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In
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, a polyhedron (plural polyhedra or polyhedrons; ) is a
three-dimensional Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...
shape with flat
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
al
faces The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...
, straight edges and sharp corners or vertices. A
convex polyhedron A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...
is the
convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
of finitely many points, not all on the same plane.
Cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...
s and
pyramids A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilate ...
are examples of convex polyhedra. A polyhedron is a 3-dimensional example of a
polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...
, a more general concept in any number of dimensions.


Definition

Convex polyhedra A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...
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 not universal agreement over which of these to choose. Some of these definitions exclude shapes that have often been counted as polyhedra (such as the self-crossing polyhedra) or include shapes that are often not considered as valid polyhedra (such as solids whose boundaries are not
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s). As
Branko Grünbaum Branko Grünbaum ( he, ברנקו גרונבאום; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentvertices (corner points), edges (line segments connecting certain pairs of vertices),
faces The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...
(two-dimensional
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
s), and that it sometimes can be said to have a particular three-dimensional interior
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
. 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 In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that involves concepts such as length, angles, continuity, betweenness, and incidence. An ''incide ...
. * 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 polygon In geometry, a simple polygon is a polygon that does not intersect itself and has no holes. That is, it is a flat shape consisting of straight, non-intersecting line segments or "sides" that are joined pairwise to form a single closed path. If ...
s, 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.Cromwell (1997), pp. 206–209. For instance, defines a polyhedron as a union of
convex polygon In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a ...
s (its faces), arranged in space so that the intersection of any two polygons is a shared vertex or edge or the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
and so that their union is a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
. 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 angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the uni ...
s between them. Somewhat more generally, Grünbaum defines an ''acoptic polyhedron'' to be a collection of 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 In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on t ...
'' 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 (the faces) whose pairwise intersections are required to be points (vertices), topological arcs (edges), or the empty set. However, there exist topological polyhedra (even with all faces triangles) 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 set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
s whose elements are the vertices, edges, and faces of a polyhedron. A vertex or edge element is less than an edge or face element (in this partial order) when the vertex or edge is part of the edge or face. Additionally, one may include a special bottom element of this partial order (representing the empty set) and a top element representing the whole polyhedron. If the sections of the partial order between elements three levels apart (that is, between each face and the bottom element, and between the top element and each vertex) 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.. (This means that each edge contains two vertices and belongs to two faces, and that each vertex on a face belongs to two edges of that face.) 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 Degeneracy, degenerate, or degeneration may refer to: Arts and entertainment * Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed * Degenerate art, a term adopted in the 1920s by the Nazi Party i ...
or unfaithful polyhedra (for instance, by mapping all vertices to a single point) and the question of how to constrain realizations to avoid these degeneracies has not been settled. In all of these definitions, a polyhedron is typically understood as a three-dimensional example of the more general
polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...
in any number of dimensions. For example, a polygon has a two-dimensional body and no faces, while a 4-polytope has a four-dimensional body and an additional set of three-dimensional "cells". 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.


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 In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
is a polyhedron with four faces, a pentahedron is a polyhedron with five faces, a
hexahedron A hexahedron (plural: hexahedra or hexahedrons) or sexahedron (plural: sexahedra or sexahedrons) is any polyhedron with six faces. A cube, for example, is a regular hexahedron with all its faces square, and three squares around each vertex. Ther ...
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 (8-sided polyhedra), dodecahedra (12-sided polyhedra), and icosahedra (20-sided polyhedra) are sometimes used without additional qualification to refer to the
Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
s, 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 A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...
paper model can each be given a different colour (although the inside colour will be hidden from view). These polyhedra are
orientable In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space i ...
. 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 In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices. Its vertex figure is a crossed quadrilateral. Its Coxeter–Dynkin dia ...
, 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 A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...
with the same incidences between its vertices, edges, and faces. A more subtle distinction between polyhedron surfaces is given by their
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
, which combines the numbers of vertices V, edges E, and faces F of a polyhedron into a single number \chi defined by the formula :\chi=V-E+F.\ 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 surface a topological sphere, it always equals 2. For more complicated shapes, the Euler characteristic relates to the number of
toroid In mathematics, a toroid is a surface of revolution with a hole in the middle. The axis of revolution passes through the hole and so does not intersect the surface. For example, when a rectangle is rotated around an axis parallel to one of its ...
al holes, handles or cross-caps in the surface and will be less than 2. All polyhedra with odd-numbered Euler characteristic are non-orientable. A given figure with even Euler characteristic may or may not be orientable. For example, the one-holed
toroid In mathematics, a toroid is a surface of revolution with a hole in the middle. The axis of revolution passes through the hole and so does not intersect the surface. For example, when a rectangle is rotated around an axis parallel to one of its ...
and the
Klein bottle In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a o ...
both have \chi = 0, with the first being orientable and the other not. For many (but not all) ways of defining polyhedra, the surface of the polyhedron is required to be a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
. This means that every edge is part of the boundary of exactly two faces (disallowing shapes like the union of two cubes that meet only along a shared edge) and that every vertex is incident to a single alternating cycle of edges and faces (disallowing shapes like the union of two cubes sharing only a single vertex). 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 In geometry, a toroidal polyhedron is a polyhedron which is also a toroid (a -holed torus), having a topological genus () of 1 or greater. Notable examples include the Császár and Szilassi polyhedra. Variations in definition Toroidal polyhedr ...
is a polyhedron whose
Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...
is less than or equal to 0, or equivalently whose
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...
is 1 or greater. Topologically, the surfaces of such polyhedra are
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...
surfaces having one or more holes through the middle.


