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Classical solids. (English) Zbl 0840.52003
The authors formulate the notion of classical solids: compact, convex bodies in Euclidean \(n\)-space, including polytopes, but also cones, cylinders, spheres and other classically studied geometric objects. More precisely, a face of a convex body \(B\) is the intersection of \(B\) with a supporting hyperplane. A strictly classical solid is a compact convex set \(B\) such that there are at most finitely many topologically connected components of faces and no additional components arise by partitioning the faces by orbits of the symmetry group of \(B\). A right circular cylinder in \(\mathbb{E}^3\), for example, is strictly classical, the two components of 2-faces being two disks, the single component of 1-faces being a cylindrical sheet, and the two components of 0-faces being two circles. A classical solid is a solid that is combinatorially equivalent (isomorphic lattices of face components) to a strictly classical solid. Products, duality and certain generating functions associated with classical solids are investigated. Moreover, the strictly classical solids in dimensions 2, 3 and 4 are completely classified. In dimension 2, a strictly classical solid is either a polygon or a circular disk; in dimension 3, a polyhedron, a 3-ball or a lantern (the result of rotating a polygon about an axis of symmetry). In dimension 4, the convex hull of the Clifford torus is an example of a classical solid.
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52B11 \(n\)-dimensional polytopes
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