Dual polyhedron
Encyclopedia
In geometry
, polyhedra
are associated into pairs called duals, where the vertices of one correspond to the face
s of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another with equivalent edges. So the regular polyhedra — the Platonic solid
s and KeplerPoinsot polyhedra — are arranged into dual pairs, with the exception of the regular tetrahedron
which is selfdual.
Duality is also sometimes called reciprocity or polarity.
the vertex
is associated with the plane
.
The vertices of the dual, then, are the poles reciprocal to the face planes of the original, and the faces of the dual lie in the polars reciprocal to the vertices of the original. Also, any two adjacent vertices define an edge, and these will reciprocate to two adjacent faces which intersect to define an edge of the dual.
Notice that the exact form of the dual will depend on what sphere we reciprocate with respect to; as we move the sphere around, the dual form distorts. The choice of center (of the sphere) is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will necessarily intersect at a single point, and this is usually taken to be the center. Failing that a circumscribed sphere, inscribed sphere, or midsphere (one with all edges as tangents) can be used.
If a polyhedron has an element passing through the center of the sphere, the corresponding element of its dual will go to infinity. Since traditional "Euclidean" space never reaches infinity, the projective equivalent, called extended Euclidean space, must be formed by adding the required 'plane at infinity'. Some theorists prefer to stick to Euclidean space and say that there is no dual. Meanwhile Wenninger (1983) found a way to represent these infinite duals, in a manner suitable for making models (of some finite portion!).
The concept of duality here is closely related to the duality
in projective geometry
, where lines and edges are interchanged; in fact it is often mistakenly taken to be a particular version of the same. Projective polarity works well enough for convex polyhedra. But for nonconvex figures such as star polyhedra, when we seek to rigorously define this form of polyhedral duality in terms of projective polarity, various problems appear. See for example Grünbaum & Shepherd (1988), and Gailiunas & Sharp (2005). Wenninger (1983) also discusses some issues on the way to deriving his infinite duals.
(or intersphere) exists tangent to every edge, such that the average position of these points is the center of the sphere, and this form is unique up to congruences.
If we reciprocate such a polyhedron about its intersphere, the dual polyhedron will share the same edgetangency points and so must also be canonical; it is the canonical dual, and the two together form a canonical dual compound.
It is worth noting that the vertices and edges of a convex polyhedron can be projected to form a graph
(sometimes called a Schlegel diagram) on the sphere or on a flat plane, and the corresponding graph formed by the dual of this polyhedron is its dual graph
.
is a certain kind of partially ordered set
(poset) of elements, such that adjacencies, or connections, between elements of the set correspond to adjacencies between elements (faces, edges, etc.) of a polyhedron. Such a poset may be represented in a Hasse diagram
. Any such poset has a dual poset. The Hasse diagram of the dual polyhedron is obtained very simply, by turning the original diagram upsidedown.
, the face of the dual polyhedron may be found from the original polyhedron's vertex figure
using the Dorman Luke construction. This construction was originally described by Cundy & Rollett (1961) and later generalised by Wenninger (1983).
As an example, here is the vertex figure (red) of the cuboctahedron
being used to derive a face (blue) of the rhombic dodecahedron
.
Before beginning the construction, the vertex figure
ABCD is (in this case) obtained by cutting each connected edge at its midpoint.
Dorman Luke's construction then proceeds:
The size of the vertex figure was chosen so that its circumcircle lies on the intersphere of the cuboctahedron, which also becomes the intersphere of the dual rhombic dodecahedron.
Dorman Luke's construction can only be used where a polyhedron has such an intersphere and the vertex figure is cyclic, i.e. for uniform polyhedra
.
whose dual is a congruent figure, though not necessarily the identical figure: for example, the dual of a regular tetrahedron is a regular tetrahedron "facing the opposite direction" (reflected through the origin).
