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Colloquial catchy statements encoding serious mathematics

As the title says, please share colloquial statements that encode (in a non-rigorous way, of course) some nontrivial mathematical fact (or heuristic). Instead of giving examples here I added them as answers so they may be voted up and down with the rest.

This is a community-wiki question: One colloquial statement and its mathematical meaning per answer please!

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Recent answers suggest this question is getting a bit long in the tooth. – S. Carnahan Jul 6 at 5:38

It's not possible to comb a hedgehog.

Or, alternatively:

You can't comb the hair on a coconut.

Both statements are referring to the fact that every continuous tangent vector field on the 2-sphere has to vanish at some point. That's the well known Hairy ball theorem.

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Armin, you obviously travel in more poetic circles than I! I'd always just heard “you can't comb a hairy ball”. – L Spice Feb 24 2010 at 0:06
Of course one can comb a hedgehog, but not without a bald point! – Konrad Waldorf Jul 5 at 8:37
Those who comb hedgehogs for a living know that they have bald bellies: goo.gl/E1HWh – Theo Johnson-Freyd Aug 25 at 13:25

A drunk man will find his way home, but a drunk bird may get lost forever.

This encodes the fact that a 2-dimensional random walk is recurrent (appropriately defined for either the discrete or continuous case) whereas in higher dimensions random walks are not. More details can be found for instance in this enjoyable blog post by Michael Lugo.

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This particular saying, by the way, is usually attributed to Shizuo Kakutani. (I don't want anybody thinking I came up with it!) – Michael Lugo Oct 31 2009 at 17:16
Uh oh. Thanks for the warning. – tweetie-bird Oct 10 at 0:42

Black holes have no hair.

This no-hair theorem "postulates that all black hole solutions of the Einstein-Maxwell equations of gravitation and electromagnetism in general relativity can be completely characterized by only three externally observable classical parameters: mass, electric charge, and angular momentum".

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This is of course true only in 4 dimensions. Higher-dimensional black holes can be hairy. – José Figueroa-O'Farrill Nov 2 2009 at 5:54

The ham sandwich theorem comes to mind: given n measurable sets in Rn, there is a hyperplane (i. e. an affine subspace of codimension 1) that bisects them all. I don't know of a colloquial way to state this, though.

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You can just say: "a ham sandwich (two pieces of bread and one of ham) can be split in half with a single cut." – Ricardo Nov 1 2009 at 14:16
Ricardo, that's true. But I feel like what's really remarkable is that this holds in all dimensions, and ham sandwich definitely brings to mind a three-dimensional picture. – Michael Lugo Nov 1 2009 at 16:39
So just take an n-dimensional ham sandwich... :) – Cam McLeman Jul 5 2010 at 5:12

There is a way to cut up a pea and rearrange the pieces to get the sun.

As a corollary of the Banach-Tarski paradox, we have that if A and B are bounded subsets of R^n (n > 2) with nonempty interior, there exists a partition of A into k pieces {A1, ..., Ak} and isometries of R^n {f1, ... , fk} so {f1A1, ... , fkAk} partitions B.

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This is not an immediate consequence, and the pea-to-sun statement is a highly misleading translation of the statement of the theorem. Both the pea and sun are roughly spherical, but the theorem shows that you can geometrically decompose a sphere into two spheres of the same radius, not into a sphere of a different radius. If you disagree, please tell me how many pieces you will use to go from a ball of radius 1 into a ball of radius 2. I would use 9 pieces to decompose a sphere into two of the same radius, although that's not minimal. – Douglas Zare Feb 19 2010 at 21:15
Maybe it shoud be formulated as: Give me one pea and I'll feed the world. :-) – Yiftach Barnea Apr 19 2011 at 8:41
@Douglas: There is a generalization of the Banach-Tarski-theorem that applies to almost arbitrary subsets of IR^n. (I think the precise condition is that they have non-empty interior) – Johannes Hahn Apr 19 2011 at 10:14
@Johannes: I think that, in addition to having non-empty interior, the sets need to be bounded. – Andreas Blass Jul 5 at 1:01
Visualize world peas? – Noam D. Elkies Jul 6 at 4:47
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Complete disorder is impossible.

This is the standard way of summing up Ramsey theory in a succinct sentence (according to that Wikipedia article, the above quote is due to Motzkin).

