To expand search, see Europe. Laterally related topics: Hungary, Greece, The Roman Empire, The Celts, Medieval Europe, Germany, England, Denmark, Switzerland, Russia, Italy, France, Spain, Holland /The Netherlands, and Austria.
The Mathematics and the Liberal Arts pages are intended to be a resource for student research projects and for teachers interested in using the history of mathematics in their courses. Many pages focus on ethnomathematics and in the connections between mathematics and other disciplines. The notes in these pages are intended as much to evoke ideas as to indicate what the books and articles are about. They are not intended as reviews. However, some items have been reviewed in Mathematical Reviews, published by The American Mathematical Society. When the mathematical review (MR) number and reviewer are known to the author of these pages, they are given as part of the bibliographic citation. Subscribing institutions can access the more recent MR reviews online through MathSciNet.
Bruins, Evert M. The division of the circle and ancient arts and sciences. Janus 63 (1976), no. 1--3, 61--84. (Reviewer: J. L. Berggren.) SC: 01A15 (01A20), MR: 57 #12015.
One Etruscan cup, made in Caere about 500 BC, and now in the Museum of Fine Arts in Budapest, has both an 11-gon and a 14-gon inscribed on it. As the author notes, one possible reason why both were given together could be that the sum of the sides of an 11-gon and of a 14-gon imperceptibly deviates from the radius of a circle inscribing them. Moreover, methods known in the old Babylonian period could be used to provide excellent approximations to the lengths of the sides. All this raises questions about the level of Etruscan mathematical development, about which little is still known (their language still being poorly understood). The author also discusses Heron's rather accurate method for approximating the area of a circle. The article is very interesting, but the reader should be forewarned that it is a bit technical. Closely related topics: Sumerians and Babylonians, The Circle, Polygons, and Heron.
Emmer, Michele. Art and mathematics: the Platonic solids. The Visual Mind, 215--220, Leonardo Book Series, MIT Press, Cambridge, Mass., 1993.
The author begins by mentioning some ancient representations of Platonic solids. These include a pair of Egyptian die from the Ptolemaic dynasty, an Etruscan dodecahedron (at least 2500 years old), two Celtic dodecahedra, and a West German dodecahedron from the 2nd century BC. The author continues with a discussion of the regular solids in Plato's Timaeus. The author notes that Dürer's Melancholia, which includes a truncated rhombohedron, is sometimes thought to show the influence of Luca Pacioli. The magic square in the painting gives some evidence for this; Dürer's engraving may be one of the earliest depictions of a magic squares in the West, but an earlier manuscript by Pacioli showed an interest in them. On the other hand, Luca Pacioli's De Divina Proportione relied heavily on, and perhaps even appropriated the work of Piero della Francesca. The book is also notable for its pictures of the regular solids, attributed to Leonardo da Vinci. Also discusses work on the regular solids due to Johannes Kepler, including Kepler's recognition of a duality and his idea of a combination of two tetrahedra called a stella octangula. The author notes that the notion of the stella octangula also appears in Pacioli's De Divina Proportione. In addition, Kepler's stellated dodecahedron occurs in mosaics in the San Macro Cathedral in Venice; this work is thought to have been done by Paolo Uccello. Regarding Uccello, the author quotes Donatello as saying to his close friend "Ah Paolo, this perspective of yours makes you neglect what we know for what we don't know. These things are no use except for marquetry." (The source is Vasari's Vita di Paolo Uccello.) The author, Michele Emmer, collaborated on the film Art and Mathematics. Closely related topics: The Regular Solids, Plato, Art, Germany in Ancient Times, The Celts, Albrecht Dürer, Luca Pacioli, Magic Squares, Piero della Francesca, Leonardo da Vinci (1452-1519), Paolo Uccello (1397-1475), Johannes Kepler (1571-1630), and Perspective.