This was obtained by examining a problem that interested me: What is the densest distribution in a positive grid such that every point on the grid can see a "screen" from (0,1) to (1,0), ie, the largest set of points such that the triangle between the point, (0,1), and (1,0) contains no other point in the set, given a specific maximum number.

The fractal appears to be about 1.61 dimensional, although it may become closer to 1d as the size increases. This image was achieved with a 2000x2000 grid. On the right side, where each peak can be assigned a rational number between 0 and 1, the horizontal position of the peak at a/b is (1-1/(a+b+2), a/b\*(1-1/(a+b+2)). I am unsure of what further analysis would be interesting.

I have not analyzed the fractals for the circular and right triangle ones, although I expect they are similar. The set of points for the square and the right triangle have the same magnitude, while the circle has slightly higher, so the circle has the highest density in the square. However, within their own area, the right triangle is the most efficient.