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Weitere Untersuchungen über das Ikosaeder.


Leipzig, B.G. Teubner, 1877. 8vo. Original printed wrappers, no backstrip and a small nick to front wrapper. In "Mathematische Annalen. Begründet durch Rudolf Friedrich Alfred Clebsch. XII. [12]. Band. 4. Heft." Entire issue offered. Internally very fine and clean. [Klein:] Pp. 503-60. [Entire issue: Pp. pp. 433-576]. ¶ Frist printing of Klein's paper on the icosahedron.

"A problem that greatly interested Klein was the solution of fifth-degree equations, for its treatment involved the simultaneous consideration of algebraic group theory, geometry, differential equations, and function theory. Hermite, Kronecker, and Brioschi had already employed transcendental methods in the solution of the general algebraic equation of the fifth degree. Klein succeeded in deriving the complete theory of this equation from a consideration of the icosahedron, one of the regular polyhedra known since antiquity. These bodies sometimes can be transformed into themselves through a finite group of rotations. The icosahedron in particular allows sixty such rotations into itself. If one circumscribes a sphere about a regular polyhedron and maps it onto a plane by stereographic projection, then to the group of rotations of the polyhedron into itself there corresponds a group of linear transformations of the plane into itself. Klein demonstrated that in this way all finite groups of linear transformations are obtained, if the so-called dihedral group is added. By a dihedron Klein meant a regular polygon with n sides, considered as rigid body of null volume." (DSB VII, p. 400).

The icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids.

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