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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?)

Fractal coast (this illustrates self similarity):

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closed as off topic by Harry Gindi, Loop Space, Victor Protsak, Andy Putman, Noah Snyder Jul 20 '10 at 5:19

Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here.If this question can be reworded to fit the rules in the help center, please edit the question.

This is a very board question. (Examples of natural phenomena that inspired mathematical theories covers almost all of mathematics.) It's up to the answerers where they will take this question. –  Gil Kalai Jul 2 '10 at 9:21
Community wiki? –  supercooldave Jul 2 '10 at 9:33
I've voted to close because it's not clear to me that there is any mathematical content here at all. meta.MO thread:… –  Harry Gindi Jul 19 '10 at 9:28
The Nautilus shell has nothing to do with the Fibonacci sequence; this popular belief was scientifically debunked a while ago (can't find a citation at the moment, but just try comparing an image of a nautilus shell with a golden spiral). It is, however, a perfectly decent logarithmic spiral. A better example of the Fibonacci sequence appearing in nature would be phyllotaxis, the arrangement of leaves on a plant stem, or of seedheads or petals on a flower - famously illustrated with the spirals seen in sunflower heads and pinecones. –  Robin Saunders Jul 19 '10 at 21:26
I closed per the discussion on meta. There might be a good question on this topic, but it would need to be much more specific. –  Noah Snyder Jul 20 '10 at 5:21

12 Answers 12

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.

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Very good and relatively simple class of examples,Micheal.Not a bad place to begin to answer this very important question.Hopefully many more to come. –  The Mathemagician Jul 2 '10 at 6:47

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.

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I wish I could vote this up several times. –  Dan Petersen Jul 2 '10 at 21:46
Thanks sigfpe, I also like your answer. However, it seems to me that it is quite out of the scope (even if the scope is very large) because rocks you find in nature never have the same size, and it is most often difficult to rearrange with ease :) ... If we can distinguish human creation and nature (OK, this is not that obvious) I guess your example is a human creation, a human abstraction ? anyway it appears in nature somehow and I vote up ! –  robin girard Jul 8 '10 at 5:55

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

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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.

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See The biological one is close to, but not exactly, the theoretically most efficient (for certain definition of efficiency). Somewhat related is the Kelvin Conjecture… . –  Willie Wong Jul 2 '10 at 12:42
The honeycomb conjecture was proved by Hales and you can find his article here –  j.c. Jul 2 '10 at 13:14

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.

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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.

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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.

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Dear Andrew, was it really necessary to call my example lame? I happen to think the calculus derivation is quite elegant, and one can use orbital motion as a fruitful pedagogical example in Hamiltonian and Lagrangian mechanics. –  S. Carnahan Jul 2 '10 at 23:24
I agree this is a HISTORICALLY important example,Scott-but it's not my favorite. Orbital motion obtains both formulations of mechanics,this is indeed in general true. I'm specifically talking about the derivation of the fact the paths of the planets around the sun is derived as having the form of a polar ellipse.It's not a BAD example-it's just so ubiqutious as to have become redundant,that's all I meant. –  The Mathemagician Jul 3 '10 at 6:06
@Scott continued: It's in every single calculus book with a several variable calculus section published in the last 40 years in one form or another and even high school students taking calculus and physics know it!!! In fact,the fact this is the only physical application in the latest edition of Spivak's CALCULUS has always irritated me.There are a lot deeper and better examples.Still,due to its' historical significance,definitely belongs on any list of this type. –  The Mathemagician Jul 3 '10 at 6:06
@Scott: Comment edited to remove questionable part. Again,I'm sorry. –  The Mathemagician Jul 3 '10 at 6:07
I think Andrew is exaggerating a quite bit. Firstly, I know some Calculus textbooks that do not include it. Secondly, I know no one who actually taught this derivation to their Calculus class (not even I myself when I had a chance to do it this past spring). Thirdly, one of the most discouraging discoveries I've made teaching Calculus is that many if not most students do $\textit{not}$ know Kepler's laws, and that includes astronomy majors. The last point has a lot to do with the reluctance of mathematics instructors to teach this material. –  Victor Protsak Jul 20 '10 at 4:15

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.

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  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

  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.

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Thanks sigolberg1. I have edited my question with your sentence: physical situations in which the underlying mathematics is especially clear, I like it ! –  robin girard Jul 8 '10 at 6:00

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)


  • Symmetries of the electron $\rightarrow$ Quaternions
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Example 1 is bullshit ;) –  darij grinberg Jul 19 '10 at 11:55
Hey, it's an idealization, ok? And anyway, if it was good enough for Fibonacci himself... ;) –  soulphysics Jul 19 '10 at 12:00
Hmm, just because it's theoretical physics doesn't make it "nature", but I am possibly misunderstanding the question. –  Gjergji Zaimi Jul 19 '10 at 12:09
Completely irrelevant. $\textit{None}$ of the items on the left (with possible exception of 1, if you believe it) inspired the corresponding theory on the right. The chronology is backwards. –  Victor Protsak Jul 20 '10 at 4:08
@Victor -- obviously. But the question asks for natural phenomena that "perfectly illustrate" a mathematical concept, not "historically inspired". Sheesh. –  soulphysics Jul 21 '10 at 12:12

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