This article explores the basic properties of inversive geometry from a computational point of view. Topics included in this part are involutions, generalized circles, and the inversion of segments, arcs, triangles, and quadrilaterals. The applications are to Nicomachus's theorem, the inversion of tilings made by regular polygons, and an inversive spirograph.

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This article systematically verifies a series of properties of an ancient figure called the arbelos. It includes some new discoveries and extensions contributed by the author. ‡ Introduction Motivated by the computational advantages offered by Mathematica, I decided some time ago to embark on collecting and implementing properties of the fascinating geometric figure called the arbelos. I have since been impressed by the large number of surprising discoveries and computational challenges that have sprung out of the growing literature concerning this remarkable object. I recall its resemblance to the lower part of the iconic canopied penny-farthing bicycle of the 1960s TV series The Prisoner, Punch's jester cap (of Punch and Judy fame), and a yin-yang symbol with one arc inverted; see Figure 1. There is now an online specialized catalog of Archimedean circles (circles contained in the arbelos) [1] and important applications outside the realm of mathematics and computer science [2] of arbelos-related properties.Many famous names are involved in this fascinating theme, among them Archimedes (killed by a Roman soldier in 212