The Self-Made Tapestry: Pattern Formation in Nature
A quick search under the phrase "patterns in nature" turn up all kinds of interesting-looking books. This one updates the original by Peter S. Stevens. (See below.)
Basically, there are eight or so patterns in nature. Understanding the mathematics behind each one helps you understand why they turn up where they do, and how their combinations in different size ranges gives such an endless supply of amazing things to see.
Though the book is very challenging to understand in its entirety, it is quite accessible as a discussion of the patterns to be found. If you can’t follow the equations, you can still see the pattern and remember how they appear in your world. Patterns have an orderly source.
At some point, a science or a math course should offer a reflection on the patterns generated by bubbles and their packing rules, waves and their interference rules, the Fibonacci series and the spirals it generates, and the branching rules that govern the growth of plants and also of rivers. These are the fundamental patterns of the universe. They turn up everywhere in nature, and Ball particularly makes the point that in the rush to embrace Darwinism, the fact that biological entities must obey the laws of physics is often ignored. Very important!
Theodore Andrea Cook
The Curves of Life is a classic work showing how math describes various patterns and curves that are found in living creatures.
The Joy of Mathematics:
of Discovering Mathematics
All Around You
You know, if a math book has Hokusai's famous Great Wave on the cover, something exciting must be inside. Math is about pattern as well as number, and a mathematician author covers both.
Peter S. Stevens
Patterns in Nature
In 1974, Peter Stevens wrote this wonderful text that is out of print. And how could it stay in print, being only black and white, while his followers have the advantage of color. But the original still has its own glories, particularly some things about architecture.
On Growth and Form
Ball’s work (above) is partly based on the earlier work of D’Arcy Thompson, called On Growth and Form, which is a little easier to read and much cheaper, being out of copyright, but also less colorful and engaging.
Still, this older book and the one by Peter Stevens offer a swath of unexpected information about patterns, including an amazing discussion of certain architectural wonders based on shell spirals.