IFS Animations

For the equilateral gasket rules, let t range from 0 to 360 in steps of 10 deg. Note the lower left corner (both blues) makes one complete rotation, the lower left corner of the lower left corner (dark blue) makes two complete rotations. The lesson here is that each subpiece is viewed relative to the larger part. The dark blue is the lower left corner of both blues, so makes one rotation relative to both blues.

Here we spin the lower left corner CCW and the lower right corner CW in 10 deg steps. To maintain the symmetry of the left and right corners, we translate the left corner by e = A = 1 - 0.5*cos(-t) and f = B = -0.5*sin(-t). What do you notice about the motion of the three subpieces of the lower corners?

Here we let t range between 0 and .375 in steps of 0.125. Note the motion in each piece is a scaled version of the motion of the whole. Is the idea of fractal motion beginning to be clear?

Here we investigate the effect of different values for theta and phi. In this example, theta = t ranges between 90 and 0 in steps of 10 deg. Can you give a complete description of the theta = 0 picture?

Here phi = t ranges between 90 and 0 in steps of 10 deg. Can you give a complete description of the phi = 0 picture?

To illustrate how a rotation can create a spiral, here we step the rotation of the middle piece from 20 deg to -20 deg in steps of 5 deg. Again, note the motion within motion of pieces within pieces.

Here the scaling factor of the upper right piece ranges between a = 0.5 and a = 0.25, the translation between b = 0.75 and b = 0.5. Throughout the animation, the unchanging part is the right isosceles Sierpinski gasket. Do you see why?

Here we move the upper right corner from a translation of b = 0.75 to b = 0.25 in steps of 1/16 = 0.625. Note how the reflection of the lower right and upper left pieces affects the motion within those pieces.

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