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# Directions:

• To move the circles, Drag the square in the middle of the circle.
• To view an approximation of the fractal curve that is the infinite intersection of nested circles click "View Curve" in the upper left hand corner of the applet. This button will change to a "View Circles" button which will return you to viewing the nested circles.
• "Zoom Out" will zoom out from the midpoint by a factor of two
• To zoom in, click the "Zoom In" button which will turn blue, then click anywhere on the screen and the picture will zoom in at this point a factor of two.
• "Move" does the same thing as "Zoom in" but merely changes the location of the image.
• "Add Circle" and "Remove Circle" add or subtract circles from the picture. This resets the image as well. (you cannot have less then 3 circles)
• When in Curve mode, a "More Detail" button will appear. This button will draw the curve in more detail, but take longer. "Less Detail" does the opposite.
• The "Reset" button return the circles to standard location. If you do something stupid like drag the circles inside each other and chaos appears, click this button.

# What's going on:

•  Each circle represents a map from C->C given by translating the center to the origin dilating to the unit circle then applying the map z |-> 1/z then reflecting about the real line and dilating and translating back. Anyway this map is a Mobius transformation (Takes circles to circles)
• By applying each one of these maps to each of the other circles we obtain a ring of tangent circles with n*(n-1) circles, where n is the number we started with. Each iteration has n-1 times more circles with the radii of the circles going to zero.
• The limit of this operation is a fractal loop, Provided the circles do not intersect.
• If some of the circles are tangent in more than two locations, the loop is self intersecting. (making it more interesting to look at)
• If the circles intersect, the operation does not produce smaller and smaller circles, instead the operation will erupt into a chaotic mess. But the chaos is kind of fun to watch.