 Animations for Chaos Under Control

Gasket construction (22K) shows the iterative construction of the right isosceles Sierpinski gasket, starting from a right isosceles triangle. The iterated process is this: for each filled-in triangle, locate the midpoint of each side, connect these midpoints, and remove the "middle triangle" formed.

Square gasket construction (22K) builds the right isosceles Sierpinski gasket by applying the rules T1(x,y) = (x/2,y/2), T2(x,y) = (x/2,y/2) + (1/2,0), and T3(x,y) = (x/2,y/2) + (0,1/2) iteratively, here starting with the unit square.

Steve gasket (22K) shows the first few stages of turing my brother into a gasket.

Gasket Zoom (44K) demonstrates the self-similarity of the gasket by continually magnfying about the lower left corner.

Spin Left (132K) shows how the equilateral gasket changes when the lower left corner is rotated about the origin in 10 degree increments. Can you identify the fixed poits of this motion?

Spin Both Ways (110K) shows how the equilateral gasket changes when the lower left corner is rotated about the origin, and the lower right corner is rotated about (1,0) in 10 degree increments. Can you identify the fixed poits of this motion?

Koch (22K) illustrates the standard constructon of the Koch curve.

Koch thickening (22K) shows how the apparent "thickness" of the Koch curve changes as a function of the size of the pieces. How does the similarity dimension change as a function of the size? As a function of the angle?

The next two animations illustrate the possibilities for non-rigid rotations in IFS rules. The base IFS is
R S Theta Phi E F
.5 .5 0 0 0 0
.5 .5 0 0 0 .5
.5 .5 90 90 1 0

Phi squash (44K) shows the effect of changing phi of the third rule from 90 to 0 in steps of 10. Does the last picture in the sequence make sense?

Theta squash (22K) shows the effect of changing theta of the third rule from 90 to 0 in steps of 10. Does the last picture in the sequence make sense?

Four Corner Chaos Game Slide (44K) starts with the attractor for this IFS
R S Theta Phi E F
.5 .5 0 0 0 0
.5 .5 0 0 0 .5
.5 .5 0 0 .5 0
.5 .5 0 0 x .5

with x = 0.5 (the unit square), and steps x back to 0 (a right isosceles Sierpinski gasket).

Spiral Morph (22K) and Square Morph (66K) animate changes in spiral and square patterns as one of the IFS rules is changed. Can you find the IFS rules and corresponding changes?

Side Slide (22K) and Corner Glide (44K) animate changes in other IFS rules. Again, can you find the IFS rules and corresponding changes?

Newton Promendae (66K) illustrates how the basins of attraction of the roots of a cubic polynomial change as the positions of the roots vary across the complex plane.