Sierpinski Attractor - notes

OBS. 1 - " the first triangle, the colour of each point is the colour given by the (last) dice roll..."

Actually, the statement should be seen for "all" points but three, in which the a priori colour could be each one of two: they are the vertices common to blue and green triangles, blue and red triangles, and green and red triangles. In these, it is not clear which is the colour in the picture. That ambiguity corresponds to some apparent impossibility of determining its history, event recent, only from the position of the point. For example, the first of these three points - common vertex to blue and green triangles - can both be obtained with a initial blue roll and a second green (case in which the last roll was green) as well as with a initial green roll and a second blue (case in which the last roll was blue).

OBS. 2 - "...for every point of th small green triangle marked with the arrow, the last roll was red, the penultimate was blue and the antepenult green. Therefore, all the points of that marked small green triangle have the same late history."

In fact, also here, "all" should exclude a priori, for the same reasons, the vertices of the marked small triangle. Relatively to these three vertices, the situation is the following:

  1. there is no ambiguity for either, with respect to the colour of the last rolls: they were red;
  2. there is no ambiguity with respect to the colour of the penultimate roll, for the left and top vertices: they were blue, but the one for the right vertex seems that may have been blue or green;
  3. apparently there is ambiguity in the colour of the antepenult rolls for the three vertices: the left one seems may have been green or blue, as well as the right one, while the top one may have been green or red.

If the arrow were to signal another small triangle, the circumstances could be different, for instance, it could apparently be even ambiguous for the last roll.

OBS. 3 - "In order to know the older history of rolls, from the final position of those points (of the marked triangle), we may observe how the points of that green triangle are coloured in the two triangles that follow it (second row of the table)."

Actually, once more, watching these two big triangles of the table allow us to broaden the knowledge of rolls history up to two more generations, for all points (less a priori the three vertices) of the small marked (with the arrow) green triangle, with some (12=3+9) apparent new exceptions - the vertices of the new smaller triangles.

OBS. 4 - The constraints presented in the previous remarks about the apparent exceptional points (vertices), for which it was not possible to know a priori the history of the rolls that originated them, did not use an additional knowledge that is sufficient to undo any ambiguity. This is the somewhat subtle reasoning that we expose next.

In the game, one starts from one of the vertices of the initial triangle - blue, green or red -, drawn by lot: suppose that it is the one on the left (blue); in case it is another one, the reasoning is analogous. If the first is blue, no point in the opposite side (having the green and red points at the ends) can be achieved (it suffices to notice that the middle point of a line segment that links a point of the triangle that is not in that side, to one of the ends of that side, is not in that side). Then, it is not hard to see that also no point of any other parallel side of the small triangles (next in the table), can be achieved. Therefore, in each small triangle of the ones that are successively formed by the labelling of points, there is at most a marked vertex and this vertex holds the same position, relatively to the adjacent triangle with the same colour, as the initial point (blue, in the case we are considering) hold relatively to the initial triangle. (In the case we are considering - the initial roll was blue - a marked vertex has the colour of the one of the two adjacent triangles that is on the right of the vertex and there is always such a small triangle, since no marked point is located in a parallel side to the right side of the big triangle.)
In summary, there is no ambiguity provided that one knows the first marked point. But, by the above considerations, this point is precisely the only vertex of the big triangle that was flagged.