Pappus Chain

Pappus Chain

Pappus chain consists of all the black circles in the pink region.

Basic Description

A chain of inscribed circles is called a Pappus Chain when its first circle $C_1$ is tangent to the three semicircles forming the Arbelos, and all the subsequent circles $C_n$ are tangent to one another and to the boundaries of the arbelos. The Pappus Chain is named after Pappus of Alexandria, a great Greek mathematician who studied and wrote about it in the 4th century A.D.

The figure above shows all of the three variations of the chain: leftward, rightward, and downward. The default position is the one that extends to the right.

A More Mathematical Explanation

Note: understanding of this explanation requires: *Geometry

Properties

Height

Figure 5-Height
In the Papp [...]

Properties

Height

Figure 5-Height

In the Pappus Chain, the height of the center of the nth circle is n times the diameter of that circle.

The proof Pappus gave is long and complicated, containing Euclidean geometry, similar triangles, and the Pythagorean Theorem. We won’t list his proof here. Instead, here’s a simpler, more modern proof using the concept of inversion:

It’s fine if you don't know much about “circle inversion.” Basically, "inversion is a type of transformation that moves points from the inside of a circle to the outside and from the outside of a circle to the inside using a specific rule" [1]. The basic properties of inversion are as follows. Go to Inversion to get a more complete understanding.

• The inverse of a line not passing through the center of the circle is a circle;
• The inverse of a circle not passing through the center of the circle is a circle;
• The inverse of a circle passing through the center of the circle is a line.

We want to prove that the height of the center of the nth circle above $AB$ is n times the diameter of that circle, so we are inverting over the nth circle in the chain. To do this, first invert circle $AC$ and circle $BC$ with respect to a circle centered at $C$. Because both circle $AC$ and circle $BC$ pass through the center of the circle $C$, they become two vertical lines according to the third property listed above (see the figure on the left). Circle $BC$ inverts to the left line because when $BC$ has a smaller radius than $AC$, it inverts to a further line. To help you understand it better, treat the circle centered at $C$ as a mirror. If the circle is closer to the center point $C$, it will be reflected further away.

Second, invert the nth circle in the Pappus Chain and the two circles that are tangent to it. The inverse of the nth circle is itself. The inverse of the subsequent circles inscribed in the chain are circles tangent to the two parallel, vertical lines below or above the nth circle. Because the subsequent circles are tangent to circle $AC$ and $BC$, it makes sense that they are still tangent to the inverse of the two circles.

Now we are done with the inversions. Let's take a look at the right figure. Since those inversed circle are identical, it is not hard to conclude that the height of the center of the nth circle is n times the diameter of that circle. For example, $EF = 4r = 2d$ for the second circle.

Ellipse

Figure 6-Ellipse

The centers of all the circles $C_n$ in the Pappus Chain lie on an ellipse.

Let $M$ be the center of circle $AC, N$ be the center of the circle $BC, C_n$ represent the centers of all of the circles in the Pappus Chain, and $r_n$ represent the radii of those circles.

In order to prove this property, the first thing to do is to know the definition of an ellipse. In what conditions will a point lie on an ellipse in two-dimensions? First, the sum of the distances from a point on the ellipse to two fixed points is a constant. Second, the constant is greater than the distance between the two fixed points.

We need to show that all of the centers of the circles in the Pappus Chain satisfy the two conditions. The two fixed points are $M, N$.

Because

$C_n M = MD - C_n D =\frac{1}{2} - r_n$

and

$C_n N= C_n E + EN = r_n + \frac{1- r}{2}$,

we get

$C_n M + C_n N = (\frac{1}{2} - r_n) + (r_n + \frac{1- r}{2})= 1 - \frac{r}{2}= constant.$

For the reason that $MN = MC - NC = \frac{1}{2} - \frac{1- r}{2} = \frac{r}{2} < 1 - \frac{r}{2}$ (because $r < 1$ )

Thus, the centers of all the circles $C_n$ in the Pappus Chain lie on an ellipse.

Pappus Chain and Steiner Chain

The Pappus Chain is also a Steiner Chain (see Steiner's Porism), a chain formed by circles that are tangent to two circles, one inscribed within the other.

References

[1] van Lamoen, Floor and Weisstein, Eric W. Pappus Chain. From MathWorld--A Wolfram Web Resource. Retrieved from http://mathworld.wolfram.com/PappusChain.html.

1. More mathematical explanation about Pappus Chain
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