![Polyhedron pair 6-8](https://upload.wikimedia.org/wikipedia/commons/d/de/Polyhedron_pair_6-8.png)
In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, every
polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on th ...
is associated with a second dual structure, where the
vertices of one correspond to the
faces
The face is the front of an animal's head that features the eyes, nose and mouth, and through which animals express many of their emotions. The face is crucial for human identity, and damage such as scarring or developmental deformities may affe ...
of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or
abstract polyhedra, but not all can also be constructed as geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron.
Duality preserves the
symmetries
Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
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 polyhedrathe (convex)
Platonic solid
In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...
s and (star)
Kepler–Poinsot polyhedraform dual pairs, where the regular
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
is
self-dual
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a Injective function, one-to-one fashion, often (but not always) by means of an Involution (mathematics), involutio ...
. The dual of an
isogonal polyhedron (one in which any two vertices are equivalent under symmetries of the polyhedron) is an
isohedral
In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...
polyhedron (one in which any two faces are equivalent
.., and vice versa. The dual of an
isotoxal polyhedron (one in which any two edges are equivalent
.. is also isotoxal.
Duality is closely related to ''polar reciprocity'', a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron.
Kinds of duality
![Ioanniskepplerih00kepl 0271 crop](https://upload.wikimedia.org/wikipedia/commons/1/13/Ioanniskepplerih00kepl_0271_crop.jpg)
There are many kinds of duality. The kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality.
Polar reciprocation
In
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, the dual of a polyhedron
is often defined in terms of
polar reciprocation
In geometry, a pole and polar are respectively a point and a line that have a unique reciprocal relationship with respect to a given conic section.
Polar reciprocation in a given circle is the transformation of each point in the plane into it ...
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.
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 standard
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
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.
For each face plane of
described by the linear equation
the 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 a
center of symmetry A fixed point of an isometry group is a point that is a fixed point for every isometry in the group. For any isometry group in Euclidean space the set of fixed points is either empty or an affine space.
For an object, any unique centre and, more ...
, it is common to use a sphere centered on this point, as in the
Dorman Luke construction
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. ...
(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 in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
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, 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
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pro ...
, 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.
Because of the definitional issues for geometric duality of non-convex polyhedra, argues that any proper definition of a non-convex polyhedron should include a notion of a dual polyhedron.
Canonical duals
![Dual compound 6-8 max](https://upload.wikimedia.org/wikipedia/commons/d/d8/Dual_compound_6-8_max.png)
Any convex polyhedron can be distorted into a
canonical form
In mathematics and computer science, a canonical, normal, or standard form of a mathematical object is a standard way of presenting that object as a mathematical expression. Often, it is one which provides the simplest representation of an obje ...
, in which a unit
midsphere
In geometry, the midsphere or intersphere of a polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex poly ...
(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.
Dorman Luke construction
For a
uniform polyhedron
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent.
Uniform polyhedra may be regular (if also fa ...
, each face of the dual polyhedron may be derived from the original polyhedron's corresponding
vertex figure
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...
by using the
Dorman Luke construction
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. ...
.
Topological duality
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 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 discre ...
(the
1-skeleton of the polyhedron), embedded on the surface of the polyhedron (a topological sphere). This graph can be projected to form a
Schlegel diagram
In geometry, a Schlegel diagram is a projection of a polytope from \mathbb^d into \mathbb^ through a point just outside one of its facets. The resulting entity is a polytopal subdivision of the facet in \mathbb^ that, together with the origina ...
on a flat plane. The graph formed by the vertices and edges of the dual polyhedron is the
dual graph
In the mathematical discipline of graph theory, the dual graph of a plane graph is a graph that has a vertex for each face of . The dual graph has an edge for each pair of faces in that are separated from each other by an edge, and a self-loop ...
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.
An
abstract polyhedron
In mathematics, an abstract polytope is an algebraic partially ordered set which captures the dyadic property of a traditional polytope without specifying purely geometric properties such as points and lines.
A geometric polytope is said to be ...
is a certain kind of
partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
(poset) of elements, 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 a
Hasse diagram
In order theory, 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. Concretely, for a partially ordered set ''(S, ≤)'' one represents ea ...
, 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.
