polyhedron(redirected from Convex polyhedra)
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polyhedron(pŏl'ēhē`drən), closed solid bounded by plane faces; each face of a polyhedron is a polygonpolygon,
closed plane figure bounded by straight line segments as sides. A polygon is convex if any two points inside the polygon can be connected by a line segment that does not intersect any side. If a side is intersected, the polygon is called concave.
..... Click the link for more information. . A cube is a polyhedron bounded by six polygons (in this case squares) meeting at right angles. Although regular polygons are possible for any number of sides, there are only five possible regular polyhedrons, having congruent faces, each a regular polygon and meeting at equal angles. The five regular polyhedrons are also known as the Platonic solids, although they were known to the Greeks before the time of Plato. They are the tetrahedron, bounded by four equilateral triangles; the hexahedron, or cubecube,
in geometry, regular solid bounded by six equal squares. All adjacent faces of a cube are perpendicular to each other; any one face of a cube may be its base. The dimensions of a cube are the lengths of the three edges which meet at any vertex.
..... Click the link for more information. , bounded by six squares; the octahedron, bounded by eight equilateral triangles; the dodecahedron, bounded by twelve regular pentagons; and the icosahedron, bounded by twenty equilateral triangles. The 18th-century Swiss mathematician Leonhard Euler showed that for any simple polyhedron, i.e., a polyhedron containing no holes, the sum of the number of vertices V and the number of faces F is equal to the number of edges E plus 2, or V+F=E+2.
(in three-dimensional space), a finite system of plane polygons arranged in space in such a way that (1) exactly two polygons meet (at an angle) at every side and (2) it is possible to get from every polygon to every other polygon by way of a sequence of adjacent polygons (that is, polygons sharing a side). The component polygons are called faces, and the sides and vertices are called, respectively, the edges and vertices of the polyhedron.
The meaning of “polyhedron” depends on that of “polygon.” If by a polygon we mean a plane, closed, broken (possibly self-intersecting) line, then we arrive at one definition of polyhedron (problems connected with polyhedrons thus defined will be considered at the end of the article). The discussion in most of this article is based on another definition of polyhedron, in which the faces are polygons construed as parts of the plane bounded by broken lines. From this standpoint, a polyhedron is a surface made up of polygonal pieces. If this surface does not intersect itself, then it is the complete surface of some geometric solid, which is also called a polyhedron. This leads to a third view of polyhedrons as geometric solids. These solids may have “holes,” that is, they need not be simply connected.
A polyhedron is said to be convex if it lies entirely on one side of any plane containing one of its faces; then the faces are also convex. A convex polyhedron decomposes space into two regions—the interior and the exterior of the polyhedron. The interior is a convex solid. Conversely, if the surface of convex solid is polyhedral, then the solid is a convex polyhedron.
The following are the most important theorems of the general theory of convex polyhedrons (considered as surfaces)
(1) Euler’s theorem (1758). The number of vertices minus the number of edges plus the number effaces of a convex polyhedron —its Euler characteristic—equals two; in symbols, V — E + F = 2.
(2) Cauchy’s theorem (1812). The modern form of Cauchy’s theorem states that if two convex polyhedrons are mutually isometric (that is, if there exists a one-to-one length-preserving mapping of one polyhedron to the other), then, apart from a possible reflection, the second polyhedron can be obtained from the first by moving it as a rigid body. It follows, then, that if the faces of a convex polyhedron are rigid, then the polyhedron itself is rigid, even if its faces are joined along the edges by means of hinges. This result was taken for granted by Euclid and is known by anyone who has glued together cardboard models of polyhedrons. It was first proved by Cauchy 2,000 years after Euclid.
(3) A. D. Aleksandrov’s theorem (1939). It is clear that a set of polygons that form the pattern of a closed convex polygon satisfies the following conditions: (1) exactly two faces meet at each edge and three or more faces meet at each vertex; (2) the number of vertices, faces, and edges satisfies Euler’s condition; (3) we can go from a polygon to any other polygon by way of adjacent polygons; (4) the sides common to two polygons have equal length; and (5) the sum of the plane angles of the faces at a common vertex is less than 277. Aleksandrov proved that these conditions are sufficient for a system of polygons to be a pattern for a convex polyhedron. Aleksandrov’s theorem is an existence theorem, that is, it shows what patterns yield convex polyhedrons, and Cauchy’s theorem is for it a uniqueness theorem, that is, it shows that, apart from a direct motion or a reflection, a pattern determines a unique convex polyhedron.