A selfdual polyhedron must have the same number of vertices as faces. We can distinguish between structural (topological
) duality and geometrical
duality. The topological structure of a selfdual polyhedron is also selfdual. Whether or not such a polyhedron is also geometrically selfdual will depend on the particular geometrical duality being considered. For example, every polygon is topologically selfdual (it has the same number of vertices as edges, and these are switched by duality), but will not in general be geometrically selfdual (up to rigid motion, for instance) – regular polygons are geometrically selfdual (all angles are congruent, as are all edges, so under duality these congruences swap), but irregular polygons may not be geometrically selfdual.
The most common geometric arrangement is where some convex polyhedron is in its canonical form, which is to say that the all its edges must be tangent to a certain sphere whose centre coincides with the centre of gravity (average position) of the tangent points. If the polar reciprocal of the canonical form in the sphere is congruent to the original, then the figure is selfdual.
There are infinitely many selfdual polyhedra. The simplest infinite family are the pyramids
of n sides and of canonical form. Another infinite family consists of polyhedra that can be roughly described as a pyramid sitting on top of a prism
(with the same number of sides). Add a frustum (pyramid with the top cut off) below the prism and you get another infinite family, and so on.
There are many other convex, selfdual polyhedra. For example, there are 6 different ones with 7 vertices, and 16 with 8 vertices.
Nonconvex selfdual polyhedra can also be found, for example there is one among the facetting
s of the regular dodecahedron (and hence by duality also among the stellation
s of the icosahedron
).
, being a compound of two tetrahedra
is also selfdual, as well as four other regulardual compounds.
s; in 2dimensions these are called dual polygon
s.
The vertices of one polytope correspond to the (n − 1)dimensional elements, or facets, of the other, and the j points that define a (j − 1)dimensional element will correspond to j hyperplanes that intersect to give a (n − j)dimensional element. The dual of a honeycomb
can be defined similarly.
In general, the facets of a polytope's dual will be the topological duals of the polytope's vertex figures. For regular
and uniform
polytopes, the dual facets will be the polar reciprocals of the original's facets. For example, in four dimensions, the vertex figure of the 600cell is the icosahedron
; the dual of the 600cell is the 120cell, whose facets are dodecahedra, which are the dual of the icosahedron.
s with palindromic Schläfli symbols. All regular polygons, {a} are selfdual, polyhedra
of the form {a,a}, 4polytopes
of the form {a,b,a}, 5polytope
s of the form {a,b,b,a}, etc.
The selfdual regular polytopes are:
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of premodern mathematics, the other being the study of numbers ....
, polyhedra
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
are associated into pairs called duals, where the vertices of one correspond to the face
Face (geometry)
In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...
s of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another with equivalent edges. So the regular polyhedra — the Platonic solid
Platonic solid
In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...
s and KeplerPoinsot polyhedra — are arranged into dual pairs, with the exception of the regular tetrahedron
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
which is selfdual.
Duality is also sometimes called reciprocity or polarity.
Kinds of duality
There are many kinds of duality. The kinds most relevant to polyhedra are: Polar reciprocity
 Topological duality
 Abstract duality
Polar reciprocation
Duality is most commonly defined in terms of polar reciprocation about a concentric sphere. Here, each vertex (pole) is associated with a face plane (polar plane or just polar) so that the ray from the center to the vertex is perpendicular to the plane, and the product of the distances from the center to each is equal to the square of the radius. In coordinates, for reciprocation about the spherethe vertex
is associated with the plane
.
The vertices of the dual, then, are the poles reciprocal to the face planes of the original, and the faces of the dual lie in the polars reciprocal to the vertices of the original. Also, any two adjacent vertices define an edge, and these will reciprocate to two adjacent faces which intersect to define an edge of the dual.
Notice that the exact form of the dual will depend on what sphere we reciprocate with respect to; as we move the sphere around, the dual form distorts. The choice of center (of the sphere) is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will necessarily intersect at a single point, and this is usually taken to be the center. Failing that a circumscribed sphere, inscribed sphere, or midsphere (one with all edges as tangents) can be used.
