Sierpinski Attractor - rules
There are three non-collinear points (let us call them the INITIAL points), a coloured disk (blue, green, red) in each initial point, and a dice with coloured faces (blue, green, red,
opposite faces with the same colour).
One constructs a (random) sequence of points in the following way:
- The first point is chosen among the given initial points^{*}, occasionally given by a roll of the dice;
- given a point P in the sequence, the next one is constructed by tossing the dice and then by taking the middle point of the line segment linking P with the initial point that has the colour given by the dice.
In the next applet, one may see which point would follow point P according to the colour given by the dice, by clicking in one of the colours (blue, green, or red).
Some remarks:
- If you want to know in which region the new point will arise when the dice roll gives the blue colour, you may click at Blue and at Show blue triangle and then move the point P inside the bigger triangle. You will see that the new point never goes out of the blue area.
- You may also observe that if P is in the interior of the big triangle, then the next point (and, therefore, all the subsequent points) are also in the interior (and not on one side).
- In order that the new point arises on one of the triangle sides, it is necessary that point P is on that side and that the dice roll gives one of the colours of the side vertices.
- Therefore, one gets a point located in one of the sides of the big triangle only if all its preceding points were also located there, that is, all dice rolls so far produced only the two colours of the vertices of that side.
^{*}Even if we would delete the rule "the first point is chosen among the three initial points", allowing the start with any point in the same plane, we would get a similar picture as before provided that we would suppress the marking of the first obtained points. Is precisely this property - of the existence of a set that attracts all the random sequences produced by the described recursive process -that motivates the name
"Sierpinski attractor".
More specifically, the
Sierpinski attractor is the name that mathematicians give to the
adherence of any of the sequences described above. It coincides (this is a theorem!) with the so called Sierpinski triangle, defined by the initial triangle, in which one deletes the interior of the "middle" triangle as well as the interiors of all smaller triangles that will remain...