Ford Sr (1886-1967), is related to ideas about mutually

tangent circles that were studied by, among others, Apollonius of Perga in the third century BC and by Rene Descartes in the 17th century (Wikipedia, 2015).

According to Ibn Sinan's autobiography, his work on the second level of analysis, which treated the conditions under which solutions could exist, was found in his lost treatise

Tangent Circles.

Researchers discovered a remarkable formula relating the curvatures and coordinates of

tangent circles packed within a circle (159: 254 (*)).

Where they begin with four equicircles, we begin with four tangent circles, attached to the corners of a rectangle based on the right triangle.

These six points of tangency support a second set of four tangent circles, a general fact which Coxeter (1989) ascribes to a schoolteacher named Beecroft.

Some readers may be curious to know how the four tangent circles come to satisfy a celebrated equation found by Descartes and rediscovered by Beecroft and Soddy.

With this convention, the Cartesian condition for four tangent circles with (signed) curvatures (x, y, z, w) is

So the same four circles, moved around, can be a set of tangent circles, a second system of tangent circles (Figure 2), or the set of equicircles for [increment of ABC].

Given four mutually tangent circles with curvatures a, b, c, and d, the Descartes circle equation specifies that ([a.

The new formula looks like the original Descartes equation for four mutually tangent circles, provided the coordinates of the centers are expressed as so-called complex numbers.

The existence of this relationship among the centers of tangent circles allows anyone to derive a relation that makes it very easy to plot exquisitely detailed tangent-circle patterns on the computer.

Lagarias recognized the pattern as an example of an Apollonian packing--a type of arrangement named after a Greek mathematician who studied tangent circles more than 2,000 years ago.