Internet explorer users should click empty spaces to start the applet

Take 2 circles one inside the other. (The red ones in the applet) Try to fill it with a chain of sequentially tangential circles.(the black ones in the applet)

Steiner found that if you could not fill a particular circle pair starting at one spot you never could from any starting spot.(button 5+)

However if you could fill a particular circle pair it works from any starting spot.(buttons 4, 6 and 10)

It is obvious why this is so if the starting circles are concentric but not so if they are not concentric. The reason is that every "Steiner circle set" can be generated from a geometric transformation called an inversion of a starting pair of concentric circles with the filler circles having a constant diameter. Under geometric inversions circles remain circles and touch points remain touch points. You can see the concentric circles and the filler circles in light gray if you click [Show Inversion], the circle of inversion is green . It is clearest with 10 circles showing.

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