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By the way, Motzkin's remark, which is in a paper in J. Comb. Theory 3:244-252 (1967) made an interesting comparison between disorder in physics and combinatorics: "Whereas the entropy theorems of probability and mathematical physics imply that, in a large universe, disorder is probable, certain combinatorial theorems imply that complete disorder is impossible." – John Stillwell Jan 31 2010 at 6:04

After stirring a cup of coffee, at least one point will end up in the exact same position as it was before.

Being an approximate statement of Brouwer's fixed point theorem.

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I've always had issues with this one since I'm not convinced the "stirring" function would be continuous... – Elisha Peterson Nov 1 2009 at 20:11

Take a map of wherever you are and lay it on the ground. There will be exactly one point on the map that is directly above the point it represents on the ground.

This refers to Banach's fixed point theorem.

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This should also follow from Brouwer's fixed-point theorem, right? If so, it's the best statement of Brouwer's fixed-point theorem I've ever heard, and I'll definitely be using it in the future! – Vectornaut Nov 17 2009 at 19:02
You need Banach's fixed point for the uniqueness, but the existence follows Brouwer's fixed point. – Josiah Sugarman Dec 10 2009 at 15:59
i actually did this in class to demonstrate the fixed point theorem. there was a satisfying gasp when I picked up my lecture notes and scrunched them :) – Suresh Venkat Dec 10 2009 at 20:36
With this formulation one may say that even the proof somewhat made its way thorugh and reached literature. In Borges "Partial Enchantments of the Quixote" a story contained in Other Inquisitions, an apocryphal quote states that in a perfect map, a copy of the map should be contained, and such copy would contain another copy of the map and so on at infinity, which is basically the proof of the Theorem, if you add to it existence of the limit... – Nicola Ciccoli Jul 5 at 7:24

Time Average is the Space Average

which is the the Ergodic Theorem.

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It seems like a reasonable way of summarizing it catchily, but I'd prefer a little extra context -- as stated it reads like it could be something from Time Cube... – Harrison Brown Nov 1 2009 at 15:43

"The shortest path between two truths in the real domain passes through the complex domain." -- Jacques Hadamard

I guess he meant that often the best proof of a theorem about real numbers uses complex analysis.

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You Can’t Unscramble Scrambled Eggs

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I understand the others, but how does this relate to P vs. NP? – Michael Lugo Nov 3 2009 at 15:56
Ah, but theorem: Consider a compact pan with some unscrambled eggs in a closed (no "external" i.e. time-varying physics) classical Newtonian (energy is kinetic, which is positive-def quadratic in velocity, plus potential, which depends only on position) universe. The eggs may be in the process of scrambling. Then at some time in the future (indeed, after some precisely integer number of years, where how long you have to wait can be given an explicit absolute bound in terms of epsilon), the eggs will be within epsilon of unscrambled. – Theo Johnson-Freyd Dec 24 2009 at 20:43
Non-mathematically, this reminds me of the cryptic crossword clue: gegs (9,4). – Andrew Lobb Feb 22 2010 at 21:40

"Can you hear the shape of a drum?"

This was Kac's famous way of asking whether the shape of a two-dimensional domain could be reconstructed from the spectrum of the Laplacian on that domain. (The answer, by the way, is "no", at least if one allows the domain to have corners.)

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"Chewy on the outside, crunchy on the inside."

The way that Dick Canary would characterize hyperbolic 3-manifolds with nonempty (and nonrigid) conformal boundary.

"Crunchy on the outside, chewy on the inside."

Description of Rich Schwartz's complex hyperbolic manifold with conformal boundary a real hyperbolic manifold.

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It's better to be lucky than good.

P != NP. A nondeterministic polynomial is one which is always "lucky".

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Given that P!=NP is unproven, perhaps I hope it's better to be lucky than good. – Ilya Nikokoshev Nov 1 2009 at 20:48
Or "better lucky than smart" – vonjd Feb 23 2010 at 14:24

Fields medalist René Thom famously wrote:

Ce qui limite le vrai n'est pas le faux, mais l'insignifiant. (Approx translation: "What limits truth is not wrongfulness, it's meaninglessness.")

This refers to the basic mathematical issue that one must only consider well-formed formulas rather than arbitrary ones.

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Is that really what he meant? That seems like a pretty narrow interpretation. – Darsh Ranjan Nov 2 2009 at 2:48

Odd is better, but three is best.

This refers to the fact that "sound" propagation in arbitrary dimensions (using the wave equation on an initial pulse) is only possible in odd-dimensional spaces (in even dimensions there's an ever-lasting echo), and that dispersion-free propagation only happens for dimension three.

