Circle Inversion Fractals

Exercises

Open the Circle Inversion Limit Set software.

Under the Circles menu, change the centers and radii to

x = 1.0	y = 1.0	radius = 1.0
x = -1.0	y = 1.0	radius = 1.0
x = -1.0	y = -1.0	radius = 1.0
x = 1.0	y = -1.0	radius = 1.0
x = 0.0	y = 0.0	radius = 0.41

then click OK.

1. Describe the main features of the limit set.

2. Return to the Circles menu and add a sixth circle with center at x = -2.0, y = 0.0 and radius = 0.41. Click OK before exiting the Circles menu. How does the limit set change?

3. Return to the Circles menu and add a seventh circle with center at x = 2.0, y = 0.0 and radius = 0.41. Click OK before exiting the Circles menu. How does the limit set change?

4. Return to the Circles menu and add eighth and nonth circles with centers at x = 0.0, y = 2.0 and x = 0.0, y = -2.0, and radius = 0.41. Click OK before exiting the Circles menu. How does the limit set change?

5. Return to the original list of five circles. Move the center of the fifth circle from (0,0) to (2,0) in x-increments of 0.25. After each change, click OK before exiting the Circles menu. For what values of x is the limit set a clean shape, fractal or Euclidean?

6. Return to the original list of five circles and change the radius of the fifth circle to 0.5. Change the center from (0,0) to (2.25,0) in x-increments of 0.25. Click OK before exiting the Circles menu. For what values of x is the limit set a clean shape, fractal or Euclidean?

7. Change the center of the fifth circle to (1.0,0.5), and then to each of these, moving this circle around the inside of the circle with center (1,1) and radius 1.

x = 0.75	y = 0.57
x = 0.57	y = 0.75
x = 0.5	y = 1.0
x = 0.57	y = 1.25
x = 0.75	y = 1.43
x = 1.0	y = 1.5

For what centers is the limit set a clean shape, fractal or euclidean? Do you see a pattern?

8. Move the center of the fifth circle to x = 1.35 and y = 1.35. Add three more circles, all of radius 0.5, and with centers

x = -1.35	y = 1.35
x = -1.35	y = -1.35
x = 1.35	y = -1.35

Zoom out (select 0.25 in the zoom menu). Describe the limit set from this perspective.

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