tetrahedron octahedron icosahedron cube dodecahedron (Fire) (Air) (Water) (Earth) (Universe)

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1 Platonic Solids A regular polyhedron is one whose faces are identical regular polygons. The solids as drawn in Kepler s Mysterium Cosmographicum: tetrahedron octahedron icosahedron cube dodecahedron (Fire) (Air) (Water) (Earth) (Universe) 2006 Spring Semester 1

2 Faces around a vertex Only five regular solids are possible. Schläfli symbol {p, q} means: the faces are regular p-gons, q surrounding each vertex. {4, 3} {5, 3} P P {3, 5} P P P {3, 3} {3, 4} 2006 Spring Semester 2

3 Archimedean Polyhedra The 13 Archimedean solids are the convex polyhedra that have a similar arrangement of nonintersecting regular convex polygons of two or more different types arranged in the same way about each vertex with all sides the same length (Cromwell 1997, pp ). chimedeansolid.html 2006 Spring Semester 3

4 Geometrical Constructions 2 Archimedean Polyhedra 2006 Spring Semester 4 Pál Ledneczki

5 Fullerains (named after Buckminster Fuller) A highlight of one of the pentagonal rings A highlight of one of the hexagonal rings The Royal Swedish Academy of Sciences has awarded the 1996 Nobel Prize for Chemistry jointly to: Professor Robert F. Curl, Jr., Rice University, Houston, USA Professor Sir Harry W. Kroto FRS, University of Sussex, Brighton, UK Professor Richard E. Smalley, Rice University, Houston, USA For their Discovery of Fullerenes. In 1985 one of the greatest new discoveries in science was made when chemists Richard Smalley and Harold Kroto discovered the existence of a third form of carbon. Unlike the two other forms of carbon, diamond and graphite, this amazing 60-atom cage molecule was shaped like a soccer ball. Both Kroto and Smalley felt it most appropriate to name it, "buckminsterfullerene" for its striking resemblance to a geodesic dome. A new family of these molecules have since been found called "fullerenes." (Note: Diamond is a molecular network crystal with each carbon bonded to four others in a tetrahedral configuration. Graphite is formed in flat sheets with each carbon bonded to three others in a hexagonal configuration.) Buckminster Fuller's Dome - Expo '67 Montreal 2006 Spring Semester 5

6 Regular Star Polyhedra Two star polyhedra were discovered by Poinsot in The others were discovered about 200 years before that by Johannes Kepler ( ), the German astronomer and natural philosopher noted for formulating the three laws of planetary motion, now known as Kepler's laws, including the law that celestial bodies have elliptical, not circular orbits. Stellation is the process of constructing polyhedron by extending the facial planes past the polyhedron edges of a given polyhedron until they intersect (Wenninger 1989). The set of all possible polyhedron edges of the stellations can be obtained by finding all intersections on the facial planes. The Kepler-Poinsot solids consist of the three dodecahedron stellations and one of the icosahedron stellations, and these are the only stellations of Platonic solids which are uniform polyhedra Spring Semester 6

7 Art and Science JACOPO DE 'BARBERI: Luca Pacioli, c This painting shows Fra Luca Pacioli and his student, Guidobaldo, Duke of Urbino. In the upper left is a rhombi-cuboctahedron, and on the table is a dodecahedron on top ofa copy of Euclid's Elements. Leonardo's Illustrations for Luca's book. Da Divina Proportione Luca Pacioli wrote a book called Da Divina Proportione (1509) which contained a section on the Platonic Solids and other solids, which has 60 plates of solids by none other than his student Leonardo da Vinci Spring Semester 7

8 M. C. ESCHER ( ) Escher made a set of nested Platonic Solids. When he moved to a new studio he have away most of his belongings but took his beloved model. Stars, 1948 Note the similarity between this polyhedron and Leonardo's illustrations for Pacioli's book 2006 Spring Semester 8

9 Models 2006 Spring Semester 9

10 Links Spring Semester 10