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In
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, a polyhedron (: polyhedra or polyhedrons; ) is a
three-dimensional figure Solid geometry or stereometry is the geometry of three-dimensional Euclidean space (3D space). A solid figure is the region of 3D space bounded by a two-dimensional closed surface; for example, a solid ball consists of a sphere and its inte ...
with flat
polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
al faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a
solid figure Solid geometry or stereometry is the geometry of three-dimensional Euclidean space (3D space). A solid figure is the region of 3D space bounded by a two-dimensional closed surface; for example, a solid ball consists of a sphere and its inte ...
or to its boundary
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
. The terms solid polyhedron and polyhedral surface are commonly used to distinguish the two concepts. Also, the term ''polyhedron'' is often used to refer implicitly to the whole
structure A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...
formed by a solid polyhedron, its polyhedral surface, its faces, its edges, and its vertices. There are many definitions of polyhedron. Nevertheless, the polyhedron is typically understood as a generalization of a two-dimensional
polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
and a three-dimensional specialization 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
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
s. Polyhedra have several general characteristics that include the number of faces, topological classification by
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's ...
, duality,
vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
s,
surface area The surface area (symbol ''A'') of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the d ...
,
volume Volume is a measure of regions in 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) ...
, interior lines, Dehn invariant, and
symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
. The symmetry of a polyhedron means that the polyhedron's appearance is unchanged by the transformation such as rotating and reflecting. The ''convex polyhedron'' is well-defined with several equivalent standard definitions, one of which is a polyhedron that is a
convex set In geometry, a set of points is convex if it contains every line segment between two points in the set. For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is n ...
, or the polyhedral surface that bounds it. Every convex polyhedron is the
convex hull In geometry, the convex hull, 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 its vertices, and the convex hull of a finite set of points is a polyhedron. There are many families of convex polyhedra, and the most common examples are
cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
and the family of
pyramids A pyramid () is a Nonbuilding structure, structure whose visible surfaces are triangular in broad outline and converge toward the top, making the appearance roughly a Pyramid (geometry), pyramid in the geometric sense. The base of a pyramid ca ...
.


Definition

Convex polyhedra In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary surfa ...
are well-defined, with several equivalent standard definitions. However, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of "polyhedron" have been given within particular contexts,. some more rigorous than others, and there is no universal agreement over which of these to choose. Some of these definitions exclude shapes that have often been counted as polyhedra (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 (; 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 (two-dimensional
polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
s), and that it sometimes can be said to have a particular three-dimensional interior
volume Volume is a measure of regions in 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) ...
. 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 Intersection (Euclidean geometry), intersect itself and has no holes. That is, it is a Piecewise linear curve, piecewise-linear Jordan curve consisting of finitely many line segments. The ...
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.; for definitions of polyhedra, see pp. 206–209; for polyhedra with equal regular faces, see p. 86. 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 ...
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 or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
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 angles 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 (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...
'' 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 partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...
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 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 In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: Vertex (geometry), vertices, Edge (geo ...
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.


General characteristics


Number of faces

Polyhedra may be classified and are often named according to the number of faces. The naming system is based on Classical Greek, and combines a prefix counting the faces with the suffix "hedron", meaning "base" or "seat" and referring to the faces. For example a
tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...
is a polyhedron with four faces, a pentahedron is a polyhedron with five faces, a
hexahedron A hexahedron (: hexahedra or hexahedrons) or sexahedron (: 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. There are seven ...
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 (eight-sided polyhedra), dodecahedra (twelve-sided polyhedra), and icosahedra (twenty-sided polyhedra) are sometimes used without additional qualification to refer to the
Platonic solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...
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 paper model can each be given a different colour (although the inside colour will be hidden from view). These polyhedra are orientable. The same is true for non-convex polyhedra without self-crossings. Some non-convex self-crossing polyhedra can be coloured in the same way but have regions turned "inside out" so that both colours appear on the outside in different places; these are still considered to be orientable. However, for some other self-crossing polyhedra with simple-polygon faces, such as the tetrahemihexahedron, it is not possible to colour the two sides of each face with two different colours so that adjacent faces have consistent colours. In this case the polyhedron is said to be non-orientable. For polyhedra with self-crossing faces, it may not be clear what it means for adjacent faces to be consistently coloured, but for these polyhedra it is still possible to determine whether it is orientable or non-orientable by considering a topological
cell complex In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generali ...
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's ...
, 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 the surface of 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 characteristics 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 mathematics, the Klein bottle () is an example of a Orientability, non-orientable Surface (topology), surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the ...
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 topology (Mathematics), topological Genus (mathematics), genus () of 1 or greater. Notable examples include the Császár polyhedron, Császár a ...
is a polyhedron whose Euler characteristic is less than or equal to 0, or equivalently whose
genus Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...
is 1 or greater. Topologically, the surfaces of such polyhedra are
torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
surfaces having one or more holes through the middle. One of the notable example is the Szilassi polyhedron, which has the geometrically ralizes the Heawood map.


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. 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 partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable ...
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 general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
, 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 vertex. For the
Platonic solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...
s and other highly-symmetric polyhedra, this slice may be chosen to pass through the midpoints of each edge incident to the vertex, but other polyhedra may not have a plane through these points. For convex polyhedra, and more generally for polyhedra whose vertices are in convex position, this slice can be chosen as any plane separating the vertex from the other vertices. When the polyhedron has a center of symmetry, it is standard to choose this plane to be perpendicular to the line through the given vertex and the center; with this choice, the shape of the vertex figure is determined up to scaling. When the vertices of a polyhedron are not in convex position, there will not always be a plane separating each vertex from the rest. In this case, it is common instead to slice the polyhedron by a small sphere centered at the vertex. Again, this produces a shape for the vertex figure that is invariant up to scaling. All of these choices lead to vertex figures with the same combinatorial structure, for the polyhedra to which they can be applied, but they may give them different geometric shapes.


