### Sierpinski Attractor

If you don't know the game, see its description in the pages
(for the moment, this webpage is only in Portuguese) about the exhibit in the Exhibition
Matemática Viva, for the case of 3 colours, or use the *applet*
made with *JavaSketchpad*.

Each one of the pictures above has a pointer to an applet that simulates the roll of a dice with three colours (first 5 triangles), four colours (2 squares) and five colours (2 pentagons). Each time you click an applet, some thousands of rolls are simulated and the corresponding points are marked. Consecutive "cliks" do not produce exactly the same picture, as you may find after a careful look, but the consecutive pictures are all very similar, in each case.

Colours represent the following: in the first triangle,
each point is coloured with the colour determined by the (last) dice roll that originated it, in the second triangle it is coloured by the colour determined by the penultimate dice roll, etc. For example, observing the three triangles of the first table line, we conclude that in the third of those triangles, i.e., the rightmost one, for every
points of the green small triangle marked by the arrow, the last roll
was red, the last but one was blue and the antepenult was green. Therefore,
all points of that small green triangle marked with an arrow have the same recent "history". In order to know, from the final position of those
points, the older history of rolls, we may observe how
the points of that green triangle are coloured in the two triangles that follow it (second line of the table).

Since different histories are equally probable, and each one is associated to the position of the point by the shape suggested by the pictures, we may understand the reason why the points are
*equally* distributed in the
Sierpinski triangle, that is, why the probability that a point goes to some region only depends on the
"size" of that region.

In the case of 4 points and 5 points with similar rules, one realizes some differences to the simpler case of 3 points: there are no "holes" in the squares and, in the five points case, the holes are back, but there are darker regions, of overlap. Try to find the reason for these differences and then conjecture what will happen in the case of 6 or 7 points. What do you think it will happen, in all these cases, to the probability of the points ending up in the different regions of the picture?