Pappus chain

In geometry, the Pappus chain is a ring of circles between two tangent circles investigated by Pappus of Alexandria in the 3rd century AD.

A Pappus chain

Construction

The arbelos is defined by two circles, C_U and C_V, which are tangent at the point A and where C_U is enclosed by C_V. Let the radii of these two circles be denoted as r_U and r_V, respectively, and let their respective centers be the points U and V. The Pappus chain consists of the circles in the shaded grey region, which are externally tangent to C_U (the inner circle) and internally tangent to C_V (the outer circle). Let the radius, diameter and center point of the n^th circle of the Pappus chain be denoted as r_n, d_n and P_n, respectively.

Properties

Ellipse

All the centers of the circles in the Pappus chain are located on a common ellipse, for the following reason. The sum of the distances from the n^th circle of the Pappus chain to the two centers U and V of the arbelos circles equals a constant

{\overline {\mathbf {P} _{n}\mathbf {U} }}+{\overline {\mathbf {P} _{n}\mathbf {V} }}=\left(r_{U}+r_{n}\right)+\left(r_{V}-r_{n}\right)=r_{U}+r_{V}

Thus, the foci of this ellipse are U and V, the centers of the two circles that define the arbelos; these points correspond to the midpoints of the line segments AB and AC, respectively.

Coordinates

If r = AC/AB, then the center 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)

Radii of the circles

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

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

Circle inversion

Under a particular inversion centered on A, the four initial circles of the Pappus chain are transformed into a stack of four equally sized circles, sandwiched between two parallel lines. This accounts for the height formula h_n = n d_n and the fact that the original points of tangency lie on a common circle.

The height h_n of the center of the n^th circle above the base diameter ACB equals n times d_n.[1] This may be shown by inverting in a circle centered on the tangent point A. The circle of inversion is chosen to intersect the n^th circle perpendicularly, so that the n^th circle is transformed into itself. The two arbelos circles, C_U and C_V, are transformed into parallel lines tangent to and sandwiching the n^th circle; hence, the other circles of the Pappus chain are transformed into similarly sandwiched circles of the same diameter. The initial circle C₀ and the final circle C_n each contribute ½d_n to the height h_n, whereas the circles C₁–C_n−1 each contribute d_n. Adding these contributions together yields the equation h_n = n d_n.

The same inversion can be used to show that the points where the circles of the Pappus chain are tangent to one another lie on a common circle. As noted above, the inversion centered at point A transforms the arbelos circles C_U and C_V into two parallel lines, and the circles of the Pappus chain into a stack of equally sized circles sandwiched between the two parallel lines. Hence, the points of tangency between the transformed circles lie on a line midway between the two parallel lines. Undoing the inversion in the circle, this line of tangent points is transformed back into a circle.

Steiner chain

In these properties of having centers on an ellipse and tangencies on a circle, the Pappus chain is analogous to the Steiner chain, in which finitely many circles are tangent to two circles.

References

Ogilvy, pp. 54–55.

Bibliography

Ogilvy, C. S. (1990). Excursions in Geometry. Dover. pp. 54–55. ISBN 0-486-26530-7.
Bankoff, L. (1981). "How did Pappus do it?". In Klarner, D. A. (ed.). The Mathematical Gardner. Boston: Prindle, Weber, & Schmidt. pp. 112–118.
Johnson, R. A. (1960). Advanced Euclidean Geometry: An elementary treatise on the geometry of the triangle and the circle (reprint of 1929 edition by Houghton Miflin ed.). New York: Dover Publications. pp. 116–117. ISBN 978-0-486-46237-0.
Wells, D. (1991). The Penguin Dictionary of Curious and Interesting Geometry. New York: Penguin Books. pp. 5–6. ISBN 0-14-011813-6.

External links

Floer van Lamoen and Eric W. Weisstein. "Pappus Chain". MathWorld.
Tan, Stephen. "Arbelos".

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[1] Ogilvy, pp. 54–55.