Polyhedron
In geometry, a polyhedron is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary surface. The terms solid polyhedron and polyhedral surface are commonly used to distinguish the two concepts. Also, the term polyhedron is often used to refer implicitly to the whole structure formed by a solid polyhedron, its polyhedral surface, its faces, its edges, and its vertices.
There are many definitions of polyhedra, not all of which are equivalent. Under any definition, polyhedra are typically understood to generalize two-dimensional polygons and to be the three-dimensional specialization of polytopes. Polyhedra have several general characteristics that include the number of faces, topological classification by Euler characteristic, duality, vertex figures, surface area, volume, interior lines, Dehn invariant, and symmetry. A symmetry of a polyhedron means that the polyhedron's appearance is unchanged by the transformation such as rotating and reflecting.
The convex polyhedra are a well defined class of polyhedra with several equivalent standard definitions. Every convex polyhedron is the convex hull of its vertices, and the convex hull of a finite set of points is a polyhedron. Many common families of polyhedra, such as cubes and pyramids, are convex.
Definition
are well-defined, with several equivalent standard definitions. However, the formal mathematical definition of polyhedra that are not required to be convex has been problematic.Many definitions of "polyhedron" have been given within particular contexts, some more rigorous than others, and there is no universal agreement over which of these to choose.
Some of these definitions exclude shapes that have often been counted as polyhedra or include
shapes that are often not considered as valid polyhedra. And although several of these definitions explicitly require the number of faces of a polyhedron to be finite, shapes with infinitely many faces such as the skew apeirohedra have also been called polyhedra. As Branko Grünbaum observed,
Nevertheless, there is general agreement that a polyhedron is a solid or surface that can be described by its vertices, edges,
faces, and that it sometimes can be said to have a particular three-dimensional interior volume.
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.
- 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. Natural refinements of this definition require the solid to be bounded, to have a connected interior, and possibly also to have a connected boundary. 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. 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.
- Definitions based on the idea of a bounding surface rather than a solid are also common. For instance, defines a polyhedron as a union of finitely many convex polygons, 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. 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. Somewhat more generally, Grünbaum defines an acoptic polyhedron to be a collection of finitely many simple polygons that form an embedded manifold, with each vertex incident to at least three edges and each two faces intersecting only in shared vertices and edges of each. Cromwell's Polyhedra gives a similar definition but without the restriction of at least three edges per vertex. Again, this type of definition does not encompass the self-crossing polyhedra. Similar notions form the basis of topological definitions of polyhedra, as subdivisions of a topological manifold into topological disks whose pairwise intersections are required to be points, topological arcs, or the empty set. However, there exist topological polyhedra that cannot be realized as acoptic polyhedra.
- One modern approach is based on the theory of abstract polyhedra. These can be defined as partially ordered sets whose elements are the vertices, edges, and faces of a polyhedron. A vertex or edge element is less than an edge or face element when the vertex or edge is part of the edge or face. Additionally, one may include a special bottom element of this partial order and a top element representing the whole polyhedron. If the sections of the partial order between elements three levels apart have the same structure as the abstract representation of a polygon, then these partially ordered sets carry exactly the same information as a topological polyhedron. However, these requirements are often relaxed, to instead require only that sections between elements two levels apart have the same structure as the abstract representation of a line segment. Geometric polyhedra, defined in other ways, can be described abstractly in this way, but it is also possible to use abstract polyhedra as the basis of a definition of geometric polyhedra. A realization of an abstract polyhedron is generally taken to be a mapping from the vertices of the abstract polyhedron to geometric points, such that the points of each face are coplanar. A geometric polyhedron can then be defined as a realization of an abstract polyhedron. Realizations that omit the requirement of face planarity, that impose additional requirements of symmetry, or that map the vertices to higher dimensional spaces have also been considered. Unlike the solid-based and surface-based definitions, this works perfectly well for star polyhedra. However, without additional restrictions, this definition allows degenerate or unfaithful polyhedra and the question of how to constrain realizations to avoid these degeneracies has not been settled.
However, some of the literature on higher-dimensional geometry uses the term "polyhedron" to mean something else: not a three-dimensional polytope, but a shape that is different from a polytope in some way. For instance, some sources define a convex polyhedron to be the intersection of finitely many half-spaces, and a polytope to be a bounded polyhedron. The remainder of this article considers only three-dimensional polyhedra.
General characteristics
Number of faces
Polyhedra may be classified and are often named according to the number of faces. The naming system is based on Classical Greek, and combines a prefix counting the faces with the suffix "hedron", meaning "base" or "seat" and referring to the faces. For example a tetrahedron is a polyhedron with four faces, a pentahedron is a polyhedron with five faces, a hexahedron is a polyhedron with six faces, etc. For a complete list of the Greek numeral prefixes see, in the column for Greek cardinal numbers. The names of tetrahedra, hexahedra, octahedra, dodecahedra, and icosahedra are sometimes used without additional qualification to refer to the Platonic solids, and sometimes used to refer more generally to polyhedra with the given number of sides without any assumption of symmetry.Topological classification
Some polyhedra have two distinct sides to their surface. For example, the inside and outside of a convex polyhedron paper model can each be given a different colour. These polyhedra are orientable. The same is true for non-convex polyhedra without self-crossings. Some non-convex self-crossing polyhedra can be coloured in the same way but have regions turned "inside out" so that both colours appear on the outside in different places; these are still considered to be orientable. However, for some other self-crossing polyhedra with simple-polygon faces, such as the tetrahemihexahedron, it is not possible to colour the two sides of each face with two different colours so that adjacent faces have consistent colours.In this case the polyhedron is said to be non-orientable. For polyhedra with self-crossing faces, it may not be clear what it means for adjacent faces to be consistently coloured, but for these polyhedra it is still possible to determine whether it is orientable or non-orientable by considering a topological cell complex with the same incidences between its vertices, edges, and faces.
A more subtle distinction between polyhedron surfaces is given by their Euler characteristic, which combines the numbers of vertices, edges, and faces of a polyhedron into a single number defined by the formula
The same formula is also used for the Euler characteristic of other kinds of topological surfaces. It is an invariant of the surface, meaning that when a single surface is subdivided into vertices, edges, and faces in more than one way, the Euler characteristic will be the same for these subdivisions. For a convex polyhedron, or more generally any simply connected polyhedron with the surface of a topological sphere, it always equals 2. For more complicated shapes, the Euler characteristic relates to the number of toroidal holes, handles or cross-caps in the surface and will be less than 2.
All polyhedra with odd-numbered Euler characteristics are non-orientable. A given figure with even Euler characteristic may or may not be orientable. For example, the one-holed toroid and the Klein bottle both have, with the first being orientable and the other not.
For many ways of defining polyhedra, the surface of the polyhedron is required to be a manifold. This means that every edge is part of the boundary of exactly two faces and that every vertex is incident to a single alternating cycle of edges and faces. For polyhedra defined in these ways, the classification of manifolds implies that the topological type of the surface is completely determined by the combination of its Euler characteristic and orientability. For example, every polyhedron whose surface is an orientable manifold and whose Euler characteristic is 2 must be a topological sphere.
A toroidal polyhedron is a polyhedron whose Euler characteristic is less than or equal to 0, or equivalently whose genus is 1 or greater. Topologically, the surfaces of such polyhedra are torus surfaces having one or more holes through the middle. A notable example is the Szilassi polyhedron, which geometrically realizes the Heawood map. A polyhedron with the symmetries of a regular polyhedron and with genus more than one is a Leonardo polyhedron.