Vediamola allora questa gif.
E leggete, se ne avete voglia, la spiegazione in inglese:
The name of this fractal tree derives from the fact that it is a binary tree in which the size of each parent node (a) is connected to that of two children nodes, left (c) and right (b), by the Pythagorean theorem:
Each square has a side in common with a right triangle, which in turn has the other two sides in common with other two squares, and so on.
In the gif, the triangles have the acute angles equal to 30° and 60°, and the angles are arranged in the same direction.
The tree can be achieved by the composition of rotations and homotheties, with fixed centers.
Seen that each square, in each step, generates two square, the number of squares added to step n is 2^n.
At each step, the perimeter increases, in particular, by a factor:
The triangle with the acute angles equal to 30° and 60° is in fact half of an equilateral triangle that has side equal to the starting square.
Pythagoras tree therefore has infinite perimeter.
For the Pythagorean theorem, the sum of the areas of the squares added in each step is equal to the starting square area, which for convenience we can set equal to 1.
The area thus seems to become infinite, when the number of steps tends to infinity; instead already from the fifth passage the figure folds in on itself, remaining enclosed in a limited area.
Pythagoras tree therefore has infinite perimeter and limited area.
The structure that we observe in normal scale is repeated many times within the smaller scale, and we can find it whatever the power of the magnifying glass we use (auto-similarity).
The endless vertices of Pythagoras tree, when the length of the sides tends to zero, they are all angular points, so you can not find any "regular" area that admits tangent.
For English speakers, read more:
Per i lettori italiani, approfondire a questi link:
Al seguente link, una bella animazione con le prime fasi dello sviluppo dell'albero di Pitagora:
A quest'altro link potete creare, muovere e variare alberi di Pitagora grazie ad un programma Java, realizzato dal mio amico Guzman Tierno:
Concludo con il bel filmato "A Year in the Life of a Pythagoras Tree":