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, r_V, respectively, and let their respective centers be the points U, 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, P_n, respectively.

Properties

Centers of the circles

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, V of the arbelos circles equals a constant

$${\overline {P_{n}U}}+{\overline {P_{n}V}}=(r_{U}+r_{n})+(r_{V}-r_{n})=r_{U}+r_{V}$$

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

If $r={\tfrac {\overline {AC}}{\overline {AB}}},$ then the center of the nth circle in the chain is:

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

Radii of the circles

If $r={\tfrac {\overline {AC}}{\overline {AB}}},$ 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 = nd_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. 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 ⁠1/2⁠d_n to the height h_n, whereas the circles C₁ to C_n−1 each contribute d_n. Adding these contributions together yields the equation h_n = nd_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, 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.

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 Mifflin 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".