The Banach–Tarski paradox

26.07.10

I've come across several funky theorems in my time studying mathematics, but this one by far blows my mind the most. The theorem goes by the name of the Banach–Tarski paradox, and the reason it is called a paradox will soon become evident - believe me.

The theorem basically states that a solid ball in a 3-dimensional space can be split into a finite number of non-overlapping pieces and then be reassembled into two copies of the original ball. The reassembling can only be done by moving and rotating the pieces; no stretching or bending is allowed.

Now this immediately creates a paradox. In geometry, the act of splitting and rotating are volume preserving. This would make it impossible to end up with two units of volume when you only started with one unit. It is now we must ask the question; in what field of mathematics does this apply? And the answer is set theoretic geometry. Now, I have no thorough understanding of this subject but I will try to explain parts of the theorem in so called lay-mathematics' terms.

If two geometric figures can be transformed into each other, they are called congruent. In set theory we describe this by letting the geometric figures be a group (say X) and we let G be all the isometries of X. This can be extended to the idea of two sets being equidecomposable with respect to G, which means that both sets can be partitioned into the same finite number of respectively G-congruent pieces. Formally we can say if
$$A = \bigcup_{i=1} ^k A_i, \,\,\,\,\, B = \bigcup_{i=1} ^k B_i, \,\,\,\,\, A_i \cap A_j = B_i \cap B_j = \emptyset \,\,\,\,\, \forall i,j \,\,\,\, 1 \leq i < j \leq k,$$
and there are elements
$$g_1,...,g_k \in G$$
such that for each i between 1 and k,
$$g_i(A_i) = B_i,$$
then we say that A and B are G-equidecomposable using k pieces. Furthermore, if a set E has two disjoint subsets A and B such that A and E, as well as B and E, are G-equidecomposable then E is called paradoxical. Using this we can reformulate the theorem to the following:

A three-dimensional Euclidean ball is equidecomposable with two copies of itself.

This theorem is somewhat hard to believe, but there exist proofs. It only goes to show that we have to be extra careful when we define things such as "volume" in mathematics since our intuition often are contrary to what we will discover.

Exciting, in a way. :D

Euler Characteristic

23.07.10

The Euler characteristic is a number which describes a topological space's shape or structure no matter how it is bent. For the plane (and sphere), the Euler characteristic is 2 (well known for those of my classmates who took the course TMA4165 - dynamical systems) and I will now show you how you can use this number for a more… practical situation (rather than calculating the index at the point infinity).

So the scenario is this.

Say that you want to build a system of fences. You know how many pieces of fence you have available (edges, essentially) and you know how many areas you want to enclose. How many fence posts do you then need, and is this number dependent of how you arrange your fence?

We will now use the "definition" of the Euler characteristic which is
$$\chi = V - E + F$$
where V is the number of vertices, E is the number of edges and F is the number of areas you are enclosing. Note that the outside is to be included in this number.

So say now that you have 12 pieces of fence and want to enclose 5 regions. You will then need
$$2 + 12 - 5 = 9$$
posts to do so.

Try it yourself. Create a fence system and verify that the Euler characteristic is indeed two for any combination you might come up with.

Science Jokes

18.07.10

Science jokes are all fun and games and since it is summer I won't bother you with any "heavy" reading. This is a small collection of physics jokes or puns I've picked up around the magnificent information blob that is the Internet. Enjoy.



Helium walks into a bar.
The bartender says "We don't serve noble gases here" Helium doesn't react.

--

The tachyon says "That's ok, I was just leaving anyways" The bartender says "We don't serve tachyons here". A tachyon walks into a bar.

--

Schrödinger's cat walks into a bar. And it doesn't.

--

A neutrino walks into a bar.
The bartender says: "We don't serve neutrinos here!".
The neutrino says: "That's OK, I'm just passing through."

--

An infinite number of mathematicians walk into a bar. The first asks for a beer, the second asks for a half, the third asks for a quarter, etc...
The bartender goes "You're all idiots!" and taps two beers.

--

Share your jokes/puns in the comment section below and I'll update this entry for future readers. ^__^

PS: I've started writing a small article about length contraction as a follow up to this entry. I'll try to finish it up during next week.

Up North and Back Again

05.07.10

I’ve just come home from a brief three-day-trip up north. My first vacation as far as this summer is concerned, and I loved it. Nothing cheers me up like meeting up with old friends at our local pub. It was also nice to see progress happening in the local business life - we now have another bar, making bar-to-bar-trips almost useful.



Visiting the local pub was however not the sole purpose of my journey. Oh no, it was far more serious than that; it was the baptism of my niece (which by the way is adorable!).

(I am now uncle of two nephews and two nieces requiring me to be super-uncle whenever the whole family decides to meet together up north.)

The baptism ceremony went by with a preacher venturing out on very thin ice talking about religion and war in the modern world, us singing a hymn promising salvation to heathens as well as Jews (!:D), and a particularly awkward scene where I, my nephew and a bunch of other kids stood in front of the entire church singing some sort of Christian soldier song.

Strange indeed.

The rest of the week was, briefly speaking, spent eating cake and spending quality time with my family (which I by the way am extremely fond of ^__^).

So now I sit here, back in my apartment in Trondheim, without any luggage. Yes, the airline company managed to mislay my suit case somewhere, leaving me with very little to wear besides the clothes I was traveling in. Needless to say, I’ve been searching all over for dirty shirts, socks and other clothing to wash tonight so I can look somewhat presentable at work tomorrow. I sure hope SAS manages to clear this up by tomorrow.

Well, time to pop the clothes into the dryer. Take care, dear reader! :)

Edit:
The universe must be playing some sort of cruel joke on me. When I went down to open the washer, the door wouldn't open. Looks like the washer is flooded and now I have to find some way of draining it.

:D Son of a bitch.