Ingeometry, everypolyhedron is associated with a seconddual structure, wherein thevertices of one correspond to thefaces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other.[1] Such dual figures remain combinatorial orabstract polyhedra, but not all can also be constructed as geometric polyhedra.[2] Starting with any given polyhedron, the dual of its dual is the original polyhedron.
Duality preserves thesymmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a corresponding symmetry class. For example, the regular polyhedra – the (convex)Platonic solids and (star)Kepler–Poinsot polyhedra – form dual pairs, where the regulartetrahedron isself-dual. The dual of anisogonal polyhedron (one in which any two vertices are equivalent under symmetries of the polyhedron) is anisohedral polyhedron (one in which any two faces are equivalent [...]), and vice versa. The dual of anisotoxal polyhedron (one in which any two edges are equivalent [...]) is also isotoxal.
Duality is closely related topolar reciprocity, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron.
There are many kinds of duality. The kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality.
InEuclidean space, the dual of a polyhedron is often defined in terms ofpolar reciprocation about a 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.[3]
When the sphere has radius and is centered at the origin (so that it is defined by the equation), then the polar dual of a convex polyhedron is defined as
where denotes the standarddot product of and.
Typically when no sphere is specified in the construction of the dual, then the unit sphere is used, meaning in the above definitions.[4]
For each face plane of described by the linear equationthe corresponding vertex of the dual polyhedron will have coordinates. Similarly, each vertex of corresponds to a face plane of, and each edge line of corresponds to an edge line of. The correspondence between the vertices, edges, and faces of and reverses inclusion. For example, if an edge of contains a vertex, the corresponding edge of will be contained in the corresponding face.
For a polyhedron with acenter of symmetry, it is common to use a sphere centered on this point, as in theDorman Luke construction (mentioned below). Failing that, for a polyhedron with a circumscribed sphere, inscribed sphere, or midsphere (one with all edges as tangents), this can be used. However, it is possible to reciprocate a polyhedron about any sphere, and the resulting form of the dual will depend on the size and position of the sphere; as the sphere is varied, so too is the dual form. The choice of center for the sphere is sufficient to define the dual up to similarity.
If a polyhedron inEuclidean space has a face plane, edge line, or vertex lying on the center of the sphere, the corresponding element of its dual will go to infinity. Since Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may 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 ofduality here is closely related to theduality inprojective geometry, where lines and edges are interchanged. Projective polarity works well enough for convex polyhedra. But for non-convex figures such as star polyhedra, when we seek to rigorously define this form of polyhedral duality in terms of projective polarity, various problems appear.[5]Because of the definitional issues for geometric duality of non-convex polyhedra,Grünbaum (2007) argues that any proper definition of a non-convex polyhedron should include a notion of a dual polyhedron.
Any convex polyhedron can be distorted into acanonical form, in which a unitmidsphere (or intersphere) exists tangent to every edge, and such that the average position of the points of tangency is the center of the sphere. This form is unique up to congruences.
If we reciprocate such a canonical polyhedron about its midsphere, the dual polyhedron will share the same edge-tangency points, and thus will also be canonical. It is the canonical dual, and the two together form a canonical dual compound.[6]
For auniform polyhedron, each face of the dual polyhedron may be derived from the original polyhedron's correspondingvertex figure by using theDorman Luke construction.[7]
Even when a pair of polyhedra cannot be obtained by reciprocation from each other, they may be called duals of each other as long as the vertices of one correspond to the faces of the other, and the edges of one correspond to the edges of the other, in an incidence-preserving way. Such pairs of polyhedra are still topologically or abstractly dual.
The vertices and edges of a convex polyhedron form agraph (the1-skeleton of the polyhedron), embedded on the surface of the polyhedron (a topological sphere). This graph can be projected to form aSchlegel diagram on a flat plane. The graph formed by the vertices and edges of the dual polyhedron is thedual graph of the original graph.
More generally, for any polyhedron whose faces form a closed surface, the vertices and edges of the polyhedron form a graph embedded on this surface, and the vertices and edges of the (abstract) dual polyhedron form the dual graph of the original graph.
Anabstract polyhedron is a certain kind ofpartially ordered set (poset) of elements, such that incidences, or connections, between elements of the set correspond to incidences between elements (faces, edges, vertices) of a polyhedron. Every such poset has a dual poset, formed by reversing all of the order relations. If the poset is visualized as aHasse diagram, the dual poset can be visualized simply by turning the Hasse diagram upside down.
Every geometric polyhedron corresponds to an abstract polyhedron in this way, and has an abstract dual polyhedron. However, for some types of non-convex geometric polyhedra, the dual polyhedra may not be realizable geometrically.
Topologically, a polyhedron is said to beself-dual if its dual has exactly the same connectivity between vertices, edges, and faces. Abstractly, they have the sameHasse diagram. Geometrically, it is not only topologically self-dual, but its polar reciprocal about a certain point, typically its centroid, is a similar figure. For example, the dual of a regular tetrahedron is another regular tetrahedron,reflected through the origin.
Every polygon is topologically self-dual, since it has the same number of vertices as edges, and these are switched by duality. But it is not necessarily self-dual (up to rigid motion, for instance). Every polygon has aregular form which is geometrically self-dual about its intersphere: all angles are congruent, as are all edges, so under duality these congruences swap. Similarly, every topologically self-dual convex polyhedron can be realized by an equivalent geometrically self-dual polyhedron, itscanonical polyhedron, reciprocal about the center of themidsphere.
There are infinitely many geometrically self-dual polyhedra. The simplest infinite family is thepyramids.[8] Another infinite family,elongated pyramids, consists of polyhedra that can be roughly described as a pyramid sitting on top of aprism (with the same number of sides). Adding a frustum (pyramid with the top cut off) below the prism generates another infinite family, and so on. There are many other convex self-dual polyhedra. For example, there are 6 different ones with 7 vertices and 16 with 8 vertices.[9]
A self-dual non-convex icosahedron with hexagonal faces was identified by Brückner in 1900.[10][11][12] Other non-convex self-dual polyhedra have been found, under certain definitions of non-convex polyhedra and their duals.
Duality can be generalized ton-dimensional space anddualpolytopes; in two dimensions these are calleddual polygons.
The vertices of one polytope correspond to the (n − 1)-dimensional elements, or facets, of the other, and thej points that define a (j − 1)-dimensional element will correspond toj hyperplanes that intersect to give a (n −j)-dimensional element. The dual of ann-dimensional tessellation orhoneycomb 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 the polar reciprocals of theregular anduniform polytopes, the dual facets will be polar reciprocals of the original's vertex figure. For example, in four dimensions, the vertex figure of the600-cell is theicosahedron; the dual of the 600-cell is the120-cell, whose facets aredodecahedra, which are the dual of the icosahedron.
The primary class of self-dual polytopes areregular polytopes withpalindromicSchläfli symbols. All regular polygons, {a} are self-dual,polyhedra of the form {a,a},4-polytopes of the form {a,b,a},5-polytopes of the form {a,b,b,a}, etc.
The self-dual regular polytopes are:
The self-dual (infinite) regular Euclideanhoneycombs are:
The self-dual (infinite) regularhyperbolic honeycombs are: