Question

Do you have (not trivial) examples of a natural phenomenon that illustrates perfectly a mathematical concept, structure, equation or theory ? As suggested by sigoldberg1, I search physical situations in which the underlying mathematics is especially clear. This could be a second separate question: Do you have an example of a natural phenomenon that directly inspired a mathematical theory? do you have pictures ? movies ?

Nautilus Shell (this illustrates Fibonacci?) http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html

Fractal coast (this illustrates self similarity): http://www.flickr.com/photos/buggs_moran/4516938146/

Answer 1

The symmetries of crystals give rise to a lot of interesting group theory. There exists an actual theory of crystallographic groups with a very rich theory.

Answer 2

If you put $m$ rocks in a row and then another $n$, the total number is $m+n$. If you rearrange the rocks any way you like, you still get $m+n$. If you make $m$ rows of $n$ rocks each you get $mn$ rocks. Many rearrangements of rocks result in demonstrations of familiar laws like $m(n+p)=mn+mp$. I suspect that these kinds of properties (which apply to a wide range of objects, not just rocks) resulted in the theory we now call arithmetic and the generalisations that we now call algebra.

Answer 3

A classic that should be on any list like this is D'Arcy Thompson's On Growth and Form.

Answer 4

Tao's Cosmic distance ladder presentation is a very good resource.

Answer 5

I believe Charles Darwin wrote about honeycombs in his book "The Origin of the Species". He wrote that the hexagons shapes built by the bees is proved to be the most efficient way of storing the honey (I'm assuming it was proved by the mathematicians of the era, or perhaps earlier. How valid is the proof? I have no idea).

He explained how they build it - which is quite amazing. Each bee digs in rotation, to form a sphere. Whenever two bees dig long and deep enough to meet, they build a thin wall.

Again, as for the proof of this optimum and its validity - I am clueless (geometry was never my strong side). If anyone can fill this part in - I'd be happy to read it myself.

Answer 6

A very good resource for this topic is John A. Adams's Mathematics in Nature: Modeling Patterns in the Natural World (Princeton University Press, 2003). An annotated table of contents and other information about the book is available at the author's web site.

Answer 7

Tidal Bores ( http://en.wikipedia.org/wiki/Severn_bore ) as an illustration of solitons in non-linear PDEs ( http://en.wikipedia.org/wiki/Korteweg%E2%80%93de_Vries_equation#Soliton_solutions ).

Answer 8

Refraction illustrates geodesics in metric spaces (with infinitesimal distances proportional to the inverse of the speed of light).

Huge enough astronomical objects (of diameter more than 1000km) which spin not too fast illustrate sets of equidistant points with respect to a given center.

Waterfalls illustrate parabolas.

Answer 9

The orbits of the planets around the sun illustrate ... Kepler's laws of planetary motion (modulo relativistic corrections and multibody interactions). The phenomenology inspired the development of calculus.

Answer 10

A recent example is the Nature paper "The von Neumann relation generalized to coarsening of three-dimensional microstructures" by Robert D. MacPherson & David J. Srolovitz.

Von Neumann had a formula for how 2-dimensional grains evolve over time, and the authors generalize his formula to 3-dimensional grains and higher.

Answer 11

1. Here is a very beautiful example. There is a kirigami of the way leaves like maple leaves (actually all plants with leaves folded in the bud) are shaped which arises from volume constraints in the bud, and a very few other factors. A good ref is http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2776983/

2. Also, to me all of physics is just the viewing of nature through the lens of mathematics. So maybe your question restates as

physical situations in which the underlying mathematics is especially clear,

e.g. chladni figures for eigenfunctions, icosahedral or helical virus coats from limiting the number of viral coat proteins due to small viral genomes, examples mentioned above, etc.

Also zillions of minimal principles (including minimization of effort when we unconsciously plan grasping say a glass of water), rainbows, etc. Anything where one particular mathematical principle dominates.

Answer 12

A few quick examples:

• Breeding bunnies $\rightarrow$ the Fibonacci sequence
• Projectile/planetary motion $\rightarrow$ conic sections
• Natural springs and sinks $\rightarrow$ Gauss' law
• Galilean relativity + constant speed of light $\rightarrow$ non-Euclidean geometry
• Relativistic gravity $\rightarrow$ intrinsic curvature
• Invariances of Maxwell equations $\rightarrow$ Conformal transformations
• Observables in quantum mechanics $\rightarrow$ Lie groups (a la Wigner)

EDIT:

• Symmetries of the electron $\rightarrow$ Quaternions