Pappus chain

File:Pappus Chain Full.svg

A 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, Template:Mvar and Template:Mvar, which are tangent at the point Template:Mvar and where Template:Mvar is enclosed by Template:Mvar. Let the radii of these two circles be denoted as Template:Mvar, respectively, and let their respective centers be the points Template:Mvar. The Pappus chain consists of the circles in the shaded grey region, which are externally tangent to Template:Mvar (the inner circle) and internally tangent to Template:Mvar (the outer circle). Let the radius, diameter and center point of the Template:Mvar^th circle of the Pappus chain be denoted as Template:Mvar, 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 Template:Mvar^th circle of the Pappus chain to the two centers Template:Mvar 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 Template:Mvar, the centers of the two circles that define the arbelos; these points correspond to the midpoints of the line segments Template:Mvar, respectively.

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

$$(x_n,y_n)=(\frac{r(1+r)}{2[n^2(1-r{)}^2+r]},\frac{nr(1-r)}{n^2(1-r{)}^2+r})$$

Radii of the circles

If $r=\frac{\overline{AC}}{\overline{AB}},$ then the radius of the Template:Mvarth circle in the chain is: $$r_n=\frac{(1-r)r}{2[n^2(1-r{)}^2+r]}$$

Circle inversion

File:Pappus Chain Theorem.svg

Under a particular inversion centered on Template:Mvar, 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 Template:Mvar of the center of the Template:Mvar^th circle above the base diameter Template:Mvar equals Template:Mvar times Template:Mvar. This may be shown by inverting in a circle centered on the tangent point Template:Mvar. The circle of inversion is chosen to intersect the Template:Mvar^th circle perpendicularly, so that the Template:Mvar^th circle is transformed into itself. The two arbelos circles, Template:Mvar and Template:Mvar, are transformed into parallel lines tangent to and sandwiching the Template:Mvar^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 Template:Mvar each contribute Template:Sfracd_n to the height Template:Mvar, whereas the circles C₁ to C_n−1 each contribute Template:Mvar. 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 Template:Mvar transforms the arbelos circles Template:Mvar 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.