Duality

For every convex polyhedron, there exists a dual polyhedron having * faces in place of the original's vertices and vice versa, and * the same number of edges. The dual of a convex polyhedron can be obtained by the process of
polar reciprocation In geometry, a pole and polar are respectively a point and a line that have a unique reciprocal relationship with respect to a given conic section. Polar reciprocation in a given circle is the transformation of each point in the plane into it ...
. Dual polyhedra exist in pairs, and the dual of a dual is just the original polyhedron again. Some polyhedra are self-dual, meaning that the dual of the polyhedron is congruent to the original polyhedron. Abstract polyhedra also have duals, obtained by reversing the
partial order In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary ...
defining the polyhedron to obtain its dual or opposite order. These have the same Euler characteristic and orientability as the initial polyhedron. However, this form of duality does not describe the shape of a dual polyhedron, but only its combinatorial structure. For some definitions of non-convex geometric polyhedra, there exist polyhedra whose abstract duals cannot be realized as geometric polyhedra under the same definition.


Vertex figures

For every vertex one can define a
vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
, which describes the local structure of the polyhedron around the vertex. Precise definitions vary, but a vertex figure can be thought of as the polygon exposed where a slice through the polyhedron cuts off a corner.


Surface area and distances

The surface area of a polyhedron is the sum of areas of its faces, for definitions of polyhedra for which the area of a face is well-defined. The
geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
distance between any two points on the surface of a polyhedron measures the length of the shortest curve that connects the two points, remaining within the surface. By
Alexandrov's uniqueness theorem The Alexandrov uniqueness theorem is a rigidity theorem in mathematics, describing three-dimensional convex polyhedra in terms of the distances between points on their surfaces. It implies that convex polyhedra with distinct shapes from each oth ...
, every convex polyhedron is uniquely determined by the
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
of geodesic distances on its surface. However, non-convex polyhedra can have the same surface distances as each other, or the same as certain convex polyhedra.


Volume

Polyhedral solids have an associated quantity called
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
that measures how much space they occupy. Simple families of solids may have simple formulas for their volumes; for example, the volumes of pyramids, prisms, and
parallelepiped In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidea ...
s can easily be expressed in terms of their edge lengths or other coordinates. (See Volume § Volume formulas for a list that includes many of these formulas.) Volumes of more complicated polyhedra may not have simple formulas. Volumes of such polyhedra may be computed by subdividing the polyhedron into smaller pieces (for example, by
triangulation In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points. Applications In surveying Specifically in surveying, triangulation involves only angle me ...
). For example, the volume of a regular polyhedron can be computed by dividing it into congruent
pyramids A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilate ...
, with each pyramid having a face of the polyhedron as its base and the centre of the polyhedron as its apex. In general, it can be derived from the
divergence theorem In vector calculus, the divergence theorem, also known as Gauss's theorem or Ostrogradsky's theorem, reprinted in is a theorem which relates the ''flux'' of a vector field through a closed surface to the ''divergence'' of the field in the vol ...
that the volume of a polyhedral solid is given by \frac \left, \sum_F (Q_F \cdot N_F) \operatorname(F) \, where the sum is over faces of the polyhedron, is an arbitrary point on face , is the
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vecto ...
perpendicular to pointing outside the solid, and the multiplication dot is the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
. In higher dimensions, volume computation may be challenging, in part because of the difficulty of listing the faces of a convex polyhedron specified only by its vertices, and there exist specialized
algorithm In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specificat ...
s to determine the volume in these cases.


