Steven Dutch, Natural and Applied Sciences,University
of WisconMath.sin - Green Bay

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Construct two circles, one inside the other (not necessarily concentric). Now draw a circle between the two circles and tangent to both. Draw another circle tangent to those three circles. Keep on doing it until you have a complete ring of small circles between the two larger ones.

Such a chain is called a Steiner Chain. This program demonstrates them and explains the neat trick by which they can be simply constructed.

In the overwhelming majority of cases (in fact, all but an infinitesimally tiny fraction of cases), the smaller circles will not close neatly to form a complete ring. But in the rare cases when they do, it is a remarkable fact that they close regardless of where the first circle in the chain is constructed. In other words, you can draw the first circle anywhere, as long as it is tangent to the two bounding circles, and you will get a complete chain of circles.

Creating a Steiner Chain using straightforward geometry would be brutal. But using *inversion geometry*
it's very easy. If you have a circle and a point not
on the circle, construct a line from any point on the circle through the given point.
If the distance from the circle to the point is r, construct a second point at distance 1/r on the opposite side
of the point. Those points will also describe a circle. The key to constructing a Steiner chain is to draw a chain
of identical circles, each tangent to its neighbors, around the inside of a circle, then construct the circle tangent to the
chain on the inside. Then pick an arbitrary point and invert the circles.

In the animation here, "Offset" refers to the offset between the inner and outer circles. An offset of zero means the circles are concentric. An offset of 99 means the two circles are almost tangent. Having the circles tangent or intersecting is impossible, since there would be an infinite number of small circles between them, because no matter how tiny the ramaining space is, you can always get another circle into it. "Roll Circles" animates the drawing to show that the chain does indeed close regardless of where it starts.

Note: for small numbers of circles (less than 10), there is still a slight glitch that results in small errors in the circle radii.

Number of circles:

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*Created 14 February 2012, Last Update
18 March 2012
*

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