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.

Construction
The arbelos is defined by two circles, $CU$ and $CV$, which are tangent at the point $A$ and where $CU$ is enclosed by $CV$. Let the radii of these two circles be denoted as $rU, rV$, 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 $CU$ (the inner circle) and internally tangent to $CV$ (the outer circle). Let the radius, diameter and center point of the $n$th circle of the Pappus chain be denoted as $r_{n}, dn, Pn$, respectively.

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

$$ \overline{P_nU} + \overline{P_nV} = (r_U + r_n) + (r_V - r_n) = r_U + r_V $$

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 $\overline{AB}, \overline{AC}$, 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 $A$ of the center of the $hn$th circle above the base diameter $n$ equals $ACB$ times $n$. This may be shown by inverting in a circle centered on the tangent point $dn$. The circle of inversion is chosen to intersect the $A$th circle perpendicularly, so that the $n$th circle is transformed into itself. The two arbelos circles, $n$ and $CU$, are transformed into parallel lines tangent to and sandwiching the $CV$th circle; hence, the other circles of the Pappus chain are transformed into similarly sandwiched circles of the same diameter. The initial circle $hn = ndn$ and the final circle $n$ each contribute $C_{0}$ to the height $Cn$, whereas the circles $½dn$ to $C_{1}$ each contribute $hn$. Adding these contributions together yields the equation $C_{n&minus;1}$.

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 $dn$ transforms the arbelos circles $A$ 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.