Dehn invariant

In two dimensions, the Bolyai–Gerwien theorem asserts that any polygon may be transformed into any other polygon of the same area by cutting it up into finitely many polygonal pieces and rearranging them. The analogous question for polyhedra was the subject of
Hilbert's third problem The third of Hilbert's list of mathematical problems, presented in 1900, was the first to be solved. The problem is related to the following question: given any two polyhedra of equal volume, is it always possible to cut the first into finitely m ...
.
Max Dehn Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German mathematician most famous for his work in geometry, topology and geometric group theory. Born to a Jewish family in Germany, Dehn's early life and career took place in Germany. ...
solved this problem by showing that, unlike in the 2-D case, there exist polyhedra of the same volume that cannot be cut into smaller polyhedra and reassembled into each other. To prove this Dehn discovered another value associated with a polyhedron, the
Dehn invariant In geometry, the Dehn invariant is a value used to determine whether one polyhedron can be cut into pieces and reassembled (" dissected") into another, and whether a polyhedron or its dissections can tile space. It is named after Max Dehn, who us ...
, such that two polyhedra can only be dissected into each other when they have the same volume and the same Dehn invariant. It was later proven by Sydler that this is the only obstacle to dissection: every two Euclidean polyhedra with the same volumes and Dehn invariants can be cut up and reassembled into each other. The Dehn invariant is not a number, but a
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
in an infinite-dimensional vector space, determined from the lengths and
dihedral angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the uni ...
s of a polyhedron's edges. Another of Hilbert's problems, Hilbert's 18th problem, concerns (among other things) polyhedra that tile space. Every such polyhedron must have Dehn invariant zero. The Dehn invariant has also been connected to
flexible polyhedra In geometry, a flexible polyhedron is a polyhedral surface without any boundary edges, whose shape can be continuously changed while keeping the shapes of all of its faces unchanged. The Cauchy rigidity theorem shows that in dimension 3 such ...
by the strong bellows theorem, which states that the Dehn invariant of any flexible polyhedron remains invariant as it flexes.


Convex polyhedra

A three-dimensional solid is a
convex set In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex r ...
if it contains every line segment connecting two of its points. A
convex polyhedron A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...
is a polyhedron that, as a solid, forms a convex set. A convex polyhedron can also be defined as a bounded intersection of finitely many half-spaces, or as the
convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
of finitely many points. Important classes of convex polyhedra include the highly symmetrical
Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
s, the
Archimedean solids In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are compose ...
and their duals the
Catalan solid In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865. The Catalan sol ...
s, and the regular-faced
Johnson solid In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that isohedral, each face must be the same polygon, or that the same polygons join around each Vertex (geometry), ver ...
s.


Symmetries

Many of the most studied polyhedra are highly
symmetrical Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
, that is, their appearance is unchanged by some reflection or rotation of space. Each such symmetry may change the location of a given vertex, face, or edge, but the set of all vertices (likewise faces, edges) is unchanged. The collection of symmetries of a polyhedron is called its
symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambient ...
. All the elements that can be superimposed on each other by symmetries are said to form a symmetry orbit. For example, all the faces of a cube lie in one orbit, while all the edges lie in another. If all the elements of a given dimension, say all the faces, lie in the same orbit, the figure is said to be transitive on that orbit. For example, a cube is face-transitive, while a truncated cube has two symmetry orbits of faces. The same abstract structure may support more or less symmetric geometric polyhedra. But where a polyhedral name is given, such as
icosidodecahedron In geometry, an icosidodecahedron is a polyhedron with twenty (''icosi'') triangular faces and twelve (''dodeca'') pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 id ...
, the most symmetrical geometry is almost always implied, unless otherwise stated. There are several types of highly symmetric polyhedron, classified by which kind of element – faces, edges, or vertices – belong to a single symmetry orbit: * Regular: vertex-transitive, edge-transitive and face-transitive. (This implies that every face is the same
regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...
; it also implies that every vertex is regular.) * Quasi-regular: vertex-transitive and edge-transitive (and hence has regular faces) but not face-transitive. A quasi-regular dual is face-transitive and edge-transitive (and hence every vertex is regular) but not vertex-transitive. * Semi-regular: vertex-transitive but not edge-transitive, and every face is a regular polygon. (This is one of several definitions of the term, depending on author. Some definitions overlap with the quasi-regular class.) These polyhedra include the semiregular prisms and
antiprism In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation . Antiprisms are a subclass o ...
s. A semi-regular dual is face-transitive but not vertex-transitive, and every vertex is regular. *
Uniform A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, se ...
: vertex-transitive and every face is a regular polygon, i.e., it is regular, quasi-regular or semi-regular. A uniform dual is face-transitive and has regular vertices, but is not necessarily vertex-transitive. * Isogonal: vertex-transitive. * Isotoxal: edge-transitive. *
Isohedral In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...
: face-transitive. *
Noble A noble is a member of the nobility. Noble may also refer to: Places Antarctica * Noble Glacier, King George Island * Noble Nunatak, Marie Byrd Land * Noble Peak, Wiencke Island * Noble Rocks, Graham Land Australia * Noble Island, Great B ...
: face-transitive and vertex-transitive (but not necessarily edge-transitive). The regular polyhedra are also noble; they are the only noble uniform polyhedra. The duals of noble polyhedra are themselves noble. Some classes of polyhedra have only a single main axis of symmetry. These include the
pyramids A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilate ...
, bipyramids,
trapezohedra In geometry, an trapezohedron, -trapezohedron, -antidipyramid, -antibipyramid, or -deltohedron is the dual polyhedron of an antiprism. The faces of an are congruent and symmetrically staggered; they are called ''twisted kites''. With a hi ...
, cupolae, as well as the semiregular prisms and antiprisms.


