# zbMATH — the first resource for mathematics

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.
##### MSC:
 52A20 Convex sets in $$n$$ dimensions (including convex hypersurfaces) 52B11 $$n$$-dimensional polytopes
Full Text: