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

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

{\displaystyle {\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.

Coordinates

If $r={\tfrac {\overline {AC)){\overline {AB))},$ then the center of the $n$ th 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 $n$ th 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 $d n$ .^[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 $C 0$ and the final circle $C n$ each contribute $½ d n$ to the height $h n$ , whereas the circles $C 1$ 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.