Unnested Sierpinski Polygons I

by Simon Whitechapel

The construction here is to select a point, (x,y), within a polygon, then find the center of the polygon it forms with n randomly selected points of the original polygon, where diff = 1 or 0 (when diff = 1, rand(x,y)₁ <> rand(x,y)₂ ... <> rand(x,y)_n). This center becomes the (x,y) used for the next iteration of the process. When the polygon is a triangle and n=1, the familiar Sierpiński gasket appears:

vertices = 3; n = 1; diff = 0

When a point chosen in cycle_n cannot be used in cycle_n+1, the fractal is quite different:

v = 3; n = 1; cycle = 1

The original triangle can be given a central point:

v = 3 + center; n = 1; diff = 0

v = 3 + center; n = 1; cycle = 1

Here are further images created by the process (some are incomplete to prevent saturation at low resolution):

v = 3 + center; n = 2; diff = 1

Enlargements of previous image

v = 4; n = 2; diff = 1

Enlargement of previous image

v = 5; n = 2; diff = 1

v = 5; n = 3; diff = 1

Enlargement of previous image

v = 6; n = 3; diff = 1

Enlargement of previous image

v = 6; n = 4; diff = 1

Enlargement of previous image
(Note stronger central patches)

v = 3 + 3; n = 3; diff = 1

Enlargement of previous image

v = 3 + 3 + center; n = 5; diff = 1

Enlargement of previous image

© 2007 Simon Whitechapel

More Unnested Sierpinski Polygons

Nested Sierpinski Polygons