Surface area and lines inside polyhedra

The
surface area The surface area (symbol ''A'') of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the d ...
of a polyhedron is the sum of the 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 locally 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 conn ...
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 Alexandrov's theorem on polyhedra 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 othe ...
, every convex polyhedron is uniquely determined by the
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
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. When segment lines connect two vertices that are not in the same face, they form the diagonal lines. However, not all polyhedra have diagonal lines, as in the family of
pyramids A pyramid () is a Nonbuilding structure, structure whose visible surfaces are triangular in broad outline and converge toward the top, making the appearance roughly a Pyramid (geometry), pyramid in the geometric sense. The base of a pyramid ca ...
, Schönhardt polyhedron in which three diagonal lines lies entirely outside of it, and Császár polyhedron has no diagonal lines (rather, every pair of vertices is connected by an edge).


Volume

Polyhedral solids have an associated quantity called
volume Volume is a measure of regions in 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) ...
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. Three equiva ...
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. The 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 m ...
). For example, the volume of a Platonic solid can be computed by dividing it into congruent
pyramids A pyramid () is a Nonbuilding structure, structure whose visible surfaces are triangular in broad outline and converge toward the top, making the appearance roughly a Pyramid (geometry), pyramid in the geometric sense. The base of a pyramid ca ...
, 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 relating the '' flux'' of a vector field through a closed surface to the ''divergence'' of the field in the volume ...
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 F of the polyhedron, Q_F is an arbitrary point on face F , N_F is the
unit vector In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumfle ...
perpendicular to F 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 (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. N ...
. 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 Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algo ...
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 problems, 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 polyhedron, polyhedra of equal volume, is it always possible t ...
.
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. Dehn's early life and career took place in Germany. However, he was forced to retire in 1 ...
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, 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 * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...
in an infinite-dimensional vector space, determined from the lengths and dihedral angles of a polyhedron's edges. Another of Hilbert's problems, Hilbert's eighteenth 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 by the strong bellows theorem, which states that the Dehn invariant of any flexible polyhedron remains invariant as it flexes.


Symmetries

Many of the most studied polyhedra are highly
symmetrical Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
. Their appearance is unchanged by some reflection by plane or rotation around the
axes Axes, plural of ''axe'' and of ''axis'', may refer to * ''Axes'' (album), a 2005 rock album by the British band Electrelane * a possibly still empty plot (graphics) See also * Axis (disambiguation) An axis (: axes) may refer to: Mathematics ...
passing through two opposite vertices, edges, or faces in space. Each symmetry may change the location of a given element, but the set of all vertices (likewise faces and 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 amb ...
.


By elements of polyhedron

All the elements (vertex, face, and edge) that can be superimposed on each other by symmetries are said to form a symmetry orbit. If these elements lie in the same orbit, the figure may be transitive on the orbit. Individually, they are isohedral (or face-transitive, meaning the symmetry transformations involve the polyhedra's faces in orbit), See p. 247. isotoxal (or edge-transitive, which involves the edge's polyhedra), and isogonal (or vertex-transitive, which involves the polyhedra's vertices). For example, the
cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
in which all the faces are in one orbit and involving the rotation and reflections in the orbit remains unchanged in its appearance; hence, the cube is face-transitive. The cube also has the other two such symmetries. When three such symmetries belong to a polyhedron, it is known as a
regular polyhedron A regular polyhedron is a polyhedron whose symmetry group acts transitive group action, transitively on its Flag (geometry), flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In ...
. There are nine regular polyhedra: five
Platonic solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...
s (cube,
octahedron In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of i ...
,
icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical tha ...
,
tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...
, and
dodecahedron In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Po ...
—all of which have regular polygonal faces) and four Kepler–Poinsot polyhedrons. Nevertheless, some polyhedrons may not possess one or two of those symmetries: * A polyhedron with vertex-transitive and edge-transitive is said to be a quasiregular, although they have regular faces, and its dual is face-transitive and edge-transitive. * A vertex- but not edge-transitive polyhedron with regular polygonal faces is said to be a semiregular. and such polyhedrons are prisms and
antiprism In geometry, an antiprism or is a polyhedron composed of two Parallel (geometry), parallel Euclidean group, direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway po ...
s. Its dual is face-transitive but not vertex-transitive, and every vertex is regular. * A polyhedron with regular polygonal faces and vertex-transitive is said to be
uniform A uniform is a variety of costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ...
. This class may be subdivided into a regular, quasi-regular, or semi-regular polyhedron, and may be convex or starry. The dual is face-transitive and has regular vertices but is not necessarily vertex-transitive. The uniform polyhedra and their duals are traditionally classified according to their degree of symmetry, and whether they are
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
or not. * A face- and vertex-transitive (but not necessarily edge-transitive) polyhedra is said to be
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, Gr ...
. The regular polyhedra are also noble; they are the only noble uniform polyhedra. The duals of noble polyhedra are themselves noble. Some polyhedra may have no
reflection symmetry In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a Reflection (mathematics), reflection. That is, a figure which does not change upon undergoing a reflection has reflecti ...
such that they have two enantiomorph forms, which are reflections of each other. Such symmetry is known for having
chirality Chirality () is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable fro ...
. Examples include the snub cuboctahedron and snub icosidodecahedron.


