If a planar part of such a surface is not itself a convex polygon, O'Rourke requires it to be subdivided into smaller convex polygons, with flat dihedral angles between them.
For instance, O'Rourke (1993) defines a polyhedron as a union of convex polygons (its faces), arranged in space so that the intersection of any two polygons is a shared vertex or edge or the empty set and so that their union is a manifold. Definitions based on the idea of a bounding surface rather than a solid are also common.However, the polyhedra defined in this way do not include the self-crossing star polyhedra, whose faces may not form simple polygons, and some of whose edges may belong to more than two faces. The faces of such a polyhedron can be defined as the connected components of the parts of the boundary within each of the planes that cover it, and the edges and vertices as the line segments and points where the faces meet. Natural refinements of this definition require the solid to be bounded, to have a connected interior, and possibly also to have a connected boundary. A common and somewhat naive definition of a polyhedron is that it is a solid whose boundary can be covered by finitely many planes or that it is a solid formed as the union of finitely many convex polyhedra.One can distinguish among these different definitions according to whether they describe the polyhedron as a solid, whether they describe it as a surface, or whether they describe it more abstractly based on its incidence geometry. Nevertheless, there is general agreement that a polyhedron is a solid or surface that can be described by its vertices (corner points), edges (line segments connecting certain pairs of vertices),įaces (two-dimensional polygons), and that it sometimes can be said to have a particular three-dimensional interior volume. the writers failed to define what are the polyhedra".
"The Original Sin in the theory of polyhedra goes back to Euclid, and through Kepler, Poinsot, Cauchy and many others. Shapes that are often not considered as valid polyhedra (such as solids whose boundaries are not manifolds). Some of these definitions exclude shapes that have often been counted as polyhedra (such as the self-crossing polyhedra) or include Many definitions of "polyhedron" have been given within particular contexts, some more rigorous than others, and there is not universal agreement over which of these to choose. However, the formal mathematical definition of polyhedra that are not required to be convex has been problematic.
A skeletal polyhedron (specifically, a rhombicuboctahedron) drawn by Leonardo da Vinci to illustrate a book by Luca PacioliĬonvex polyhedra are well-defined, with several equivalent standard definitions.
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