Regular polyhedra

Regular polyhedra are the most highly symmetrical. Altogether there are nine regular polyhedra: five convex and four star polyhedra. The five convex examples have been known since antiquity and are called the
Platonic solids In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
. These are the triangular pyramid or
tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
,
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...
,
octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
,
dodecahedron In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
and
icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
: There are also four regular star polyhedra, known as the Kepler–Poinsot polyhedra after their discoverers. The dual of a regular polyhedron is also regular.


Uniform polyhedra and their duals

Uniform polyhedra are vertex-transitive and every face is a
regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...
. They may be subdivided into the Regular polyhedron, regular, Quasiregular polyhedron, quasi-regular, or Semiregular polyhedron, semi-regular, and may be convex or starry. The duals of the uniform polyhedra have irregular faces but are face-transitive, and every
vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
is a regular polygon. A uniform polyhedron has the same symmetry orbits as its dual, with the faces and vertices simply swapped over. The duals of the convex Archimedean polyhedra are sometimes called the
Catalan solid In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865. The Catalan sol ...
s. The uniform polyhedra and their duals are traditionally classified according to their degree of symmetry, and whether they are Convex polyhedron, convex or not.


Isohedra

An isohedron is a polyhedron with symmetries acting transitively on its faces. Their topology can be represented by a face configuration. All 5
Platonic solids In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
and 13
Catalan solid In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865. The Catalan sol ...
s are isohedra, as well as the infinite families of
trapezohedra In geometry, an trapezohedron, -trapezohedron, -antidipyramid, -antibipyramid, or -deltohedron is the dual polyhedron of an antiprism. The faces of an are congruent and symmetrically staggered; they are called ''twisted kites''. With a hi ...
and bipyramids. Some isohedra allow geometric variations including concave and self-intersecting forms.


Symmetry groups

Many of the symmetries or point groups in three dimensions are named after polyhedra having the associated symmetry. These include: * T – chiral tetrahedral symmetry; the rotation group for a regular
tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
; order 12. * Td – full tetrahedral symmetry; the symmetry group for a regular
tetrahedron In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
; order 24. * Th – tetrahedral symmetry, pyritohedral symmetry; the symmetry of a pyritohedron; order 24. * O – chiral octahedral symmetry;the rotation group of the Cube (geometry), cube and
octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
; order 24. * Oh – full octahedral symmetry; the symmetry group of the Cube (geometry), cube and
octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
; order 48. * I – chiral icosahedral symmetry; the rotation group of the
icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
and the
dodecahedron In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
; order 60. * Ih – full icosahedral symmetry; the symmetry group of the
icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
and the
dodecahedron In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
; order 120. * Cnv – Cyclic symmetries, ''n''-fold pyramidal symmetry * Dnh – Dihedral symmetry, ''n''-fold prismatic symmetry * Dnv – Dihedral symmetry, ''n''-fold antiprismatic symmetry. Those with Chirality (mathematics), chiral symmetry do not have reflection symmetry and hence have two enantiomorphous forms which are reflections of each other. Examples include the snub cuboctahedron and snub icosidodecahedron.


Other important families of polyhedra


Polyhedra with regular faces

Besides the regular and uniform polyhedra, there are some other classes which have regular faces but lower overall symmetry.


Equal regular faces

Convex polyhedra where every face is the same kind of regular polygon may be found among three families: * Triangles: These polyhedra are called deltahedra. There are eight convex deltahedra: three of the Platonic solids and five non-uniform examples. * Squares: The cube is the only convex example. Other examples (the polycubes) can be obtained by joining cubes together, although care must be taken if coplanar faces are to be avoided. * Pentagons: The regular dodecahedron is the only convex example. Polyhedra with congruent regular faces of six or more sides are all non-convex. The total number of convex polyhedra with equal regular faces is thus ten: the five Platonic solids and the five non-uniform deltahedra. There are infinitely many non-convex examples. Infinite sponge-like examples called skew apeirohedron, infinite skew polyhedra exist in some of these families.


