1543.-Three theorems on the water and his absence of form

Teor 1: Imagenemos a river, and let's put a hexagonal network in a part of the watercourse, between two coasts. We choose hexagons at random, and place in them a cement column (if two are nearby, they join hermetically).

Only there are two possible results: or a water course remained open, or the columns form a dike. Finally, all Hex game ends with the victory of one of the players.

* * *

Teor 2. In a flat area we place columns some next to other one, hermetically joined the column i with i-1 and i+1 (and only with them), such that the first one and the last one also stick. There stay two separated regions, a nursery, which we can fill with water without it happening to other one.

* * *

Teor 3: In a square pond, we wave the water (gently) and at least a point will not move (or the waters are opened).

Warning: the next paragraphs contain the demonstration. It can be omitted in the first, second, third one... reading. The simbolito √ in front of some numbers 2 refers to 'root'. The html is not math-friendly.

Let's see: if any point displaced at least a distance h, let's cover the surface with a network of triangles of diameter d minor to h (we will say in the end who are h and d).

Let's paint the apexes of red if the coordinate x changed h / √ 2 or more. If not, we paint it of green (the coordinate and it changed h / √ 2 or more).

the way is only an indicative, it might be any other

There will stay a way of red apexes or one of green (I replaced a triangle by a column if it has at least two red apexes: it has a dike, or the water happens). Let's suppose that the way is red, it does not matter. It begins in to *, and the coordinate x increased at least h / √ 2. It comes to b *, where it diminished at least h / √ 2. Then, in some moment, a change of signs took place: in two apexes of the same triangle 2 jumped at least 2 h / √ (that is major to h).

Now, we need a little analysis to say who are h and d:

If T is the transformation that waves the water, and the norm euclídea of T (x, y) - (x, y) it is not annulled, is bigger than one h (since it is continuous).

The images of two points of the square for T will be over a distance minor than h if they were over a distance minor to one d' (since T uniformly continuous).

Then, if d it is minor to d', there are two apexes of the same triangle where the function jumps more than h Ridiculously!

* * *

These three big theorems are equivalent between themselves. It is not difficult to demonstrate any more or less general version once one has the idea.

On them, let's say that the first one identifies it to Jordan. Gauss used it several decades earlier, without demonstrating, considering it to be 'clear'.

Brouwer, intuicionista, was pushing the 'traditional' mathematics back, but his theorem of fixed point demonstrated to be of big utility in the classic mathematics. An application that it did was to generalize that of Jordan to Rn, changing curves into hypersurfaces. It seems that Poincaré knew by intuition it: his idea was that if one was throwing sugar in a cup of coffee and it was stirred, some granite in the surface was not changing place (his theorem ergódico is one of so many people ramifications of this so simple idea).

The Hex was invented by Nash (a few years earlier Piet Hein had invented it). The relation of this game with the theorem of Jordan is clear, it is not difficult to prove the equivalence. With Brouwer it is more difficult, but it goes out: the existence of a balance of Nash is a direct consequence of this theorem (and it cost him a Nobel Prize). For another implication, it is necessary only to check what we did above, an idea of David Gale.

* * *

The idea that connects them is that the water has no form, but it reveals to us the local form that it occupies. Serve as moral, or better that it serves as base for the demonstration of some another theorem.

* * *

(special for the Mathematical, organized Carnival this time for Tito Eliatron)

Watch 90210 S02E14 Girl Fight free