By point group in three dimensions

The point group of polyhedra means a mathematical group endowed with its
symmetry operation In mathematics, a symmetry operation is a geometric transformation of an object that leaves the object looking the same after it has been carried out. For example, a turn rotation of a regular triangle about its center (geometry), center, a refle ...
s so that the appearance of polyhedra remains preserved while transforming in three-dimensional space. The indicated transformation here includes the rotation around the axes, reflection through the plane, inversion through a center point, and a combination of these three. The polyhedral group is the symmetry group originally derived from the three Platonic solids: tetrahedron, octahedron, and icosahedron. These three have point groups respectively known as
tetrahedral symmetry image:tetrahedron.svg, 150px, A regular tetrahedron, an example of a solid with full tetrahedral symmetry A regular tetrahedron has 12 rotational (or orientation-preserving) symmetries, and a symmetry order of 24 including transformations that co ...
,
octahedral symmetry A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedr ...
, and
icosahedral symmetry In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual polyhedr ...
. Each of these focuses on the rotation group of polyhedra, known as the ''chiral polyhedral group'', whereas the additional reflection symmetry is known as the ''full polyhedral group''. One point group, pyritohedral symmetry, includes the rotation of tetrahedral symmetry and additionally has three planes of reflection symmetry and some
rotoreflection In geometry, an improper rotation. (also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion) is an isometry in Euclidean space that is a combination of a Rotation (geometry), rotation about an axis and a reflection ( ...
s. Overall, the mentioned polyhedral groups are summarized in the following bullets: * chiral tetrahedral symmetry \mathrm , the rotation group for a regular tetrahedron and has the order of twelve. * full tetrahedral symmetry \mathrm_\mathrm , the symmetry group for a regular tetrahedron and has the order of twenty-four. * pyritohedral symmetry \mathrm_\mathrm , the symmetry of a pyritohedron and has the order of twenty-four. * chiral octahedral symmetry \mathrm , the rotation group of both cube and regular octahedron and has the order twenty-four. * full octahedral symmetry \mathrm_\mathrm , the symmetry group of both cube and regular octahedron and has order forty-eight. * chiral icosahedral symmetry \mathrm , the rotation group of both regular icosahedron and regular dodecahedron and has the order of sixty. * full icosahedral symmetry \mathrm_\mathrm , the symmetry group of both regular icosahedron and regular dodecahedron and has the order of a hundred-twenty. Point groups in three dimensions may also allow the preservation of polyhedra's appearance by the circulation around an axis. There are three various of these point groups: * pyramidal symmetry C_ , allowing rotate the axis passing through the
apex The apex is the highest point of something. The word may also refer to: Arts and media Fictional entities * Apex (comics) A-Bomb Abomination Absorbing Man Abraxas Abyss Abyss is the name of two characters appearing in Ameri ...
and its base, as well as reflection relative to perpendicular planes passing through the bisector of a base. This point group symmetry can be found in pyramids,
cupola In architecture, a cupola () is a relatively small, usually dome-like structure on top of a building often crowning a larger roof or dome. Cupolas often serve as a roof lantern to admit light and air or as a lookout. The word derives, via Ital ...
s, and rotundas. *
prismatic symmetry In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as an abstract group is a dihedral group Dih''n'' (for ''n'' ≥ 2). Types Ther ...
D_ , similar to the pyramidal symmetry, but with additional transformation by reflecting it across a horizontal plane. This may be achieved from the family of prisms and its dual
bipyramid In geometry, a bipyramid, dipyramid, or double pyramid is a polyhedron formed by fusing two Pyramid (geometry), pyramids together base (geometry), base-to-base. The polygonal base of each pyramid must therefore be the same, and unless otherwise ...
s. * antiprismatic symmetry D_ , which preserves the symmetry by rotating its half bottom and reflection across the horizontal plane. Examples can be found in antiprisms. A point group C_ consists of rotating around the axis of symmetry and reflection on the horizontal plane. In the case of n = 1 , the symmetry group only preserves the symmetry by a full rotation solely, ordinarily denoting C_s . Polyhedra may have rotation only to preserve the symmetry, and the symmetry group may be considered as the
cyclic group In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...
C_n . Polyhedra with the rotoreflection and the rotation by the cyclic group is the point group S_n .


