Pappus chain

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A Pappus chain
A Pappus chain

In geometry, the Pappus chain was created by Pappus of Alexandria in the 3rd century AD.

[edit] Construction

A Pappus chain is created by three semicircles that form an arbelos. A chain of circles Pi are created, each one tangent to the outside semicircle, the larger inside semicircle, and the circle before it.

[edit] Characteristics

If r = AB/AC, then the center of the of the nth circle in the chain is:

\left(x_n,y_n\right)=\left(\frac {r(1+r)}{2[n^2(1-r)^2+r]}~,~\frac {nr(1-r)}{n^2(1-r)^2+r}\right)

If r = AB/AC, then the radius of the of the nth circle in the chain is:

r_n=\frac {(1-r)r}{2[n^2(1-r)^2+r]}

All of the circles' centers are located on a common ellipse, with the foci being the midpoint AB and the midpoint of AC.

[edit] References

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