If a polyhedron has an element passing through the center of the sphere, the corresponding element of its dual will go to infinity. Since traditional "Euclidean" space never reaches infinity, the projective equivalent, called extended Euclidean space, must be formed by adding the required 'plane at infinity'. Some theorists prefer to stick to Euclidean space and say that there is no dual. Meanwhile Wenninger (1983) found a way to represent these infinite duals, in a manner suitable for making models (of some finite portion!).
The concept of duality here is closely related to the duality
Duality (projective geometry)
A striking feature of projective planes is the "symmetry" of the roles played by points and lines in the definitions and theorems, and duality is the formalization of this metamathematical concept. There are two approaches to the subject of duality, one through language and the other a more...
in projective geometry
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
, where lines and edges are interchanged; in fact it is often mistakenly taken to be a particular version of the same. Projective polarity works well enough for convex polyhedra. But for nonconvex figures such as star polyhedra, when we seek to rigorously define this form of polyhedral duality in terms of projective polarity, various problems appear. See for example Grünbaum & Shepherd (1988), and Gailiunas & Sharp (2005). Wenninger (1983) also discusses some issues on the way to deriving his infinite duals.
Canonical duals
Any convex polyhedron can be distorted into a canonical form, in which a midsphereMidsphere
In geometry, the midsphere or intersphere of a polyhedron is a sphere which is tangent to every edge of the polyhedron. That is to say, it touches any given edge at exactly one point...
(or intersphere) exists tangent to every edge, such that the average position of these points is the center of the sphere, and this form is unique up to congruences.
If we reciprocate such a polyhedron about its intersphere, the dual polyhedron will share the same edgetangency points and so must also be canonical; it is the canonical dual, and the two together form a canonical dual compound.
Topological duality
We can distort a dual polyhedron such that it can no longer be obtained by reciprocating the original in any sphere; in this case we can say that the two polyhedra are still topologically dual.It is worth noting that the vertices and edges of a convex polyhedron can be projected to form a graph
Graph theory
In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices or 'nodes' and a collection of edges that connect pairs of...
(sometimes called a Schlegel diagram) on the sphere or on a flat plane, and the corresponding graph formed by the dual of this polyhedron is its dual graph
Dual graph
In mathematics, the dual graph of a given planar graph G is a graph which has a vertex for each plane region of G, and an edge for each edge in G joining two neighboring regions, for a certain embedding of G. The term "dual" is used because this property is symmetric, meaning that if H is a dual...
.
Abstract duality
An abstract polyhedronAbstract polytope
In mathematics, an abstract polytope, informally speaking, is a structure which considers only the combinatorial properties of a traditional polytope, ignoring many of its other properties, such as angles, edge lengths, etc...
is a certain kind of partially ordered set
Partially ordered set
In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...
(poset) of elements, such that adjacencies, or connections, between elements of the set correspond to adjacencies between elements (faces, edges, etc.) of a polyhedron. Such a poset may be represented in a Hasse diagram
Hasse diagram
In order theory, a branch of mathematics, a Hasse diagram is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction...
. Any such poset has a dual poset. The Hasse diagram of the dual polyhedron is obtained very simply, by turning the original diagram upsidedown.
Dorman Luke construction
For a uniform polyhedronUniform polyhedron
A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertextransitive...
, the face of the dual polyhedron may be found from the original polyhedron's vertex figure
Vertex figure
In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.Definitions  theme and variations:...
using the Dorman Luke construction. This construction was originally described by Cundy & Rollett (1961) and later generalised by Wenninger (1983).
As an example, here is the vertex figure (red) of the cuboctahedron
Cuboctahedron
In geometry, a cuboctahedron is a polyhedron with eight triangular faces and six square faces. A cuboctahedron has 12 identical vertices, with two triangles and two squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such it is a quasiregular polyhedron,...
being used to derive a face (blue) of the rhombic dodecahedron
Rhombic dodecahedron
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 rhombic faces. It is an Archimedean dual solid, or a Catalan solid. Its dual is the cuboctahedron.Properties:...
.
Before beginning the construction, the vertex figure
Vertex figure
In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.Definitions  theme and variations:...