Convex polyhedra

As mentioned above, the convex polyhedra are well-defined, with several equivalent standard definitions. They are often defined as bounded intersections of finitely many half-spaces, or as the
convex hull In geometry, the convex hull, 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, restricted in either case to intersections or hulls that have nonzero volume. Important classes of convex polyhedra include the family of
prismatoid In geometry, a prismatoid is a polyhedron whose vertex (geometry), vertices all lie in two parallel Plane (geometry), planes. Its lateral faces can be trapezoids or triangles. If both planes have the same number of vertices, and the lateral faces ...
s, the
Platonic solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...
s, the
Archimedean solid The Archimedean solids are a set of thirteen convex polyhedra whose faces are regular polygon and are vertex-transitive, although they aren't face-transitive. The solids were named after Archimedes, although he did not claim credit for them. They ...
s and their duals the
Catalan solid The Catalan solids are the dual polyhedron, dual polyhedra of Archimedean solids. The Archimedean solids are thirteen highly-symmetric polyhedra with regular faces and symmetric vertices. The faces of the Catalan solids correspond by duality to ...
s, and the
Johnson solid In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two Solid geometry, s ...
s. Prismatoids are the polyhedra whose vertices lie on two parallel planes and their faces are likely to be trapezoids and triangles.. Examples of prismatoids are
pyramid A pyramid () is a structure whose visible surfaces are triangular in broad outline and converge toward the top, making the appearance roughly a pyramid in the geometric sense. The base of a pyramid can be of any polygon shape, such as trian ...
s, wedges, parallelipipeds, prisms,
antiprism In geometry, an antiprism or is a polyhedron composed of two Parallel (geometry), parallel Euclidean group, direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway po ...
s,
cupola In architecture, a cupola () is a relatively small, usually dome-like structure on top of a building often crowning a larger roof or dome. Cupolas often serve as a roof lantern to admit light and air or as a lookout. The word derives, via Ital ...
s, and
frustum In geometry, a ; (: frusta or frustums) is the portion of a polyhedron, solid (normally a pyramid (geometry), pyramid or a cone (geometry), cone) that lies between two parallel planes cutting the solid. In the case of a pyramid, the base faces a ...
s. Platonic solids are the five ancient polyhedra—
tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...
,
octahedron In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of i ...
,
icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical tha ...
,
cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
, and
dodecahedron In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Po ...
—described by
Plato Plato ( ; Greek language, Greek: , ; born  BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...
in the ''Timaeus''. Archimedean solids are the class of thirteen polyhedra whose faces are all regular polygons and whose vertices are symmetric to each other; their dual polyhedra are the Catalan solids.. Johnson solids are the class of convex polyhedra whose faces are all regular polygons.. These include the convex deltahedra, strictly convex polyhedra whose faces are all equilateral triangles.. Convex polyhedra can be categorized into elementary polyhedra or composite polyhedra. Elementary polyhedra are convex regular-faced polyhedra that cannot be produced into two or more polyhedrons by slicing them with a plane. Quite opposite to composite polyhedra, they can be alternatively defined as polyhedra constructed by attaching more elementary polyhedra. For example, triaugmented triangular prism is composite since it can be constructed by attaching three equilateral square pyramids onto the square faces of a
triangular prism In geometry, a triangular prism or trigonal prism is a Prism (geometry), prism with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a ''right triangular prism''. A right triangul ...
; the square pyramids and the triangular prism are elementaries.. Some convex polyhedra possess a
midsphere In geometry, the midsphere or intersphere of a convex polyhedron is a sphere which is tangent to every Edge (geometry), edge of the polyhedron. Not every polyhedron has a midsphere, but the uniform polyhedron, uniform polyhedra, including the reg ...
, a sphere
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
to each of their edges, which is intermediate in radius between the insphere and
circumsphere In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing, by analogy with the term ''circumcircle' ...
for polyhedra for which all three of these spheres exist. Every convex polyhedron is combinatorially equivalent to a ''canonical polyhedron'', a polyhedron that has a midsphere whose center coincides with the
centroid In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the figure. The same definition extends to any object in n-d ...
of its tangent points with edges. The shape of the canonical polyhedron (but not its scale or position) is uniquely determined by the combinatorial structure of the given polyhedron. By forgetting the face structure, any polyhedron gives rise to a
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...
, called its
skeleton A skeleton is the structural frame that supports the body of most animals. There are several types of skeletons, including the exoskeleton, which is a rigid outer shell that holds up an organism's shape; the endoskeleton, a rigid internal fra ...
, with corresponding vertices and edges. Such figures have a long history:
Leonardo da Vinci Leonardo di ser Piero da Vinci (15 April 1452 - 2 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially rested o ...
devised frame models of the regular solids, which he drew for
Pacioli Luca Bartolomeo de Pacioli, Order of Friars Minor, O.F.M. (sometimes ''Paccioli'' or ''Paciolo''; 1447 – 19 June 1517) was an Italian mathematician, Order of Friars Minor, Franciscan friar, collaborator with Leonardo da Vinci, and an early c ...
's book ''Divina Proportione'', and similar wire-frame polyhedra appear in M.C. Escher's print ''Stars''. One highlight of this approach is
Steinitz's theorem In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedron, convex polyhedra: they are exactly the vertex connect ...
, which gives a purely graph-theoretic characterization of the skeletons of convex polyhedra: it states that the skeleton of every convex polyhedron is a
planar graph In graph theory, a planar graph is a graph (discrete mathematics), graph that can be graph embedding, embedded in the plane (geometry), plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. ...
with three-connected, and every such graph is the skeleton of some convex polyhedron. Prominent ''non-convex polyhedra'' include the star polyhedra. The regular star polyhedra, also known as the Kepler–Poinsot polyhedra, are constructible via
stellation In geometry, stellation is the process of extending a polygon in two dimensions, a polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specific ...
or
faceting Stella octangula as a faceting of the cube In geometry, faceting (also spelled facetting) is the process of removing parts of a polygon, polyhedron or polytope, without creating any new Vertex (geometry), vertices. New edges of a faceted po ...
of regular convex polyhedra. Stellation is the process of extending the faces (within their planes) so that they meet. Faceting is the process of removing parts of a polyhedron to create new faces (or facets) without creating any new vertices).. A facet of a polyhedron is any polygon whose corners are vertices of the polyhedron, and is not a ''
face The face is the front of the 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 affect th ...
''. The stellation and faceting are inverse or reciprocal processes: the dual of some stellation is a faceting of the dual to the original polyhedron.


Other families of 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 equal to zero. Some honeycombs involve more than one kind of polyhedron.


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 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.