Johnson solids

Norman Johnson (mathematician), Norman Johnson sought which convex non-uniform polyhedra had regular faces, although not necessarily all alike. In 1966, he published a list of 92 such solids, gave them names and numbers, and conjectured that there were no others. Victor Zalgaller proved in 1969 that the list of these
Johnson solid In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that isohedral, each face must be the same polygon, or that the same polygons join around each Vertex (geometry), ver ...
s was complete.


Pyramids

Pyramids include some of the most time-honoured and famous of all polyhedra, such as the four-sided Egyptian pyramids.


Stellations and facettings

Stellation of a polyhedron is the process of extending the faces (within their planes) so that they meet to form a new polyhedron. Faceting is the process of removing parts of a polyhedron to create new faces, or facets, without creating any new vertices.Bridge, N.J. Facetting the dodecahedron, ''Acta crystallographica'' A30 (1974), pp. 548–552. A facet of a polyhedron is any polygon whose corners are vertices of the polyhedron, and is not a ''Face (geometry), face''. Stellation and faceting are inverse or reciprocal processes: the dual of some stellation is a faceting of the dual to the original polyhedron.


Zonohedra

A zonohedron is a convex polyhedron in which every face is a
polygon In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
that is symmetric under rotations through 180°. Zonohedra can also be characterized as the Minkowski sums of line segments, and include several important space-filling polyhedra.


Space-filling polyhedra

A space-filling polyhedron packs with copies of itself to fill space. Such a close-packing or space-filling is often called a tessellation of space or a honeycomb. Space-filling polyhedra must have a #Dehn invariant, Dehn invariant equal to zero. Some honeycombs involve more than one kind of polyhedron.


Lattice polyhedra

A convex polyhedron in which all vertices have integer coordinates is called a convex lattice polytope, lattice polyhedron or integral polyhedron. The Ehrhart polynomial of a lattice polyhedron counts how many points with integer coordinates lie within a scaled copy of the polyhedron, as a function of the scale factor. The study of these polynomials lies at the intersection of combinatorics and commutative algebra. There is a far-reaching equivalence between lattice polyhedra and certain Algebraic variety, algebraic varieties called Toric variety, toric varieties. This was used by Stanley to prove the Dehn–Sommerville equations for Simplicial polytope, simplicial polytopes.


Flexible polyhedra

It is possible for some polyhedra to change their overall shape, while keeping the shapes of their faces the same, by varying the angles of their edges. A polyhedron that can do this is called a flexible polyhedron. By Cauchy's theorem (geometry), Cauchy's rigidity theorem, flexible polyhedra must be non-convex. The volume of a flexible polyhedron must remain constant as it flexes; this result is known as the bellows theorem.


Compounds

A polyhedral compound is made of two or more polyhedra sharing a common centre. Symmetrical compounds often share the same vertices as other well-known polyhedra and may often also be formed by stellation. Some are listed in the list of Wenninger polyhedron models.


Orthogonal polyhedra

An orthogonal polyhedron is one all of whose faces meet at right angles, and all of whose edges are parallel to axes of a Cartesian coordinate system. (Jessen's icosahedron provides an example of a polyhedron meeting one but not both of these two conditions.) Aside from the rectangular cuboids, orthogonal polyhedra are nonconvex. They are the 3D analogs of 2D orthogonal polygons, also known as rectilinear polygons. Orthogonal polyhedra are used in computational geometry, where their constrained structure has enabled advances on problems unsolved for arbitrary polyhedra, for example, unfolding the surface of a polyhedron to a polygonal net. Polycubes are a special case of orthogonal polyhedra that can be decomposed into identical cubes, and are three-dimensional analogues of planar polyominoes.


Embedded regular maps with planar faces

Regular maps are flag transitive abstract 2-manifolds and they have been studied already in the nineteenth century. Some of them have 3-dimensional polyhedral embeddings like the one that represents Klein's quartic.


Canonical polyhedra

Every convex polyhedron is combinatorially equivalent to an essentially unique Midsphere, canonical polyhedron, a polyhedron which has a midsphere tangent to each of its edges.


Generalisations of polyhedra

The name 'polyhedron' has come to be used for a variety of objects having similar structural properties to traditional polyhedra.


Apeirohedra

A classical polyhedral surface has a finite number of faces, joined in pairs along edges. The apeirohedron, apeirohedra form a related class of objects with infinitely many faces. Examples of apeirohedra include: * tilings or tessellations of the plane, and * sponge-like structures called skew apeirohedron, infinite skew polyhedra.