ABCD is (in this case) obtained by cutting each connected edge at its midpoint.
Dorman Luke's construction then proceeds:

 Draw the circumcircle (tangent to every corner).
 Draw lines tangent to the circumcircle at each corner A, B, C, D.
 Mark the points E, F, G, H, where each line meets the adjacent line.
 The polygon EFGH is a face of the dual polyhedron.
The size of the vertex figure was chosen so that its circumcircle lies on the intersphere of the cuboctahedron, which also becomes the intersphere of the dual rhombic dodecahedron.
Dorman Luke's construction can only be used where a polyhedron has such an intersphere and the vertex figure is cyclic, i.e. for uniform polyhedra
Uniform polyhedron
A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertextransitive...
.
Selfdual polyhedra
A selfdual polyhedron is a polyhedronPolyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
whose dual is a congruent figure, though not necessarily the identical figure: for example, the dual of a regular tetrahedron is a regular tetrahedron "facing the opposite direction" (reflected through the origin).
A selfdual polyhedron must have the same number of vertices as faces. We can distinguish between structural (topological
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
) duality and geometrical
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of premodern mathematics, the other being the study of numbers ....
duality. The topological structure of a selfdual polyhedron is also selfdual. Whether or not such a polyhedron is also geometrically selfdual will depend on the particular geometrical duality being considered. For example, every polygon is topologically selfdual (it has the same number of vertices as edges, and these are switched by duality), but will not in general be geometrically selfdual (up to rigid motion, for instance) – regular polygons are geometrically selfdual (all angles are congruent, as are all edges, so under duality these congruences swap), but irregular polygons may not be geometrically selfdual.
The most common geometric arrangement is where some convex polyhedron is in its canonical form, which is to say that the all its edges must be tangent to a certain sphere whose centre coincides with the centre of gravity (average position) of the tangent points. If the polar reciprocal of the canonical form in the sphere is congruent to the original, then the figure is selfdual.
There are infinitely many selfdual polyhedra. The simplest infinite family are the pyramids
Pyramid (geometry)
In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle. It is a conic solid with polygonal base....
of n sides and of canonical form. Another infinite family consists of polyhedra that can be roughly described as a pyramid sitting on top of a prism
Prism (geometry)
In geometry, a prism is a polyhedron with an nsided polygonal base, a translated copy , and n other faces joining corresponding sides of the two bases. All crosssections parallel to the base faces are the same. Prisms are named for their base, so a prism with a pentagonal base is called a...
(with the same number of sides). Add a frustum (pyramid with the top cut off) below the prism and you get another infinite family, and so on.
There are many other convex, selfdual polyhedra. For example, there are 6 different ones with 7 vertices, and 16 with 8 vertices.
Nonconvex selfdual polyhedra can also be found, for example there is one among the facetting
Facetting
In geometry, facetting is the process of removing parts of a polygon, polyhedron or polytope, without creating any new vertices.Facetting is the reciprocal or dual process to stellation...
s of the regular dodecahedron (and hence by duality also among the stellation
Stellation
Stellation is a process of constructing new polygons , new polyhedra in three dimensions, or, in general, new polytopes in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again...
s of the icosahedron
Icosahedron
In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....
).
Tetrahedron Tetrahedron In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids... 
Square pyramid Square pyramid In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry. Johnson solid :... 
Pentagonal pyramid Pentagonal pyramid In geometry, a pentagonal pyramid is a pyramid with a pentagonal base upon which are erected five triangular faces that meet at a point . Like any pyramid, it is selfdual.... 
Hexagonal pyramid Hexagonal pyramid In geometry, a hexagonal pyramid is a pyramid with a hexagonal base upon which are erected six triangular faces that meet at a point... 
Elongated triangular pyramid Elongated triangular pyramid In geometry, the elongated triangular pyramid is one of the Johnson solids . Norman Johnson discovered elongated triangular pyramids. As the name suggests, it can be constructed by elongating a tetrahedron by attaching a triangular prism to its base... 
