Cool Math Theorems

[betazed]

With all this heavy duty philosophic stuff about God and all I thought maybe we can spend some time with something much easier. Math. Believe it or not it is a whole lot easier than grappling with the tenous concepts of philosophy. So here I want all math enthusiasts to mention cool math theorems that everybody can enjoy.

I will start with the morning coffee theorem.

Every morning many of you make take a cup and drink coffee. After you pour the coffee and pour some sugar in it you stir it with a spoon or some other stirrer. Your motions go round and round and create a swirl in the coffee. The intent is to make sure that all parts of the coffee gets stirred and the sugar is evenly distributed throughout. You would be surprised to know that irrespective of how long you keep doing the motion there is always one point in the coffee that has not moved at all. This surprising non-intuitive result is called in mathematics as the Brouwer Fixed Point Theorem (there are many versions of the fixed point theorems but all of them include a continuous map from one set to another set).

The only way to make sure all points have moved is to bring the motion to a stop and then stir again or abruptly change the stirring motion. No regular motion will do.

Do you know of any other cool theorems?

 

[col]

No matter how you comb your hair, there is always a bald spot - the hairy ball theorem.

http://www.math.hmc.edu/funfacts/ffiles/20005.7.shtml

 

[Guildenstern]

It's not really so much that the point hasn't moved at all, but that there is a point which is now in its original place. Of course, it is always possible that there are no water/coffee molecules located at this point, in which case it is possible to stir the coffee thoroughly, if you get lucky.

Gödel's First Incompleteness theorem is the best theorem I can think of right now, especially when combined with Turing's proof of the unsolvability of the Halting Problem, and with Chaitin's proof of the existence of numbers which have precisely-defined values, but which cannot be calculated. All three of these theorems are interrelated, and are equivalent to one another.

Gödel's theorem states that there are statements in number theory which are true, but which cannot be proven from the axioms of number theory alone. He proves this fact by observing that the symbols of number theory can be encoded as numbers, and the manipulation of these symbols can be translated to arithmetic operations on these numbers; and thus, number theory is capable of talking about itself. He then describes how to construct a sentence in number theory which states, "This sentence cannot be proven from the axioms of number theory." If the sentence is false, then it can be proven; but because we know the axioms of number theory are consistent, proving a sentence implies that the sentence is true; therefore, this is a contradiction, and the sentence must be true, yet unprovable.

Turing's theorem states that there is no Turing machine which can, in a finite number of steps, decide in general whether any given Turing machine will halt on any given input. Turing proves this, like Gödel, with self-reference: he asks his machine to decide whether it will halt itself; but it will only halt if it does not halt, and will only fail to halt if it does halt, and therefore it cannot decide the problem.

Chaitin's number is defined as a nonterminating binary fraction (like 0.0110101101011...) as follows: Take all possible Turing machines and order them by the length of their descriptions in some standard encoding (like ASCII). Then, for each binary place in the fraction, there is one corresponding Turing machine. Set this binary digit to 1 if the Turing machine halts on the blank input, and 0 if it does not halt. Thus, Chaitin's number has a precisely-defined value (the Turing machines are well-ordered, and each one must either halt or not), but there is no algorithm which can compute it (because such an algorithm must be able to decide whether any Turing machine will halt on a given input, and Turing proved that no such algorithm exists).

If I come up with any cooler theorems I'll post them.

 

[betazed]

No matter how you comb your hair, there is always a bald spot - the hairy ball theorem.

The Hairy-Ball theorem says that you cannot have a nowhere zero vector field on a sphere. So to apply it to hair we have to say precisely that if we comb our hair in a regular pattern everywhere then you must have a bald spot. Otherwise you can comb half your hair in one way the other half in another way and have two vector fields defines on your head and hence cover every spot.

edit: Corrected after Guildenstern showed the error. Thanks.

 

[betazed]

It's not really so much that the point hasn't moved at all, but that there is a point which is now in its original place. .

Correct Sir. Thanks for making that statement precise.

 

[col]

The Hairy-Ball theorem IIRC, says that you cannot have a nowhere zero vector field on a n-ball. So to apply it to hair we have to say precisely that if we comb our hair in a regular pattern everywhere then you must have a bald spot. Otherwise you can comb half your hair in one way the other half in another way and have two vector fields defines on your head and hence cover every spot.

