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The Polyhedrists: Art and Geometry in the Long Sixteenth Century

Noam Andrews
The MIT Press
Publication Date: 
Number of Pages: 
[Reviewed by
Robin Hartshorne
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The sixteenth century in Europe was a time of great change and development. Just think of Copernicus and the heliocentric model of the universe. Or Martin Luther and his revolt against the Catholic Church for the practice of selling indulgences. Or painters using the theory of perspective to put a suggestion of three-dimensionality into their two-dimensional paintings. Or the role of printing in the dissemination of knowledge after Gutenberg's discovery of movable type.
By the end of the fifteenth century, most of the classical Greek texts on mathematics and astronomy had been located and assembled in the libraries of wealthy and educated people. Some time around 1471 the great astronomer and mathematician Regiomontanus, after studies in Italy and Hungary, moved to the imperial city of Nuremberg and established an observatory and a printing press in his house. His Programme of 1474 listing the books he planned to translate and print in Latin is described by Rose as the "formal beginning of the renaissance of mathematics." Unfortunately, Regiomontanus died on a trip to Rome before he could carry out his plan.
The book under review is a revised and updated version of Andrews's thesis at Harvard in the history of science. It opens a window on this turbulent sixteenth century through the perspective of polyhedra--their role as an intermediary between the academic world of mathematics and the practical world of artists and craftsmen. Copiously illustrated in black and white and color, this book is a visual feast. We meet many talented craftsmen (all men, by the way), and learn about the details of their work.
The best known polyhedra are the five regular solids, the tetrahedron, the cube, the octahedron, the icosahedron, and the dodecahedron, They are also called Platonic solids because they are described in Plato's dialogue Timaeus (ca 350 BC), where they are associated with the four classical elements fire, earth, water, air, and the ethereal universe. Then there are the thirteen semi-regular, or Archimedean, solids, so named because they are credited to Archimedes in the Mathematical Collection of Pappus (ca. 300 AD). This work of Pappus was not known in Europe until it was translated into Latin by Commandino and published in 1588. It seems to be a mystery (not explained in this book) how the awareness of these Archimedean solids came to Europe in the late 15th and early 16th centuries. One possibility is that many of them were rediscovered by the artist and mathematician Piero della Francesca and and written in his book, De Corporibus Regularibus, which had a wide circulation in manuscript, and later reappeared in an Italian translation (unattributed) in Luca Pacioli's De Divina Proportione (1509).
A pivotal figure in this story is the artist Albrecht Dürer (1471-1528), of Nuremberg. At an early age, he was sent as an apprentice to a nearby goldsmith, but excelled at print making, so that by the early 1600s he became the best known woodblock and engraving printmaker in Europe. As a commoner he did not have a classical education, but through an improbable friendship with the wealthy and well connected Willibald Pirckheimer, a patrician of Nuremberg, he presumably got whatever help he needed to study the Latin Euclid. Towards the end of his life, in 1525, he published a book, Underweisung der Messung, giving practical methods for artists to make exact drawings with ruler and compass for their art work. One of the innovations of this book was a method of creating paper models of polyhedra from plane nets that could be cut out and folded up. Thus Dürer's book became a crucial link between the scholarly world of academia and the practical world of artisans. Still Dürer's book was too difficult for many common people, so it was followed by a number of practical manuals or Lehrbücher with easier to understand pictures.
Soon polyhedra began showing up in many contexts. There is a painting by Ugo da Carpi with a polyhedron (p. 30), which Andrews describes as follows: "Archimedes hesitates, transfixed by the rhombicuboctahedron hovering on the edge of the page in the foreground, like a strange species never before encountered in nature." Another paining (p.45, and front and back cover) shows the scholar Fra Luca Pacioli with a glass rhombicuboctahedron hanging in the background. For me an iconic example of use of polyhedra in art is Dürer's engraving Melencolia I (unfortunately not reproduced in this book). Here, two irregular pentagonal faces hide the back of a strange, unknown polyhedron, echoing the uncertainty and insecurity of the female figure representing melancholy.
Among craftsmen, the polyhedra became popular in intarsia, the art of making designs with inlaid wood or stone of different colors. The famed cabinet makers of Augsburg made their Meisterstücke with many designs of inlaid polyhedra. The master goldsmith Wenzel Jamnitzer of Nuremberg wrote a book Perspectiva Corporum Regularium with many drawings in perspective of the regular solids and their variations. The artist Lorentz Stöer published a portfolio of fantastic landscapes with polyhedra in them, perhaps the wildest expression of this mania for new strange and irregular shapes. The interest in polyhedra even penetrated the refined world of ivory turning. Clever artists managed to make ivory balls in the shape of open polyhedra, containing other polyhedral balls within, which in turn contained even more inside them.
One of the most spectacular applications of polyhedra in the 16th century was Kepler's use of the regular solids to describe the orbits of the planets around the sun. He imagined a cube, a tetrahedron, a dodecahedron an icosahedron, and an octahedron successively inscribed and circumscribed around six spheres. Then the radii of those spheres would be proportional to the distances of Mercury, Venus, Earth, Mars, Jupiter and Saturn from the sun. Near the end of this book (p. 253), is a fold-out oversize illustration, a copy of Kepler's design, taken from his Mysterium Cosmographicum of 1596.
The historical facts in this book, are mostly accurate, as far as I can tell. However, Andrews does occasionally wander off into fantasy land. When discussing the first edition of Euclid's Elements, published by Erhardt Ratdolt in Venice (1482), the author says that Ratdolt had invented the diagrams to make the book more attractive to buyers. However, those diagrams were already present in all previous manuscripts, even the earliest extant ones, the MS gr190 at the Vatican, and the MS D'Orville 301, at the Bodleian library in Oxford, both from the ninth century. These manuscripts have diagrams for the whole text very similar to those in Ratdolt's Euclid. In fact, the diagrams are essential to understanding the text of Euclid's proofs. Otherwise, for example, in the proof of XIII.17 shown on p.35, when the text says let b be the midpoint of the line a.d, how is the reader supposed to know which line is meant without the marked letters in the diagram? The suggestion that Ratdolt had added diagrams to improve the sales of the book might more properly be applied to Luca Pacioli, who invited his friend Leonardo da Vinci to provide illustrations for his book Divina Proportione of 1509.


Robin Hartshorne is  Professor of Mathematics Emeritus at the University of California, Berkeley. He is the author of Algebraic Geometry, (1977), used as a basic text all over  the world, and Geometry: Euclid and Beyond (2010), a rigorous introduction to Euclidean and non-Euclidean  geometry.