Pappus Chain

The Pappus chain extends across at least two millennia of mathematics. Its origins trace back to the ancient Greek mathematician Archimedes and his studies of circles inscribed within the figure of an arbelos (or shoemaker's knife). The inversive geometry trick for efficiently computing the positions of pairwise tangent inscribed circles, or a Pappus chain, is apparently a modern invention. The Apollonian gasket or curvilinear Sierpinski sieve is constructed by the same iterative process of inscribing a circle in triplets of tangent circles. Thus the Pappus chain construction anticipates the class-2 nested behavior of many elementary cellular automata.


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Peter Ruane has provided an excellent short history of problems involving sequences of touching circles.

P. Ruane, "The Curious Rectangles of Rollett and Rees," The Mathematical Gazette, 85(503), 2001 pp. 208–225. DOI:10.2307/3622006

Ruane's exposition was the Mathematical Gazette's article of the year in 2001. A sample copy may be found on the website of the Highland District of the Scottish Bridge Union, whose chairman, Bill Richardson, is the production editor of the Gazette.

In the late 1920s the poet Robert Graves spoke of endless enclosure within enclosure, a parcel, blocks of slate, dappled red and green, yellow tawny nets, acres of dominoes, a fruit's kernel, an island tree, again a parcel, and on in endless enclosure within enclosure. Today we might call this class-2 nested behavior with hints of gnarly class-4 behavior.

R. Graves, "Warning to Children," The Complete Poems, Penguin Books, 2003 p. 297.

Grave's poem was featured on the Saturday January 5, 2008 Writer's Almanac with Garrison Keillor. A transcript and an audio recording are available.
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