(4) Minkowski’s (existence) theorem (1896). There exists a convex polyhedron with faces of prescribed area and with external normals of prescribed direction, provided that the sum of vectors that have the direction of the respective normals and lengths equal to the areas of the corresponding faces is zero and that these vectors do not all lie in the same plane. These conditions are necessary.
(5) Minkowski’s (uniqueness) theorem (1896). A convex polyhedron is completely determined by the areas of its faces and by the directions of their external normals.
(6) Aleksandrov’s (uniqueness) theorem, which extends Minkowski’s (uniqueness) theorem: Two convex polyhedrons with pairwise parallel faces are not equal only if there exists a translation that properly imbeds one of a pair of parallel faces with identically directed external normals in the other.
(7) Steinitz’ theorem (1917). Two polyhedrons are said to be of the same topological type if they are topologically identical, that is, if one of them differs from the other only in the length of the edges and in the size of the angles between pairs of edges. Steinitz proved that a polyhedron of any type can be realized as a convex polyhedron.
If we project a convex polyhedron from an exterior point that is sufficiently close to an interior point of one of the faces, then the image is a convex polygon—the image of the selected face— filled without gaps or overlapping with small polygons that are the images of the remaining faces. Convex polyhedrons of the same type yield plane projections of the same type. The number m of types of polyhedrons with a given number n of faces is limited: namely, if n = 4, 5, 6, 7, 8, … , then m = 1, 2, 7, 34, 257,… Figure 1 shows all types of plane projections of polyhedrons with n = 4, 5, 6 faces.
The following special convex polyhedrons are the most important.
(1) Regular polyhedrons (platonic solids). Regular polyhedrons are convex polyhedrons all of whose edges are congruent regular polygons. All polyhedral angles of a regular polyhedron
are regular and equal. It is evident upon calculating the sum of the plane angles at a vertex that there are no more than five convex regular polyhedrons. It can be proved by the method indicated below that there are only five regular polyhedrons (this was proved by Euclid): regular tetrahedron, hexahedron (cube), octahedron, dodecahedron, and icosahedron (Figure 2, a—2, e).
The cube and the octahedron are a pair of dual polyhedrons, that is, they are obtained from each other if the centroids of the faces of one are taken as the vertices of the other. The dodecahedron and icosahedron are also dual. The tetrahedron is self-dual. The regular dodecahedron is obtained from a cube by constructing “roofs” on the faces of the cube (Euclid’s method). Any four vertices of a cube that are not pairwise incident on an edge are the vertices of a tetrahedron. All other regular polyhedrons can be similarly obtained from a cube.
The radius of the circumscribed sphere, the radius of the inscribed sphere, and the volume of all regular polyhedrons are given in Table 1, where a is the length of an edge of the polyhedron.
(2) Isohedrons and isogons. An isohedron (isogon) is a convex polyhedron such that the group of its rotations and reflections about its centroid carries any of its faces (vertices) to any other face (vertex). To every isogon (isohedron) there corresponds a dual isohedron (isogon). If a polyhedron is at the same time an isogon and an isohedron, then it is a regular polyhedron. There are 13 special types and two infinite series (prisms and anti-prisms) of topologically different isohedrons (isogons). It turns out that each of these isohedrons can be realized as an isohedron whose faces are regular polygons. Polyhedrons thus obtained are called semiregular polyhedrons (Archimedean solids). The 13 polyhedrons are shown in Figure 2, j-2, v, the prism in Figure 2, w, and the antiprism in Figure 2, x).
(3) Parallelohedrons (convex, discovered by the Russian scientist E. S. Fedorov in 1881). Parallelohedrons are polyhedrons whose images under translation fill space without gaps or over-laps and thus partition space. Examples are cubes and regular hexagonal prisms. There are five topologically different parallelohedrons (Figure 2, y-2, cc). The number of their faces is 6, 8, 12, 12, and 14, respectively. In order for a polyhedron to be a parallelohedron, it is necessary and sufficient that it be a convex polyhedron of one of the five indicated topological types and that all of its faces have centers of symmetry.
If the parallelohedrons of a partition share faces, then the partition is said to be normal. The centroids of the parallelohedrons of a normal partition form a lattice, that is, the set of all points with integral coordinates in some, usually nonrectangular, Cartesian coordinate system. The set of points in space each of which is no farther from some given point O of the lattice Λ under consideration than from any other point of Λ, is called the Dirichlet domain (or Voronyi domain) D0Λ of the point O. The domain Z>oA is a convex polyhedron centered at 0. The set of the Dirichlet domains of all points in an arbitrary lattice forms a normal partition of space. The following remarkable theorem holds: An arbitrary (including n -dimensional) normal partition into parallelohedrons with n + 1 parallelohedrons at each vertex can be converted by an affine transformation into a Dirichlet partition for some lattice.
