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The Origins of Four-Dimensional Geometry

In mathematician Felix Klein's posthumously published memoir Developments of Mathematics in the Nineteenth Century (1926), Klein says of Hermann Grassmann that unlike ''we academics [who] grow in strong competition with each other, like a tree in the midst of a forest which must stay slender and rise above the others simply to exist and to conquer its portion of light and air, he who stands alone can grow on all sides'' (161). Grass-mann never had a university position, taught only in German gymnasiums, and was consequently allowed to be a generalist: a philosopher, physicist , naturalist, and philologist who specialized in the Rig Veda, a Hindu classic. Grassmann's mathematics was outside the mainstream of thought; read by few, his great work Die lineale Ausdehn-ungslehre (The Theory of Linear Extension, 1844) was described even by Klein as ''almost unread-able.'' Yet this book, more philosophy than mathematics , for the first time proposed a system whereby space and its geometric components and descriptions could be extrapolated to other dimensions. Grassmann was not completely alone in his philosophical musings. August Möbius speculated that a left-handed crystal, structured like a left-turning circular staircase, could be turned into a right-handed crystal by passing it through a fourth dimension. Arthur Cayley published a paper on four-dimensional analytic geometry in 1844, at age twenty-two, and a few others worked on the idea of a general four-dimensional geometry. But these disparate musings lacked both a critical mass and a specific geometric interpretation. In the second half of the nineteenth century, however, four-dimensional geometry advanced rapidly with the discovery and description of the four-dimensional analogs of the platonic solids, the geometric building blocks of space. In three dimensions there are five platonic solids: tetrahe-dron, cube, octahedron, icosahedron, and do-decahedron (fig. 1.1). They are ''platonic'' because they are regular: not only is every two-dimensional bounding face the same, but also each ver-tex is identical. In four dimensions, however, there are six platonic solids, also called polytopes (fig. 1.2). According to the great Canadian geometer Harold Scott MacDonald Coxeter, the credit for the discovery of the platonic solids in four-dimensional space should go to Ludwig Schläfli (1814– 1895). His book Theorie der vielfachen Kontinuität (Theory of Continuous Manifolds, 1852), with a title and a spirit so much like Grassmann's but with an intensely analytic approach, went far beyond what had been done before. In calculus, an integral computes the area under …