Ideal polyhedron

Convex polyhedra can be defined in three-dimensional
hyperbolic space In mathematics, hyperbolic space of dimension ''n'' is the unique simply connected, ''n''-dimensional Riemannian manifold of constant sectional curvature equal to −1. It is homogeneous, and satisfies the stronger property of being a symme ...
in the same way as in Euclidean space, as the
convex hull In geometry, the convex hull, 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 and the points 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.


Lattice polyhedron

Convex polyhedra in which all vertices have integer coordinates are called lattice polyhedra or integral polyhedra. The Ehrhart polynomial of lattice a polyhedron counts how many points with
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
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 Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
and
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...
. An example is Reeve tetrahedron. There is a far-reaching equivalence between lattice polyhedra and certain
algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
called
toric varieties In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be ...
. This was used by Stanley to prove the Dehn–Sommerville equations for simplicial polytopes.


Polyhedral compound

A
polyhedral compound In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common Centroid, centre. They are the three-dimensional analogs of star polygon#Regular compounds, polygonal compounds such as the hexagram. The oute ...
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.


Zonohedron

A
zonohedron In geometry, a zonohedron is a convex polyhedron that is point symmetry, centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski addition, Minkows ...
is a convex polyhedron in which every face is a
polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...
that is symmetric under
rotation Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
s through 180°. Zonohedra can also be characterized as the
Minkowski sum In geometry, the Minkowski sum of two sets of position vectors ''A'' and ''B'' in Euclidean space is formed by adding each vector in ''A'' to each vector in ''B'': A + B = \ The Minkowski difference (also ''Minkowski subtraction'', ''Minkowsk ...
s of line segments, and include several important space-filling polyhedra.


Orthogonal polyhedron

Polyhedra are said to be
orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...
because all of their edges are parallel to the axes of a Cartesian coordinate system. This implies that all faces meet at
right angle In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
s, but this condition is weaker: Jessen's icosahedron has faces meeting at right angles, but does not have axis-parallel edges. Aside from the
rectangular cuboid A rectangular cuboid is a special case of a cuboid with rectangular faces in which all of its dihedral angles are right angles. This shape is also called rectangular parallelepiped or orthogonal parallelepiped. Many writers just call these ...
s, orthogonal polyhedra are nonconvex. They are the three-dimensional analogs of two-dimensional orthogonal polygons, also known as rectilinear polygons. Orthogonal polyhedra are used in computational geometry, where their constrained structure has enabled advances in problems unsolved for arbitrary polyhedra, for example, unfolding the surface of a polyhedron to a polygonal net.
Polycube image:tetracube_categories.svg, upAll 8 one-sided tetracubes – if chirality is ignored, the bottom 2 in grey are considered the same, giving 7 free tetracubes in total image:9L cube puzzle solution.svg, A puzzle involving arranging nine L tricube ...
s are a special case of orthogonal polyhedra that can be decomposed into identical cubes, and are three-dimensional analogues of planar
polyomino A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling. Polyominoes have been used in popu ...
es.


Generalisations

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 apeirohedra form a related class of objects with infinitely many faces. Examples of apeirohedra include: * tilings or
tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ...
s of the plane, and * sponge-like structures called infinite skew polyhedra.


Complex polyhedra

There are objects called complex polyhedra, for which the underlying space is a
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
rather than real Euclidean space. Precise definitions exist only for the regular complex polyhedra, whose symmetry groups are
complex reflection group In mathematics, a complex reflection group is a Group (mathematics), finite group acting on a finite-dimensional vector space, finite-dimensional complex numbers, complex vector space that is generated by complex reflections: non-trivial elements t ...
s. The complex polyhedra are mathematically more closely related to 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. * When the surface of a sphere is divided by finitely many
great arc In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
s (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 hosohedra) have no flat-faced analogue. * 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 bubbles, and forms used in architecture.


Higher-dimensional polyhedra

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. A polyhedron has been defined as a set of points in real
affine Affine may describe any of various topics concerned with connections or affinities. It may refer to: * Affine, a Affinity_(law)#Terminology, relative by marriage in law and anthropology * Affine cipher, a special case of the more general substi ...
(or 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 Linear programming (LP), also called linear optimization, is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements and objective are represented by linear function#As a polynomia ...
.


History


Before the Greeks

Polyhedra appeared in early architectural forms such as cubes and cuboids, with the earliest four-sided
Egyptian pyramids The Egyptian pyramids are ancient masonry structures located in Egypt. Most were built as tombs for the pharaohs and their consorts during the Old Kingdom of Egypt, Old and Middle Kingdom of Egypt, Middle Kingdom periods. At least 138 identi ...
dating from the
27th century BC The 27th century BC was a century that lasted from the year 2700 BC to 2601 BC. Events *c. 2750–2600 BC: Early Dynastic II Period in Mesopotamia. *c. 2700 BC: The beginning of statuary in Egypt, with shale and limestone statues of Khasekhemw ...
. The
Moscow Mathematical Papyrus The Moscow Mathematical Papyrus, also named the Golenishchev Mathematical Papyrus after its first non-Egyptian owner, Egyptologist Vladimir Golenishchev, is an ancient Egyptian mathematical papyrus containing several problems in arithmetic, ge ...
from approximately 1800–1650 BC includes an early written study of polyhedra and their volumes (specifically, the volume of a
frustum In geometry, a ; (: frusta or frustums) is the portion of a polyhedron, solid (normally a pyramid (geometry), pyramid or a cone (geometry), cone) that lies between two parallel planes cutting the solid. In the case of a pyramid, the base faces a ...
). The mathematics of the
Old Babylonian Empire The Old Babylonian Empire, or First Babylonian Empire, is dated to , and comes after the end of Sumerian power with the destruction of the Third Dynasty of Ur, and the subsequent Isin-Larsa period. The chronology of the first dynasty of Babylon ...
, from roughly the same time period as the Moscow Papyrus, also included calculations of the volumes of
cuboid In geometry, a cuboid is a hexahedron with quadrilateral faces, meaning it is a polyhedron with six Face (geometry), faces; it has eight Vertex (geometry), vertices and twelve Edge (geometry), edges. A ''rectangular cuboid'' (sometimes also calle ...
s (and of non-polyhedral
cylinder A cylinder () has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infinite ...
s), and calculations of the height of such a shape needed to attain a given volume. The
Etruscans The Etruscan civilization ( ) was an ancient civilization created by the Etruscans, a people who inhabited Etruria in List of ancient peoples of Italy, ancient Italy, with a common language and culture, and formed a federation of city-states. Af ...
preceded the Greeks in their awareness of at least some of the regular polyhedra, as evidenced by the discovery of an Etruscan
dodecahedron In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Po ...
made of
soapstone Soapstone (also known as steatite or soaprock) is a talc-schist, which is a type of metamorphic rock. It is composed largely of the magnesium-rich mineral talc. It is produced by dynamothermal metamorphism and metasomatism, which occur in sub ...
on Monte Loffa. Its faces were marked with different designs, suggesting to some scholars that it may have been used as a gaming die.


