of the evolution of course notes, and sometimes that shows up. A bit more editing would have handled the occasional repetion and re-presentation of a concept that had already appeared in a previous chapter. This origin might also explain the limited depth of analysis - there is only so much one can tell in a single lecture, and expanding it into a book still carries with it a bit of limitation.

Overall, Mathematics in Nature is a wonderful complement to the plethora of books looking at the world in terms of dynamical systems and their resulting chaos and fractals. This is a book anchored in "classical mathematics" and classical physics, showing that they still are highly relevant.

And what about the refrigerator? Atmospheric pressure is about 1 kg/cm^2 (about 10 N/cm^2) and a fridge door about one square meter. Hence a total vacuum inside would produce a force greater than 100,000 N. The world's strongest man can lift on the order of a few hundred kilograms (human muscle generates 16-30 N/cm^2, so with a bulging 10x10 cm muscle one can get about 3000 N). That is still about two orders of magnitude too little. So an absolute zero refridgerator would be unopenable (as long as it doesn't leak).

But my fridge is not that cool. Recalling the ideal gas law PV=kT (http://hyperphysics.phy-astr.gsu.edu/hbase/kinetic/idegas.html) The volume is about a cubic meter, and at 0 degrees C k is about 371. Using this, I get a pressure difference of 7140 N/m^2 between my 21 degree room and the fridge. 7000 N is equivalent to a 700 kg weight. So how come I can open my fridge without being the Hulk?

Posted by Anders at June 22, 2004 05:44 PM

Intwesting. Is it due to lack of air-tightness, so that air seeps in while the door is closed mostly removing the pressure difference? Or is the force imparted just nudging the door open enough to let air rush in? (I never hear a whooossh when i open the fridge)

Posted by: Alejandro Dubrovsky at June 22, 2004 07:24 PM

I think it is partially because air leaks in when the door is opened, but also because the fridge is not airtight in the first place. Another factor could be that usually the door is only open for a short time each time it is used, which means that much cold air remains inside. To this some more warm air is added to equalize the pressure, and when the door is closed this warm air cools - but now the fridge contains more moles of air than it would have if I had just filled it with warm air and let it cool.

Overall, it is fascinating how even everyday objects hide complex physics. Processor fan aerodynamics, paper wrinkling, bookshelf stability, shower curtain movements... you name it.

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