Pappus chain · Lablypedia

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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.

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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

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

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.

Coordinates

If $r = \frac{\bar{A C}}{\bar{A B}},$ then the center of the nth circle in the chain is:

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

Radii of the circles

If $r = \frac{\bar{A C}}{\bar{A B}},$ 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

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.