Ancient Greece

Ancient Greek mathematicians discovered and studied the convex regular polyhedra, which came to be known as the
Platonic solid In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...
s. Their first written description is in the '' Timaeus'' of
Plato Plato ( ; Greek language, Greek: , ; born  BC, died 348/347 BC) was an ancient Greek philosopher of the Classical Greece, Classical period who is considered a foundational thinker in Western philosophy and an innovator of the writte ...
(circa 360 BC), which associates four of them with the
four elements The classical elements typically refer to earth, water, air, fire, and (later) aether which were proposed to explain the nature and complexity of all matter in terms of simpler substances. Ancient cultures in Greece, Angola, Tibet, India, a ...
and the fifth to the overall shape of the universe. A more mathematical treatment of these five polyhedra was written soon after in the '' Elements'' of
Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...
. An early commentator on Euclid (possibly
Geminus Geminus of Rhodes (), was a Greek astronomer and mathematician, who flourished in the 1st century BC. An astronomy work of his, the ''Introduction to the Phenomena'', still survives; it was intended as an introductory astronomy book for students ...
) writes that the attribution of these shapes to Plato is incorrect:
Pythagoras Pythagoras of Samos (;  BC) was an ancient Ionian Greek philosopher, polymath, and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graecia and influenced the philosophies of P ...
knew the
tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...
,
cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
, and
dodecahedron In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Po ...
, and Theaetetus (circa 417 BC) discovered the other two, the
octahedron In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of i ...
and
icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical tha ...
. Later,
Archimedes Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...
expanded his study to the convex uniform polyhedra which now bear his name. His original work is lost and his solids come down to us through Pappus.


Ancient China

Both cubical dice and 14-sided dice in the shape of a
truncated octahedron In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...
in China have been dated back as early as the
Warring States period The Warring States period in history of China, Chinese history (221 BC) comprises the final two and a half centuries of the Zhou dynasty (256 BC), which were characterized by frequent warfare, bureaucratic and military reforms, and ...
. By 236 AD,
Liu Hui Liu Hui () was a Chinese mathematician who published a commentary in 263 CE on ''Jiu Zhang Suan Shu ( The Nine Chapters on the Mathematical Art).'' He was a descendant of the Marquis of Zixiang of the Eastern Han dynasty and lived in the state ...
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.


