|
A polyhedron is a geometric shape which in mathematics is defined by three related meanings. In the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. Further generalizing the latter, there are topological polyhedra. Classical polyhedron In classical mathematics, a polyhedron (from Greek πολυεδρον, from poly-, stem of πολυς, "many," + -edron, form of εδρον, "base", "seat", or "face") is a three-dimensional shape that is made up of a finite number of polygonal faces which are parts of planes, the faces meet in edges which are straight-line segments, and the edges meet in points called vertices. Cubes, prisms and pyramids are examples of polyhedra. The polyhedron surrounds a bounded volume in three-dimensional space; sometimes this interior volume is considered to be part of the polyhedron. A polyhedron is a three-dimensional analog of a polygon. The general term for polygons, polyhedra and even higher dimensional analogs is polytope. Names of polyhedra by number of faces are tetrahedron, pentahedron, hexahedron, octahedron, decahedron, etc. Such terms are in particular used with "regular" in front or implied (in the five cases in which this is applicable) because for each there are different types which have not much in common except having the same number of faces. For a tetrahedron this applies to a much lesser extent, it is always a triangular pyramid. Classical polyhedra include the five regular convex polyhedra: tetrahedron (4 sides), cube (6 sides), octahedron (8 sides), dodecahedron (12 side) and icosahedron (20 sides), four regular non convex polyhedra (the Kepler-Poinsot solids), thirteen convex Archimedean solids and the 53 remaining uniform polyhedra. Dual polyhedra of the classical polyhedra can also be considered classical. Characteristics A polyhedron is: The Euler characteristic relates the number of edges E, vertices V, and faces F of a simply connected polyhedron: V - E + F = 2. Symmetry Many polyhedra are highly symmetric, their symmetry groups are all point groups and include: Those with chiral symmetry do not have reflection symmetry and hence have two enantiomorphous forms which are reflections of each other. The snub polyhedra have this property. Uniform polyhedra Main article Uniform polyhedron. Uniform polyhedra are vertex uniform and every face is a regular polygon. They are either regular, quasi-regular, or semi-regular but not necessarily convex. The Uniform polyhedra include all the polyhedra mentioned above. As conjectured by H. S. M. Coxeter et al. in 1954 and later confirmed by J. Skilling, there are exactly 75 uniform polyhedra, plus an infinite number of prisms and antiprisms. Some of the antiprisms are non-convex. The full list of uniform polyhedra contains details of all uniform polyhedra and List of uniform polyhedra by vertex figure exhibits some relations between the polyhedra. Of the 39 non-convex semiregular polyhedra 17 are stellations of Archimedean solids. Two examples of non-convex semiregular polyhedra are the Platonic solids There are exactly five regular convex polyhedra. These have been known since ancient times, and are called the Platonic solids: Kepler-Poinsot solids There are exactly four regular non-convex polyhedra, known as the Kepler-Poinsot solids: Semi-regular convex polyhedron Semi-regular means vertex-uniform but not edge-uniform. The convex ones consist of the prisms and antiprisms and the Archimedean solids. Non-convex semi-regular are listed below. Prisms and antiprisms There are infinitely many semi-regular convex polyhedra in two infinite series: Archimedean solid There are 13 Archimedean solids. Two are quasi-regular convex polyhedra, which have the additional property of being edge-uniform: The 11 others are also convex polyhedra: No other convex edge-uniform polyhedra composed of regular polygons exist than the five regular and two quasi-regular convex polyhedra, so edge uniformity and face regularity with convexity implies vertex-uniformity. (There are two other edge-uniform convex polyhedra, the rhombic dodecahedron and the rhombic triacontahedron, but they are not face-regular and not vertex-uniform. These are the duals of the quasi-regular convex polyhedra, and are both members of the Catalan solids.) Polyhedron duals For every polyhedron there is a dual polyhedron which can be obtained, for regular polyhedra, by connecting the midpoints of the faces. For an arbitrary polyhedron, the more complicated process of