TAILIEUCHUNG - The Eightfold Way: A Mathematical Sculpture by Helaman Ferguson

The selection of materials for outdoor sculpture commissioned today may range from traditional bronze, granite or copper to ephem- eral materials like plastics, plants and electronic equipment. Over time these materials interact with each other and their environment, leading to inevitable deterioration. Metals corrode, plastics discolor or become brittle, painted surfaces chip and fade. In addition, ele- ments in the environment—water, chemical pollutants, extreme temperatures and ultraviolet light—accelerate these processes of deterioration. Sculptures have been removed from view because of neglected maintenance | The Eightfold Way MSRI Publications Volume 35 1998 The Eightfold Way A Mathematical Sculpture by Helaman Ferguson WILLIAM P. THURSTON This introduction to The Eightfold Way and the Klein quartic was written for the sculpture s inauguration. On that occasion it was distributed together with the illustration on Plate 2 to a public that included not only mathematicians but many friends of MSRI and other people with an interest in mathematics. Thurston was the Director of MSRI from 1992 to 1997. Mathematics is full of amazing beauty yet the beauty of mathematics is far removed from most people s everyday experience. The Mathematical Sciences Research Institute is committed to the search for ways to convey the beauty and spirit of mathematics beyond the circles of professional mathematicians. As a step in this effort MSRI pronounced Emissary has installed a first mathematical sculpture The Eightfold Way by Helaman Ferguson. The sculpture represents a beautiful mathematical construction that has been studied by mathematicians for more than a century from many points of view geometry symmetry group theory algebraic geometry topology number theory complex analysis. The surface depicted by the sculpture was discovered along with many of its amazing properties by the German mathematician Felix Klein in 1879 and is often referred to as the Klein quartic or the Klein curve in his honor. The abstract surface is impossible to render exactly in three-dimensional space so the sculpture should be thought of as a kind of topological sketch. Ridges and valleys carved into the white marble surface divide it into 24 regions. Each region has 7 sides and represents the ideal of a regular heptagon 7-gon . The L 24 heptagons fit together in triples at 56 vertices. It is A the pattern of the division of the surface into heptagons that carries the essence of the mathematics. The Klein J quartic thus is an extension of the concept of a regular polyhedron of which the dodecahedron the cube and

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