Medieval Islam

After the end of the Classical era, scholars in the Islamic civilisation continued to take the Greek knowledge forward (see
Mathematics in medieval Islam Mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, was built upon syntheses of Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta). Important developments o ...
). The 9th century scholar
Thabit ibn Qurra Thabit () is an Arabic name Arabic names have historically been based on a long naming system. Many people from Arabic-speaking and also non-Arab Muslim countries have not had given name, given, middle name, middle, and family names but rather a ...
included the calculation of volumes in his studies, and wrote a work on the
cuboctahedron A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertex (geometry), vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edge (geometry), edges, each separating a tr ...
. Then in the 10th century 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 The Renaissance ( , ) is a Periodization, period of history and a European cultural movement covering the 15th and 16th centuries. It marked the transition from the Middle Ages to modernity and was characterized by an effort to revive and sur ...
. Artists constructed skeletal polyhedra, depicting them from life as a part of their investigations into perspective.
Toroidal polyhedra In geometry, a toroidal polyhedron is a polyhedron which is also a toroid (a -holed torus), having a topology (Mathematics), topological Genus (mathematics), genus () of 1 or greater. Notable examples include the Császár polyhedron, Császár a ...
, made of wood and used to support headgear, became a common exercise in perspective drawing, and were depicted in
marquetry Marquetry (also spelled as marqueterie; from the French ''marqueter'', to variegate) is the art and craft of applying pieces of wood veneer, veneer to a structure to form decorative patterns or designs. The technique may be applied to case furn ...
panels of the period as a symbol of geometry.
Piero della Francesca Piero della Francesca ( , ; ; ; – 12 October 1492) was an Italian Renaissance painter, Italian painter, mathematician and List of geometers, geometer of the Early Renaissance, nowadays chiefly appreciated for his art. His painting is charact ...
wrote about constructing perspective views of polyhedra, and rediscovered many of the Archimedean solids.
Leonardo da Vinci Leonardo di ser Piero da Vinci (15 April 1452 - 2 May 1519) was an Italian polymath of the High Renaissance who was active as a painter, draughtsman, engineer, scientist, theorist, sculptor, and architect. While his fame initially rested o ...
illustrated skeletal models of several polyhedra for a book by
Luca Pacioli Luca Bartolomeo de Pacioli, O.F.M. (sometimes ''Paccioli'' or ''Paciolo''; 1447 – 19 June 1517) was an Italian mathematician, Franciscan friar, collaborator with Leonardo da Vinci, and an early contributor to the field now known as account ...
, with text largely plagiarized from della Francesca. Polyhedral nets make an appearance in the work of
Albrecht Dürer Albrecht Dürer ( , ;; 21 May 1471 – 6 April 1528),Müller, Peter O. (1993) ''Substantiv-Derivation in Den Schriften Albrecht Dürers'', Walter de Gruyter. . sometimes spelled in English as Durer or Duerer, was a German painter, Old master prin ...
. Several works from this time investigate star polyhedra, and other elaborations of the basic Platonic forms. A marble tarsia in the floor of St. Mark's Basilica, Venice, designed by
Paolo Uccello Paolo Uccello ( , ; 1397 – 10 December 1475), born Paolo di Dono, was an Italian Renaissance painter and mathematician from Florence who was notable for his pioneering work on visual Perspective (graphical), perspective in art. In his book ''Liv ...
, depicts a stellated dodecahedron. As the Renaissance spread beyond Italy, later artists such as Wenzel Jamnitzer, Dürer and others also depicted polyhedra of increasing complexity, many of them novel, in imaginative etchings.
Johannes Kepler Johannes Kepler (27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, Natural philosophy, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best know ...
(1571–1630) used
star polygon In geometry, a star polygon is a type of non-convex polygon. Regular star polygons have been studied in depth; while star polygons in general appear not to have been formally defined, Decagram (geometry)#Related figures, certain notable ones can ...
s, typically
pentagram A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle around ...
s, 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. In the same period, Euler's polyhedral formula, a
linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coeffici ...
relating the numbers of vertices, edges, and faces of a polyhedron, was stated for the Platonic solids in 1537 in an unpublished manuscript by
Francesco Maurolico Francesco Maurolico (Latin: ''Franciscus Maurolycus''; Italian language, Italian: ''Francesco Maurolico''; ; Sicilian language, Sicilian: ''Francescu Maurolicu''; 16 September 1494 – 22 July 1575) was an Italian mathematician and astronomer fr ...
.


17th–19th centuries

René Descartes René Descartes ( , ; ; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and Modern science, science. Mathematics was paramou ...
, in around 1630, wrote his book '' De solidorum elementis'' studying convex polyhedra as a general concept, not limited to the Platonic solids and their elaborations. The work was lost, and not rediscovered until the 19th century. One of its contributions was Descartes' theorem on total angular defect, which is closely related to Euler's polyhedral formula.
Leonhard Euler Leonhard Euler ( ; ; ; 15 April 170718 September 1783) was a Swiss polymath who was active as a mathematician, physicist, astronomer, logician, geographer, and engineer. He founded the studies of graph theory and topology and made influential ...
, for whom the formula is named, introduced it in 1758 for convex polyhedra more generally, albeit with an incorrect proof. Euler's work (together with his earlier solution to the puzzle of the
Seven Bridges of Königsberg The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler, in 1736, laid the foundations of graph theory and prefigured the idea of topology. The city of Königsberg in Prussia ...
) became the foundation of the new field of
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
. The core concepts of this field, including generalizations of the polyhedral formula, were developed in the late nineteenth century by
Henri Poincaré Jules Henri Poincaré (, ; ; 29 April 185417 July 1912) was a French mathematician, Theoretical physics, theoretical physicist, engineer, and philosophy of science, philosopher of science. He is often described as a polymath, and in mathemati ...
,
Enrico Betti Enrico Betti Glaoui (21 October 1823 – 11 August 1892) was an Italian mathematician, now remembered mostly for his 1871 paper on topology that led to the later naming after him of the Betti numbers. He worked also on the theory of equations ...
,
Bernhard Riemann Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the f ...
, and others. In the early 19th century,
Louis Poinsot Louis Poinsot (; 3 January 1777 – 5 December 1859) was a French mathematician and physicist. Poinsot was the inventor of geometrical mechanics, showing how a system of forces acting on a rigid body could be resolved into a single force and a ...
extended Kepler's work, and discovered the remaining two regular star polyhedra. Soon after,
Augustin-Louis Cauchy Baron Augustin-Louis Cauchy ( , , ; ; 21 August 1789 – 23 May 1857) was a French mathematician, engineer, and physicist. He was one of the first to rigorously state and prove the key theorems of calculus (thereby creating real a ...
proved Poinsot's list complete, subject to an unstated assumption that the sequence of vertices and edges of each polygonal side cannot admit repetitions (an assumption that had been considered but rejected in the earlier work of A. F. L. Meister). They became known as the Kepler–Poinsot polyhedra, and their usual names were given by
Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics, and was a professor at Trinity College, Cambridge for 35 years. He ...
. Meanwhile, the discovery of higher dimensions in the early 19th century led Ludwig Schläfli by 1853 to the idea of higher-dimensional polytopes. Additionally, in the late 19th century, Russian crystallographer Evgraf Fedorov completed the classification of parallelohedra, convex polyhedra that tile space by translations.