Complex polyhedra

There are objects called complex polyhedra, for which the underlying space is a complex number, complex Hilbert space rather than real Euclidean space. Precise definitions exist only for the regular complex polyhedra, whose symmetry groups are complex reflection groups. The complex polyhedra are mathematically more closely related to configuration (polytope), configurations than to real polyhedra.


Curved polyhedra

Some fields of study allow polyhedra to have curved faces and edges. Curved faces can allow digonal faces to exist with a positive area.


Spherical polyhedra

When the surface of a sphere is divided by finitely many great arcs (equivalently, by planes passing through the center of the sphere), the result is called a spherical polyhedron. Many convex polytopes having some degree of symmetry (for example, all the Platonic solids) can be projected onto the surface of a concentric sphere to produce a spherical polyhedron. However, the reverse process is not always possible; some spherical polyhedra (such as the hosohedron, hosohedra) have no flat-faced analogue.


Curved spacefilling polyhedra

If faces are allowed to be concave as well as convex, adjacent faces may be made to meet together with no gap. Some of these curved polyhedra can pack together to fill space. Two important types are: * Bubbles in froths and foams, such as Weaire–Phelan structure, Weaire-Phelan bubbles. * Forms used in architecture.


Ideal polyhedra

Convex polyhedra can be defined in three-dimensional hyperbolic space in the same way as in Euclidean space, as the
convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
s of finite sets of points. However, in hyperbolic space, it is also possible to consider ideal points as well as the points that lie within the space. An ideal polyhedron is the convex hull of a finite set of ideal points. Its faces are ideal polygons, but its edges are defined by entire hyperbolic lines rather than line segments, and its vertices (the ideal points of which it is the convex hull) do not lie within the hyperbolic space.


Skeletons and polyhedra as graphs

By forgetting the face structure, any polyhedron gives rise to a Graph (discrete mathematics), graph, called its n-skeleton, skeleton, with corresponding vertices and edges. Such figures have a long history: Leonardo da Vinci devised frame models of the regular solids, which he drew for Pacioli's book ''Divina Proportione'', and similar Wire-frame model, wire-frame polyhedra appear in M.C. Escher's print Stars (M. C. Escher), ''Stars''. One highlight of this approach is Steinitz's theorem, which gives a purely graph-theoretic characterization of the skeletons of convex polyhedra: it states that the skeleton of every convex polyhedron is a vertex connectivity, 3-connected planar graph, and every 3-connected planar graph is the skeleton of some convex polyhedron. An early idea of #Abstract polyhedra, abstract polyhedra was developed in
Branko Grünbaum Branko Grünbaum ( he, ברנקו גרונבאום; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descent4), and various symmetry restrictions on polyhedra give rise to skeletons that are symmetric graphs.


Alternative usages

From the latter half of the twentieth century, various mathematical constructs have been found to have properties also present in traditional polyhedra. Rather than confining the term "polyhedron" to describe a three-dimensional polytope, it has been adopted to describe various related but distinct kinds of structure.


Higher-dimensional polyhedra

A polyhedron has been defined as a set of points in real number, real affine space, affine (or Euclidean space, Euclidean) space of any dimension ''n'' that has flat sides. It may alternatively be defined as the intersection of finitely many half-spaces. Unlike a conventional polyhedron, it may be bounded or unbounded. In this meaning, a
polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...
is a bounded polyhedron. Analytically, such a convex polyhedron is expressed as the solution set for a system of linear inequalities. Defining polyhedra in this way provides a geometric perspective for problems in linear programming. Many traditional polyhedral forms are polyhedra in this sense. Other examples include: * A quadrant in the plane. For instance, the region of the cartesian plane consisting of all points above the horizontal axis and to the right of the vertical axis: . Its sides are the two positive axes, and it is otherwise unbounded. * An octant in Euclidean 3-space, . * A prism of infinite extent. For instance a doubly infinite square prism in 3-space, consisting of a square in the ''xy''-plane swept along the ''z''-axis: . * Each cell (geometry), cell in a Voronoi diagram, Voronoi tessellation is a convex polyhedron. In the Voronoi tessellation of a set ''S'', the cell ''A'' corresponding to a point is bounded (hence a traditional polyhedron) when ''c'' lies in the interior (topology), interior of the
convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...
of ''S'', and otherwise (when ''c'' lies on the Boundary (topology), boundary of the convex hull of ''S'') ''A'' is unbounded.


Topological polyhedra

A topological polytope 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. Such a figure is called ''simplicial'' if each of its regions is a simplex, i.e. in an ''n''-dimensional space each region has ''n''+1 vertices. The dual of a simplicial polytope is called ''simple''. Similarly, a widely studied class of polytopes (polyhedra) is that of cubical polyhedra, when the basic building block is an ''n''-dimensional cube.


