Question

Dear Mitch, 

So many people talk about all the math in nature, but most of their examples are kind of obvious.  The teacher I have this year for math makes all the kids research something in nature that involves math, and then at the end of the first semester you have to present your thing to the class.  I know kids who had her last year and they did stuff like probability in genetics, or the way a population can increase exponentially, but those things have been done to death.  Can you tell me something you think I should check out if I want to do something interesting that not everybody knows already?

Thanks, 

Jeff P.

Answer

Dear Jeff P., 

Yes, I'd love to!

The first thing that pops into my mind that I bet will fascinate you (as it seems to really amaze almost everyone who looks into it) is something called the "logarithmic spiral."  The basic, everyday spiral that you can see if you look at the side of a roll of paper or paper-towels or anything neatly rolled around a tube for packaging purposes is called an "Archimedes spiral". While there's nothing wrong with that type of spiral, and while we need them for all kinds of things, the logarithmic spiral is very different and is truly an astonishing and fundamental part of nature. 

Basically, it is a spiral that does not just get thicker all the way around with each revolution to accommodate another layer of paper towels (or whatever material is involved) -- not at all.  In fact, what a logarithmic spiral does is this: Its radius expands with each revolution so that it maintains its shape while growing rapidly from inside to out.  If memory serves me correctly, Dorothy's Yellow Brick Road in The Wizard of Oz is a logarithmic spiral, though I can't recall if Dorothy goes inward, making smaller and smaller revolutions, or outward, expanding and expanding her trip as she walks around it...

But here's the thing:  If we allow ourselves to think of nature as a sort of 'person' with ideas and feelings, then I think we can accurately say that She is OBSESSED with logarithmic spirals.  Why do I say that?

This shape's main property -- of growing without changing its shape -- is necessary for many growth phenomena in nature.  One example that I came across a few years ago of the logarithmic spiral shape as necessary in nature was so clearly described that I don't think I will ever forget it.  I came across it in a book about something else in math and nature called the "Golden Ratio".  The book was written by the mathematician Mario Livio, and he describes the "mollusk inside the shell of the chambered nautilus", and its growth style of closing off smaller unused chambers as it creates larger ones.  In this way, the living creature inside can grow without having to expend energy adapting to a different shape or newly proportioned 'home'. 

I'll stop right here, because it wouldn't be right for me to do your project for you and cheat you out of all the fun of reading and discovering for yourself.  But I have to say, stopping here without moving on to other astounding ways this seemingly simple shape shows up throughout nature's architecture is HARD for me – because it really does become more and more fascinating as you observe its obvious advantages and nature's genius for efficiency and purpose! 

Good luck!

Hope this helps,

Mitch