20th–21st centuries

Mathematics in the 20th century dawned with
Hilbert's problems Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the pr ...
, one of which,
Hilbert's third problem The third of Hilbert's problems, 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 polyhedron, polyhedra of equal volume, is it always possible t ...
, concerned polyhedra and their dissections. It was quickly solved by Hilbert's student
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. Dehn's early life and career took place in Germany. However, he was forced to retire in 1 ...
, introducing the Dehn invariant of polyhedra.
Steinitz's theorem In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedron, convex polyhedra: they are exactly the vertex connect ...
, published by Ernst Steinitz in 1992, characterized the graphs of convex polyhedra, bringing modern ideas from
graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...
and
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
into the study of polyhedra. The Kepler–Poinsot polyhedra may be constructed from the Platonic solids by a process called
stellation In geometry, stellation is the process of extending a polygon in two dimensions, a polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specific ...
. 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 Fifty-Nine Icosahedra ''The Fifty-Nine Icosahedra'' is a book written and illustrated by Harold Scott MacDonald Coxeter, H. S. M. Coxeter, Patrick du Val, P. Du Val, H. T. Flather and J. F. Petrie. It enumerates certain stellations of the regular convex or Platonic re ...
''.. Coxeter's analysis signaled 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 polyhedra and to develop the theory of 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, both
Branko Grünbaum Branko Grünbaum (; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentImre Lakatos pointed out the tendency among mathematicians to define a "polyhedron" in different and sometimes incompatible ways to suit the needs of the moment. In a series of papers, Grünbaum broadened the accepted definition of a polyhedron, discovering many new
regular polyhedra A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different eq ...
. At the close of the twentieth 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. Polyhedra have been discovered in many fields of science. The Platonic solids appeared in biological creatures, as in The '' Braarudosphaera bigelowii'' has a regular dodecahedral structure.
Ernst Haeckel Ernst Heinrich Philipp August Haeckel (; ; 16 February 1834 – 9 August 1919) was a German zoologist, natural history, naturalist, eugenics, eugenicist, Philosophy, philosopher, physician, professor, marine biology, marine biologist and artist ...
described a number of species of radiolarians, some of whose shells are shaped like various regular polyhedra.. Available as Haeckel, E. ''Art forms in nature'', Prestel USA (1998), . Online version a
Kurt Stüber's Biolib
(in German) The outer protein shells of many
virus A virus is a submicroscopic infectious agent that replicates only inside the living Cell (biology), cells of an organism. Viruses infect all life forms, from animals and plants to microorganisms, including bacteria and archaea. Viruses are ...
es form regular polyhedra. For example, HIV is enclosed in a regular icosahedron, as is the head of a typical myovirus. The regular icosahedron may also appeared in the applications of
cartography Cartography (; from , 'papyrus, sheet of paper, map'; and , 'write') is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an imagined reality) can ...
when R. Buckminster Fuller used its net to his project known as Dymaxion map, frustatedly realized that the
Greenland Greenland is an autonomous territory in the Danish Realm, Kingdom of Denmark. It is by far the largest geographically of three constituent parts of the kingdom; the other two are metropolitan Denmark and the Faroe Islands. Citizens of Greenlan ...
size is smaller than the
South America South America is a continent entirely in the Western Hemisphere and mostly in the Southern Hemisphere, with a considerably smaller portion in the Northern Hemisphere. It can also be described as the southern Subregion#Americas, subregion o ...
. Polyhedra make a frequent appearance in modern computational geometry,
computer graphics Computer graphics deals with generating images and art with the aid of computers. Computer graphics is a core technology in digital photography, film, video games, digital art, cell phone and computer displays, and many specialized applications. ...
, and geometric design with topics including the reconstruction of polyhedral surfaces or surface meshes from scattered data points, geodesics on polyhedral surfaces,
visibility In meteorology, visibility is the measure of the distance at which an object or light can be clearly discerned. It depends on the Transparency and translucency, transparency of the surrounding air and as such, it is unchanging no matter the amb ...
and illumination in polyhedral scenes,
polycube image:tetracube_categories.svg, upAll 8 one-sided tetracubes – if chirality is ignored, the bottom 2 in grey are considered the same, giving 7 free tetracubes in total image:9L cube puzzle solution.svg, A puzzle involving arranging nine L tricube ...
s and other non-convex polyhedra with axis-parallel sides, algorithmic forms of Steinitz's theorem, and the still-unsolved problem of the existence of polyhedral nets for convex polyhedra.


See also

* Chazelle polyhedron *
Equiprojective polyhedra In mathematics, a convex polyhedron is defined to be k-equiprojective if every orthogonal projection of the polygon onto a plane, in a direction not parallel to a face of the polyhedron, forms a k-gon. For example, a cube is 6-equiprojective: every ...
* Extension of a polyhedron * Goldberg polyhedron * List of books about polyhedra *
Near-miss Johnson solid In geometry, a near-miss Johnson solid is a strictly convex set, convex polyhedron whose face (geometry), faces are close to being regular polygons but some or all of which are not precisely regular. Thus, it fails to meet the definition of a John ...
* Polyhedron model * Polyhedral number * Polyhedral skeletal electron pair theory * Polyhedral space * Polyhedral symbol * Polyhedral terrain * Polytope model *
Stella (software) Stella is a computer program available in three versions (Great Stella, Small Stella and Stella4D). It was created by Robert Webb of Australia. The programs contain a large library of polyhedra which can be manipulated and altered in various w ...


Notes


References


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) 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 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. Models can be exported as 2D PNG images, or as 3D OBJ or VRML2 files.


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. {{Authority control