Rediscovering the Archimedean polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler
``Rediscovering the Archimedean Polyhedra'' suggests a process similar to the well-known recovery of the classical texts of Euclid, Apollonius, Archimedes and others. However, it is made clear by Dr. Field in the first paragraph that ``rediscovery'' here does not refer to a ``lost'' classical text but to the mathematics involved in the construction of these solids. The evidence that Archimedes had known these 13 solids, whose faces are regular polygons of more than one kind, lies in Pappus who listed them with some details of these faces. Elements, Books 14 and 15 (pseudo-Euclid) treat regular polyhedra but were not used as a source, with the possible exception of Piero Della Francesca. However, even this debt is somewhat doubtful. What is traced is a characteristically Renaissance contribution to mathematics, connected, sometimes tenuously, to the Classical past. As in other Renaissance developments, the Classical past turns out to be only a ``jumping-off'' ground and the mathematics which emerges, not surprisingly is marked by the characteristics of its own time and place. Dr. Field traces the work of `rediscovery' from Piero to Daniele Barbaro and refers briefly to Kepler's later rigorous mathematical treatment in Harmonices mundi libri V., Book 2, in which Kepler constructed the first known complete set of Archimedean solids and proved the completeness. The main content of the article is a meticulous examination of the first five authors named in the title; the methods, achievements and `shortcomings' of each; possible methods of discovery are speculated upon; methods of construction by truncation or by nets are clearly and concisely explained. A helpful table of solids is provided showing which authors are associated with each. The many illustrations are supplemented with the author's own explicatory diagrams. An Appendix on the differing conventions used in drawing the solids puts to rest what has been, for many, an over-simplified picture of perspective drawing in the Renaissance. The story, explicated in detail by Dr. Field, begins with Piero Della Francesca's construction of six of the solids in the Trattato d'abaco and Libellus de quinque corporibus regularibus, performed by truncations of regular polyhedra. However, the means by which he arrived at this are not known at present. Pacioli's treatment, in De divina proportione (1509) containing by illustrations by his friend Leonardo da Vinci, was also by truncation. Dr. Field's analysis of Leonardo's contribution is enhanced by a sense of history so that it is carefully noted that what a modern mathematician might call `distortion' in such illustrations would come under the Renaissance category of `perfecting'. Dürer's appearance on the list might occasion some surprise; his work appeared in Underweysung der Messung mit dem Zirkel und Richtscheyt (1525 and 1538) in which seven of the solids were described. Although the mathematics leaves something to be desired, we are indebted to him for the (probable) first use of nets in relation to the construction of solids. Finally, Daniele Barbaro, possibly basing himself on Piero, wrote about the solids in a book on perspective (La Pratica della perspettiva (1568)), but dealt with 11 in contrast to Piero's 6. Thus, he made an original contribution, albeit of a visual and not very mathematical character. It was Kepler, probably drawing his inspiration from Timaeus and without having read these authors who, in modern times, first produced a complete and mathematically rigorous account of the Archimedean Polyhedra. In Kepler is found the combination of mathematical ability with the visual imagination characteristic of the others, which made possible their own earlier contributions. An added bonus to the rigour, economy and clarity of the mathematical explications is the author's historical sensibility, in particular an awareness of the very real difference between rational reconstruction and the probable tangle of events. In analysing the relationships between the work of different authors, all speculations are properly qualified and each is justified by a convincing material basis.
scientific article; zbMATH DE number 1047103