Elongated square pyramid Elongated square pyramid In geometry, the elongated square pyramid is one of the Johnson solids . As the name suggests, it can be constructed by elongating a square pyramid by attaching a cube to its square base... 
Elongated pentagonal pyramid Elongated pentagonal pyramid In geometry, the elongated pentagonal pyramid is one of the Johnson solids . As the name suggests, it can be constructed by elongating a pentagonal pyramid by attaching a pentagonal prism to its base.... 
Self dual compound polyhedra
The Stella octangulaStella octangula
The stellated octahedron, or stella octangula, is the only stellation of the octahedron. It was named by Johannes Kepler in 1609, though it was known to earlier geometers...
, being a compound of two tetrahedra
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...
is also selfdual, as well as four other regulardual compounds.
Dual polytopes and tessellations
Duality can be generalized to ndimensional space and dual polytopePolytope
In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...
s; in 2dimensions these are called dual polygon
Dual polygon
In geometry, polygons are associated into pairs called duals, where the vertices of one correspond to the edges of the other.Properties:Regular polygons are selfdual....
s.
The vertices of one polytope correspond to the (n − 1)dimensional elements, or facets, of the other, and the j points that define a (j − 1)dimensional element will correspond to j hyperplanes that intersect to give a (n − j)dimensional element. The dual of a honeycomb
Honeycomb (geometry)
In geometry, a honeycomb is a space filling or close packing of polyhedral or higherdimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions....
can be defined similarly.
In general, the facets of a polytope's dual will be the topological duals of the polytope's vertex figures. For regular
Regular polytope
In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry. All its elements or jfaces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of...
and uniform
Uniform polytope
A uniform polytope is a vertextransitive polytope made from uniform polytope facets of a lower dimension. Uniform polytopes of 2 dimensions are the regular polygons....
polytopes, the dual facets will be the polar reciprocals of the original's facets. For example, in four dimensions, the vertex figure of the 600cell is the icosahedron
Icosahedron
In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....
; the dual of the 600cell is the 120cell, whose facets are dodecahedra, which are the dual of the icosahedron.
Selfdual polytopes and tessellations
The primary class of selfdual polytopes are regular polytopeRegular polytope
In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry. All its elements or jfaces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of...
s with palindromic Schläfli symbols. All regular polygons, {a} are selfdual, polyhedra
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
of the form {a,a}, 4polytopes
Polychoron
In geometry, a polychoron or 4polytope is a fourdimensional polytope. It is a connected and closed figure, composed of lower dimensional polytopal elements: vertices, edges, faces , and cells...
of the form {a,b,a}, 5polytope
5polytope
In fivedimensional geometry, a 5polytope is a 5dimensional polytope, bounded by facets. Each polyhedral cell being shared by exactly two polychoron facets. A proposed name for 5polytopes is polyteron.Definition:...
s of the form {a,b,b,a}, etc.
The selfdual regular polytopes are:
 All regular polygonRegular polygonA regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.General properties:...
s, {a}.  All regular nsimplexSimplexIn geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an nsimplex is an ndimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2simplex is a triangle, a 3simplex is a tetrahedron,...
es, {3,3,...,3}  The regular 24cell in 4 dimensions, {3,4,3}.
 All regular ndimensional cubic honeycombsHoneycomb (geometry)In geometry, a honeycomb is a space filling or close packing of polyhedral or higherdimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions....
, {4,3,...,3,4}. These may be treated as infinite polytopes.
See also
 Conway polyhedron notationConway polyhedron notationConway polyhedron notation is used to describe polyhedra based on a seed polyhedron modified by various operations.The seed polyhedra are the Platonic solids, represented by their first letter of their name ; the prisms , antiprisms and pyramids...
 Dual polygonDual polygonIn geometry, polygons are associated into pairs called duals, where the vertices of one correspond to the edges of the other.Properties:Regular polygons are selfdual....
 Selfdual graph
 Selfdual polygon
External links
 Software for displaying duals
 The Uniform Polyhedra
 Virtual Reality Polyhedra The Encyclopedia of Polyhedra