Abstract polyhedra

An abstract polytope is a
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
(poset) of elements whose partial ordering obeys certain rules of incidence (connectivity) and ranking. The elements of the set correspond to the vertices, edges, faces and so on of the polytope: vertices have rank 0, edges rank 1, etc. with the partially ordered ranking corresponding to the dimensionality of the geometric elements. The empty set, required by set theory, has a rank of −1 and is sometimes said to correspond to the null polytope. An abstract polyhedron is an abstract polytope having the following ranking: * rank 3: The maximal element, sometimes identified with the body. * rank 2: The polygonal faces. * rank 1: The edges. * rank 0: the vertices. * rank −1: The empty set, sometimes identified with the ''null polytope'' or ''nullitope''. Any geometric polyhedron is then said to be a "realization" in real space of the abstract poset as described above.


History


Ancient

;Prehistory Polyhedra appeared in early architecture, architectural forms such as cubes and cuboids, with the earliest four-sided pyramids of ancient Egypt also dating from the Stone Age. The Etruscan civilization, Etruscans preceded the Greeks in their awareness of at least some of the regular polyhedra, as evidenced by the discovery of an Etruscan civilization, Etruscan
dodecahedron In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
made of soapstone on Monte Loffa. Its faces were marked with different designs, suggesting to some scholars that it may have been used as a gaming die. ;Greek civilisation The earliest known ''written'' records of these shapes come from Classical Ancient Greece, Greek authors, who also gave the first known mathematical description of them. The earlier Greeks were interested primarily in the Regular polyhedron#History, convex regular polyhedra, which came to be known as the
Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
s. Pythagoras knew at least three of them, and Theaetetus (mathematician), Theaetetus (circa 417 B. C.) described all five. Eventually, Euclid described their construction in his ''Euclid's Elements, Elements''. Later, Archimedes expanded his study to the Uniform polyhedron, convex uniform polyhedra which now bear his name. His original work is lost and his solids come down to us through Pappus of Alexandria, Pappus. ;China Cubical gaming dice in China have been dated back as early as 600 B.C. By 236 AD, Liu Hui was describing the dissection of the cube into its characteristic tetrahedron (orthoscheme) and related solids, using assemblages of these solids as the basis for calculating volumes of earth to be moved during engineering excavations. ;Islamic civilisation After the end of the Classical era, scholars in the Islamic civilisation continued to take the Greek knowledge forward (see Mathematics in medieval Islam). The 9th century scholar Thabit ibn Qurra gave formulae for calculating the volumes of polyhedra such as truncated pyramids. Then in the 10th century Abūl Wafā' Būzjānī, Abu'l Wafa described the convex regular and quasiregular spherical polyhedra.


Renaissance

As with other areas of Greek thought maintained and enhanced by Islamic scholars, Western interest in polyhedra revived during the Italian Renaissance. Artists constructed skeletal polyhedra, depicting them from life as a part of their investigations into Perspective (graphical), perspective. Several appear in marquetry panels of the period. Piero della Francesca gave the first written description of direct geometrical construction of such perspective views of polyhedra. Leonardo da Vinci made skeletal models of several polyhedra and drew illustrations of them for a book by Pacioli. A painting by an anonymous artist of Pacioli and a pupil depicts a glass rhombicuboctahedron half-filled with water. As the Renaissance spread beyond Italy, later artists such as Wenzel Jamnitzer, Dürer and others also depicted polyhedra of various kinds, many of them novel, in imaginative etchings.


Star polyhedra

For almost 2,000 years, the concept of a polyhedron as a convex solid had remained as developed by the ancient Greek mathematicians. During the Renaissance star forms were discovered. A marble tarsia in the floor of St. Mark's Basilica, Venice, depicts a stellated dodecahedron. Artists such as Wenzel Jamnitzer delighted in depicting novel star-like forms of increasing complexity. Johannes Kepler (1571–1630) used star polygons, typically pentagrams, to build star polyhedra. Some of these figures may have been discovered before Kepler's time, but he was the first to recognize that they could be considered "regular" if one removed the restriction that regular polyhedra must be convex. Later, Louis Poinsot realised that star
vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
s (circuits around each corner) can also be used, and discovered the remaining two regular star polyhedra. Cauchy proved Poinsot's list complete, and Cayley gave them their accepted English names: (Kepler's) the small stellated dodecahedron and great stellated dodecahedron, and (Poinsot's) the great icosahedron and great dodecahedron. Collectively they are called the Kepler–Poinsot polyhedra. The Kepler–Poinsot polyhedra may be constructed from the Platonic solids by a process called stellation. Most stellations are not regular. The study of stellations of the Platonic solids was given a big push by H.S.M. Coxeter and others in 1938, with the now famous paper ''The 59 icosahedra''. The reciprocal process to stellation is called facetting (or faceting). Every stellation of one polytope is Dual polyhedron, dual, or reciprocal, to some facetting of the dual polytope. The regular star polyhedra can also be obtained by facetting the Platonic solids. listed the simpler facettings of the dodecahedron, and reciprocated them to discover a stellation of the icosahedron that was missing from the set of "59".. More have been discovered since, and the story is not yet ended.