Any motion that carries a lattice A into itself and leaves its point O fixed, transforms the domain Z>oA into itself, and conversely. The group of all such motions is called the holohedry of the lattice. There are seven such groups: the cubic, trigonal, tetragonal, orthorhombic, monoclinic, triclinic, and hexagonal.
Each of the seven groups has subgroups. There are a total of 32 groups and subgroups. They are called crystal classes. Let some crystal class be a subgroup of a certain holohedry; then we say that it belongs to this holohedry (or falls within its syngony) if this class is not a subgroup of some holohedry contained in the given holohedry. If we take a plane that does not pass through the point O and subject it to all rotations of some crystal class, then the planes obtained determine a certain isohedron centered at the point, or an infinite convex prismatic body, or a polyhedral angle. These solids are called simple forms of crystals: closed forms in the first case and open forms in the second and third. Two simple forms are considered to be identical if they are of the same topological type, are generated by the same crystal class, and the rotations of the class are in each case related to the form in the same way. There are 30 different closed forms and 17 open forms. Each form has its own name.
The first definition of polyhedron given in the beginning of the article covers four more regular nonconvex polyhedrons, called Poinsot solids (Figure 2, f-2, i), which were first discovered in 1809 by the French mathematician L. Poinsot. A proof of the nonexistence of other nonconvex regular polyhedrons was given in 1811 by the French mathematician A. Cauchy. In these polyhedrons, the faces either intersect each other or are them-selves self-intersecting polygons. In studying problems related to the surface areas and volumes of such polyhedrons, it is convenient to use the first definition of polyhedron.
If the faces of a polyhedron can be so oriented that the adja-cent faces assign opposite orientations to their common edge, then the polyhedron is said to be orientable; otherwise the polyhedron is not orientable. The concepts of surface area and volume may be introduced for an orientable polyhedron, even if it is self-intersecting and its faces are self-intersecting polygons. Thus the sum of the areas of the faces of an orientable polyhedron is called its area. To define volume we note that the set of interior portions of the faces of the polyhedron divides space into a certain number of connected portions, one of which is infinite (exterior) with respect to the polyhedron while the others are finite (interior). If a line segment is drawn from a point exterior to the polyhedron to a point of an interior portion, then the sum of the “coefficients” of the interior portions of the faces of the polyhedron that intersect this segment is called the coefficient of that interior portion of the polyhedron (it is independent of the choice of the external point <7); this coefficient can be a positive integer, a negative integer, or zero. The sum of the ordinary volumes of the interior portions of the polyhedron, multiplied by their respective coefficients, is called the volume of the polyhedron.
We may also consider n -dimensional polyhedrons. Some of the definitions and theorems given above have n -dimensional analogs. In particular, all convex regular polyhedrons are known; for n = 4, there are six, while for all greater n there are a total of three: the analogs of the tetrahedron, cube, and octahedron. On the other hand, we do not know, say, all the four-dimensional isohedrons and isogons.
The following are three unsolved problems of polyhedron theory.
(1) The German mathematician E. Steinitz gave examples demonstrating that a polyhedron that can be circumscribed about a sphere does not exist for every topological type of convex polyhedron; no general solution for this problem is known.
(2) Parallelohedrons are the convex fundamental domains of groups of translations; however, the principal types of stereohedrons that is, convex fundamental domains of arbitrary (Fedorov) discrete groups of motions, have yet to be determined.
(3) The determination of all types of four-dimensional isohedrons.
REFERENCESFedorov, E. S. Nachala ucheniia o figurakh. St. Petersburg, 1885.
Aleksandrov, A. D. Vypuklye mnogogranniki. Moscow-Leningrad, 1950.
Voronoi, G. F. Sobr. soch., vol. 2. Kiev, 1952.
Brückner, M. Vielecke und Vielflache. Theorie und Geschichte. Leipzig, 1900.
Steinitz, E. Vorlesungen über die Theorie der Polyeder unter Einschluss der Elemente der Topologie. Berlin, 1934.
Coxeter, H. S. M. Regular Polytopes, 2nd ed. London-New York, 1963.
B. N. DELONE