See this paper where I saw the quote: part one, part two, although the fact itself seems to be known for ages (I have no access but an old paper by Balazs seems relevant too), possibly this even goes back to Petrovskii according to an interview of Arnold (see p434).

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This is also known as the "strong Huygens principle". – Terry Tao Nov 1 2009 at 21:29

An anagram for "Banach Tarski" is "Banach Tarski Banach Tarski"

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I don't think this is what Armin Straub was looking for, but it certainly made me laugh. ^_^ – Vectornaut Nov 17 2009 at 19:10
I'd like to know the original source of this. I put it on a T-shirt. thenerdiestshirts.com/site/… – Douglas Zare Jul 4 at 21:49
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All primes are odd except 2, which is the oddest of all.

This has been discussed before. The quote is from "Concrete Mathematics," but it's a rephrasal of one of J.H. Conway in "The book of numbers."

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Concrete Mathematics (1989, 2nd ed. 1994) was published before The book of numbers (1996). – Michael Lugo Nov 3 2009 at 23:21

Period three implies chaos.

This summarizes a paper (of the same name) by Li and Yorke. The full statement of the main theorem is that if a continuous transformation of an interval has a point whose orbit has length three, then there exist points whose orbits are completely chaotic (in addition to points with orbits of every other possible finite length).

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One should read the interchange between Li, Yorke and Ruelle that took place in the AMS Notices about this! – Mariano Suárez-Alvarez Dec 10 2009 at 13:13
@L Spice: ams.org/notices/200906/rtx090600688p.pdf and ams.org/notices/200910/rtx091001232p.pdf – Vladimir Dotsenko Jul 5 at 7:56
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Trust, but verify.

this is the defn of NP, of course, but for a certain generation, it has Reaganesque overtones.

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According to the Wikipedia page on this catchphrase (yes, Wikipedia has a "trust, but verify" page), it originates in a Russian proverb, as Reagan himself introduced it. – Noam D. Elkies Aug 1 2011 at 4:34

All diagrams commute.

The actual theorem is Mac Lane's Coherence Theorem or any of a number of other coherence theorems that guarantee that all category diagrams built from certain elements commute.

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A brain tangled enough to tackle itself must be too tangled to tackle.

A neat formulation of Goedel's Incompleteness Theorem. From the novel "Galatea 2.2" by R.Powers.

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Well, it takes two to tangle. – Benjamin Dickman Aug 25 at 9:40

"Space tells matter how to move; matter tells space how to curve."

This is the quintessential colloquial expression of the Einstein field equations that govern general relativity.

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Local is global

The very first thing I heard about sheaves, from another graduate student.

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A variant (with different mathematical overtones): "Think globally, act locally". I had a button with this on it as an undergraduate, from a very non-mathematical source. – Ravi Vakil Jan 31 2010 at 5:38

There is no free lunch. Refers to risk/reward in financial investment and the fact that an efficient market moves to the point where you can only make money by taking risk.

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Numbers are mutually friendly if they share their abundancy

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Truth is undefinable,

which is a statement of Tarski's theorem. More precisely,

Truth in a context where one can do arithmetic is undefinable in that context.

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There is a pair of antipodal points on the surface of the Earth at equal temperature and equal pressure.

The Borsuk-Ulam theorem for n=2. Suppose $f: \mathbb{S}^n \rightarrow \mathbb{R}^n$ is a continuous map. Then $\exists x: f(x) = f(-x)$.

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While the pressure field should be continuous, I can see no reason why the temperature field have to be continuous? – Kjetil B Halvorsen Aug 26 at 4:32

"If it walks like a sphere and it quacks like a sphere then it is a sphere."

A professor at my university explained the Poincare Conjecture to his 1st semester abstract algebra students this way. I think it is a great explanation!

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A city is compact if it can be patrolled by finitely many nearsighted police officers.

I believe this is due to Peter Lax. Of course one must take some care with quantifiers to make this a correct definition, but I think it captures the spirit nicely.

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I think this is a dangerous metaphor: it encourages the misconception that "a space is compact iff it has a finite open cover". – Rasmus May 10 2011 at 8:04
This is attributed to Weyl in Wilson Sutherland's text Introduction to Metric and Topological Spaces (section 5.2). Sutherland gives it as "If a city is compact, it can be guarded by a finite number of arbitrarily near-sighted policemen". Exercise 5.10.15 asks you to make precise, and discuss the accuracy of, Weyl's statement. – Tom Leinster Aug 5 at 21:40
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