Euler's formula and topology

Two other modern mathematical developments had a profound effect on polyhedron theory. In 1750 Leonhard Euler for the first time considered the edges of a polyhedron, allowing him to discover his polyhedron formula relating the number of vertices, edges and faces. This signalled the birth of topology, sometimes referred to as "rubber sheet geometry", and Henri Poincaré developed its core ideas around the end of the nineteenth century. This allowed many longstanding issues over what was or was not a polyhedron to be resolved. Max Brückner summarised work on polyhedra to date, including many findings of his own, in his book "Vielecke und Vielflache: Theorie und Geschichte" (Polygons and polyhedra: Theory and History). Published in German in 1900, it remained little known. Meanwhile, the discovery of higher dimensions led to the idea of a polyhedron as a three-dimensional example of the more general polytope.


Twentieth-century revival

By the early years of the twentieth century, mathematicians had moved on and geometry was little studied. Coxeter's analysis in ''The Fifty-Nine Icosahedra'' introduced modern ideas from graph theory and combinatorics into the study of polyhedra, signalling a rebirth of interest in geometry. Coxeter himself went on to enumerate the star uniform polyhedra for the first time, to treat tilings of the plane as polyhedra, to discover the regular skew polyhedron, regular skew polyhedra and to develop the theory of complex polytope, complex polyhedra first discovered by Shephard in 1952, as well as making fundamental contributions to many other areas of geometry. In the second part of the twentieth century, Grünbaum published important works in two areas. One was in convex polytopes, where he noted a tendency among mathematicians to define a "polyhedron" in different and sometimes incompatible ways to suit the needs of the moment. The other was a series of papers broadening the accepted definition of a polyhedron, for example discovering many new Regular polyhedron#History, regular polyhedra. At the close of the 20th century these latter ideas merged with other work on incidence complexes to create the modern idea of an abstract polyhedron (as an abstract 3-polytope), notably presented by McMullen and Schulte.


In nature

For natural occurrences of regular polyhedra, see . Irregular polyhedra appear in nature as crystals.


See also

* defect (geometry), Defect * Deltohedron * Extension of a polyhedron * Goldberg polyhedron * List of books about polyhedra * List of small polyhedra by vertex count * Near-miss Johnson solid * Polyhedron models * Schlegel diagram * Spidron * Lists of shapes * Stella (software)


References


Notes


Sources

* . * . * . * .


External links


General theory

*
Polyhedra Pages

Uniform Solution for Uniform Polyhedra by Dr. Zvi Har'El



Lists and databases of polyhedra



– The Encyclopedia of Polyhedra.

– Contains a peer reviewed selection of polyhedra with unusual properties.

– Virtual polyhedra.
Paper Models of Uniform (and other) PolyhedraPolyhedra Viewer
– Web-based tool for visualizing the relationships between the convex, regular-faced polyhedra.


Free software



– An interactive and free collection of polyhedra in Java. Features includes nets, planar sections, duals, truncations and stellations of more than 300 polyhedra.

– Explorer java applet, includes a variety of 3d viewer options.
openSCAD
– Free cross-platform software for programmers. Polyhedra are just one of the things you can model. The b:OpenSCAD User Manual, openSCAD User Manual is also available.
OpenVolumeMesh
– An open source cross-platform C++ library for handling polyhedral meshes. Developed by the Aachen Computer Graphics Group, RWTH Aachen University.
Polyhedronisme
– Web-based tool for generating polyhedra models using Conway polyhedron notation, Conway Polyhedron Notation. Models can be exported as 2D PNG images, or as 3D OBJ or VRML2 files. The 3D files can be opened in CAD software, or uploaded for 3D printing at services such a
Shapeways


Resources for making physical models


Paper Models of Polyhedra
Free nets of polyhedra.
Simple instructions for building over 30 paper polyhedra


– Polyhedra models constructed without use of glue.
Adopt a Polyhedron
- Interactive display, nets and 3D printer data for all combinatorial types of polyhedra with up to nine vertices. {{Authority control Polyhedra,