I know - but its cooler my way - and makes people look at the reference

 

[the mormegil]

'Cool' and 'Math Theorems' don't really seem to fit together. But it seems they're cool to CFCers. Keep 'em coming.

 

[Guildenstern]

The Hairy-Ball theorem IIRC, says that you cannot have a nowhere zero vector field on a n-ball. So to apply it to hair we have to say precisely that if we comb our hair in a regular pattern everywhere then you must have a bald spot. Otherwise you can comb half your hair in one way the other half in another way and have two vector fields defines on your head and hence cover every spot.

The only case where I know the theorem works is on a 2-sphere (i.e., a standard sphere in 3-space, the boundary of a 3-ball). The ball itself is three-dimensional and you can easily just throw a constant nonzero vector field in there with no problem.

On a 1-sphere (a circle), it is also pretty simple to have a continuous nonzero vector field (constant magnitude, always pointing clockwise will suffice).

I'm not sure about a 3-sphere. But it seems clear that on an n-ball, it is always possible to throw in an n-dimensional constant vector field.

And, because your hair does not cover your entire head, it is possible to comb all of it flat. But you will always have bald spots because your hair is discrete, and there is empty space between the individual hairs.

 

[betazed]

You cannot draw a map that requires more than 4 colors. This is the famous Four color theorem. It was not proved for a long time until a computer proof was given. Many mathematicians still think that this theorem has not been proven since they do not deem a computer proof good enough.

@Guildenstern: Thanks for correcting me again. My math knowledge is sloppy

 

[The Last Conformist]

I hate the four colour theorem, or rather the way it's usually presented, refering to maps. It assumes that the "countries" have to be continuous, which of course isn't true for real countries.

The world map of 1914 "disproves" it; the UK, France, Germany, Belgium and the Netherlands (incl their colonies) all have common borders with each of the others, meaning you need five colours just for them.

 

[betazed]

I hate the four colour theorem, or rather the way it's usually presented, refering to maps. It assumes that the "countries" have to be continuous, which of course isn't true for real countries.

The world map of 1914 "disproves" it; the UK, France, Germany, Belgium and the Netherlands (incl their colonies) all have common borders with each of the others, meaning you need five colours just for them.

The point of this thread is not just to state abstruse theorems in math but to put them in a way that is understandable to all even to people who have absolutely no math bakground. Stating the above theorem using maps acheives that objective.

Also in the particular example that you are talking about you are making a assumption (albeit a reasonable one, but a assumption nonetheless). That the regions of the same country must be colored using the same color. This essentially turns the map from a 2-dimensional one to a three dimensional one and the theorem no longer applies.

Also let me ask you something else. In what other way can you put the four color theorem as eloquently.

 

[col]

Every even number is the sum of two primes.

Sadly there isnt room here in this box for the proof so I'll upload it and link to it tomorrow.

 

[betazed]

Ah! the famous Goldbach Conjecture. I too know a proof. I just have to write it down.

 

[Perfection]

Well, I consider myself mathematicly inclined, but this is just over my head,
I get Fixed Point Theorem, but not Brouwer Fixed Point Theorem, what the hell is that wierd B thingy?

 

[Perfection]

Every even number is the sum of two primes.

Sadly there isnt room here in this box for the proof so I'll upload it and link to it tomorrow.

What about 2?

 

[betazed]

what the hell is that wierd B thingy?

It says in the link the B thingy is a n-Ball. It also defines a n-ball. And there are links there to define a ball and in that there are links to define the things used to define a ball, and in that.... ad infinitum

 

[sourboy]

I don't know if this is the kind of theory you're referring to, per se, but how about the old 1 + 1 = 3 ? To keep math formulas out of this one for the non-mathematicians, basically the formula takes the numbers and alters them in proven ways until the answer changes from 2 to 3. This would be like saying 'green' is the same as a mix of 'yellow & blue' - then finding equals for them & so on, until you get something along the lines of 'green' is the same as 'red.'

I guess you could throw in the "women are the root of all that's evil" thing, but I'm sure we all know that one (on paper, and in reality ).

 

[Guildenstern]

True, the four-color theorem requires contiguous countries to work with regards to real maps. If countries can be disjoint (i.e., with colonies and such), I believe any number of colors may be required, but I could be wrong.

And if the Earth were a torus, then contiguous country maps could require up to 7 colors.

 

[col]

What about 2?

Er 1+1 = 2.

 

[nihilistic]

Er 1+1 = 2.

1 isn't prime. It is a unit.