spherical reciprocation is required (see dual polyhedron). Face-uniformity of a polyhedron corresponds to vertex-uniformity of the dual and conversely, and edge-uniformity of a polyhedron corresponds to edge-uniformity of the dual. Thus the regular polyhedra come in natural pairs: the dodecahedron with the icosahedron, the cube with the octahedron, and the tetrahedron with itself. In most duals of uniform polyhedra, faces are irregular polygons. The exceptions are: Quasi-regular duals The duals of the quasi-regular polyhedra are edge- and face-uniform. These are, correspondingly: There are 13 other nonconvex duals. Pyramids and prisms Semi-regular duals Main article Semiregular polyhedron. Stellations Main article Stellation. Stellation of a polyhedron is the process of extending the faces (within their planes) so that they meet to form a new polyhedron. It the exact reciprocal to the process of facetting which is the removal of parts of a polyhedron without creating any new vertices. Compounds Polyhedral compounds are formed as compounds of two or more polyhedra. These include: Johnson solids Norman Johnson sought which non-uniform polyhedra had regular faces. In 1966, he published a list of 92 convex solids, now known as the Johnson solids, and gave them their names and numbers. He did not prove there were only 92, but he did conjecture that there were no others. Victor Zalgaller in 1969 proved that Johnson's list was complete. Deltahedron A deltahedron (plural deltahedra) is a polyhedron whose faces are all equilateral triangles. There are infinitely many deltahedra, but only eight of these are convex: Other polyhedra with regular faces With regard to polyhedra whose faces are all squares: if coplanar faces are not allowed, even if they are disconnected, there is only the cube. Otherwise there is also the result of pasting six cubes to the sides of one, all seven of the same size; it has 30 square faces (counting disconnected faces in the same plane as separate). This can be extended in one, two, or three directions: we can consider the union of arbitrarily many copies of these structures, obtained by translations of (expressed in cube sizes) (2,0,0), (0,2,0), and/or (0,0,2), hence with each adjacent pair having one common cube. The result can be any connected set of cubes with positions (a,b,c), with integers a,b,c of which at most one is even. There is no special name for polyhedra whose faces are all equilateral pentagons or pentagrams. There are infinitely many of these, but only one is convex: the dodecahedron. The rest are assembled by (pasting) combinations of the regular polyhedra described earlier: the dodecahedron, the small stellated dodecahedron, the great stellated dodecahedron and the great icosahedron. There exists no polyhedron whose faces are all regular polygons with six or more sides because the vertex of three regular hexagons defines a plane. (See infinite skew polyhedron for exceptions with zig-zagging vertex figures.) Catalan solids A Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. They are face-uniform but not vertex-uniform. Zonohedron A zonohedron is a convex polyhedron where every face is a polygon with inversion symmetry or, equivalently, symmetry under rotations through 180°. General polyhedron More recently mathematics has defined a polyhedron as a set in real affine (or Euclidean) space of any dimensional n that has flat sides. It could be defined as the union of a finite number of convex polyhedra, where a convex polyhedron is any set that is the intersection of a finite number of half-spaces. It may be bounded or unbounded. In this meaning, a polytope is a bounded polyhedron. All classical polyhedra are general polyhedra, and in addition there are examples like: Topological polyhedron A topological polyhedron is a topological space given along with a specific decomposition into shapes that are topologically equivalent to convex polytopes and that are attached to each other in a regular way that needs better description. Relation with graphs Any polyhedron gives rise to a graph, called skeleton, with corresponding vertices and edges. Thus graph terminology and properties can be applied to polyhedra: History and Polytopes in nature Much of the history of polyhedra is covered in ''Regular polytope: History of discovery''. For natural occurrences of polyhedra, see Regular polytope: Polytopes in nature: Polyhedra. See also | |||||||
|
| ||||||||
 |
|
| |