Fan Fractals

Unnested Sierpinski Polygons IV

by Simon Whitechapel

In fan fractals, the direction of movement is chosen at random from a “fan” of possible angles. Below, the vertex targeted, v_n+1, rises and falls from 1 to 5 (i.e., n = 1, 2, 3, 4, 5, 4, 3, 2...), but the direction moved from (x, y)_n is either to the left of, to the right of, or directly towards v_n+1, while the distance moved is half the distance from (x, y)_n to v_n+1.

v = 5; fan = 3; θ = 0·785398163; distance = 1/2

The variable θ is the arc of the fan, i.e. the angle in radians between the far right and far left arms of the fan.

v = 4; fan = 2; θ = 0·935398164; distance = 1/2

Enlargement of previous image

v = 5; fan = 3; θ = 1·08539816; distance = 2/3

v = 5; fan = 2; θ = 1·23539816; distance = (4..7) / 7

v = 4; fan = 2; θ = 1·23539816; distance = 3/4

v = 4 + center; fan = 2; θ = 1·23539816; distance = 3/4

Here the vertex targeted rises from 1 to 4, then drops straight back to 1, rises again to 4, drops straight back again to 1, and so on:

v = 4; fan = 2; θ = 1·23539816; distance = 2/3

Enlargement of previous image

Here the vertex targeted is random:

v = rnd(3); fan = 3; θ = 0·785398163; distance = 6/7

v = rnd(3); fan = 2; θ = 0·785398163; distance = 1

And here a vertex chosen at random in cycle_n cannot be chosen again until cycle_n+t:

v = rnd(3); fan = 3; θ = 0·785398163; distance = 6/7; t = 2

v = rnd(3 + center); fan = 2; θ = (0·635398163..0·885398164); distance = 1; t = 2..3

© 2008 Simon Whitechapel

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