IFS which construct this fractal and have affine transformation only

Question

[Image updated]

Is there an IFS which construct this fractal and have affine transformation only? (I think there must be a restriction, which is not an affine transformation. Can it be proved?)

Answer 1

I do not have a definition in IFS notation but the algorithm I write below should do the work if the definition of affine transformation is as the basic definition at the Wikipedia:

"...a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation. An affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line."

Here it is:

1. Initial triangle is $T1$. Make a copy of the initial triangle, name it $T2$, rotate $T2$ $180$ degrees using as pivoting point the center of $T1$.

2. Resize $T2$ to fit into $T1$ and change the color to white (when the three vertices of $T2$ are exactly in a position that is part of the sides of $T1$).

3. Make a copy of $T2$, name it $T3$, and rotate $T3$ $180$ degrees using as rotation point the middle point of the segment of the upper side of $T3$ (once rotated you will not see it, because its sides are in the same position than the sides of the triangles $T1$ and $T2$, but it is there) and change the color to black.

4. Apply again the same algorithm from point ($1$) for $T3$, in other words, being now the $T1$ of the algorithm this last generated $T3$ (the new $T1$ will be the current $T3$).