Self-dual polyhedra
Topologically, a self-dual polyhedron is one whose dual has exactly the same connectivity between vertices, edges and faces. Abstractly, they have the same
Hasse diagram
In order theory, 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. Concretely, for a partially ordered set ''(S, ≤)'' one represents ea ...
.
A geometrically self-dual polyhedron 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 (that is, a two-dimensional polyhedron) 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 a
regular 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, its
canonical polyhedron
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. Not every polyhedron has a midsphere, but for every convex po ...
, reciprocal about the center of the
midsphere
In geometry, the midsphere or intersphere of a polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex poly ...
.
There are infinitely many geometrically self-dual polyhedra. The simplest infinite family are the canonical
pyramids
A pyramid (from el, πυραμίς ') is a structure whose outer surfaces are triangular and converge to a single step at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilate ...
of ''n'' sides. Another infinite family,
elongated pyramid
In geometry, the elongated pyramids are an infinite set of polyhedra, constructed by adjoining an pyramid to an prism. Along with the set of pyramids, these figures are topologically self-dual.
There are three ''elongated pyramids'' that are ...
s, consists of polyhedra that can be roughly described as a pyramid sitting on top of a
prism
Prism usually refers to:
* Prism (optics), a transparent optical component with flat surfaces that refract light
* Prism (geometry), a kind of polyhedron
Prism may also refer to:
Science and mathematics
* Prism (geology), a type of sedimentary ...
(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.
A self-dual non-convex icosahedron with hexagonal faces was identified by Brückner in 1900.
[Brückner, M.; ''Vielecke und Vielflache: Theorie und Geschichte'', Teubner, Leipzig, 1900.] Other non-convex self-dual polyhedra have been found, under certain definitions of non-convex polyhedra and their duals.
Dual polytopes and tessellations
Duality can be generalized to ''n''-dimensional space and dual
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 -d ...
s; in two dimension these are called
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 self-dual.
The dual of an isogonal (vertex-transitive) polygon is an isotoxal (ed ...
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 an ''n''-dimensional tessellation or
honeycomb
A honeycomb is a mass of Triangular prismatic honeycomb#Hexagonal prismatic honeycomb, hexagonal prismatic Beeswax, wax cells built by honey bees in their beehive, nests to contain their larvae and stores of honey and pollen.
beekeeping, Beekee ...
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 the
regular and
uniform
A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, se ...
polytopes, the dual facets will be polar reciprocals of the original's vertex figure. For example, in four dimensions, the vertex figure of the
600-cell
In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also known as the C600, hexacosichoron and hexacosihedroid. It is also called a tetraplex (abbreviated from " ...
is the
icosahedron
In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons".
There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
; the dual of the 600-cell is the
120-cell
In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, heca ...
, whose facets are
dodecahedra
In geometry, a dodecahedron (Greek , from ''dōdeka'' "twelve" + ''hédra'' "base", "seat" or "face") or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagon ...
, which are the dual of the icosahedron.
Self-dual polytopes and tessellations
![Infinite-order apeirogonal tiling and dual](https://upload.wikimedia.org/wikipedia/commons/0/0a/Infinite-order_apeirogonal_tiling_and_dual.png)
The primary class of self-dual polytopes are
regular polytope
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...
s with
palindromic
A palindrome is a word, number, phrase, or other sequence of symbols that reads the same backwards as forwards, such as the words ''madam'' or ''racecar'', the date and time ''11/11/11 11:11,'' and the sentence: "A man, a plan, a canal – Pana ...
Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...
s. All regular polygons, are self-dual,
polyhedra
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on t ...
of the form ,
4-polytopes of the form ,
5-polytopes of the form , etc.
The self-dual regular polytopes are:
* All
regular polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...
s, .
* Regular
tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...
:
* In general, all regular ''n''-
simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...
es,
* The regular
24-cell
In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, oct ...
in 4 dimensions, .
* The
great 120-cell and the
grand stellated 120-cell
In geometry, the grand stellated 120-cell or grand stellated polydodecahedron is a regular star 4-polytope with Schläfli symbol . It is one of 10 regular Schläfli-Hess polytopes.
It is also one of two such polytopes that is self-dual.
Rela ...
The self-dual (infinite) regular Euclidean
honeycombs are:
*
Apeirogon
In geometry, an apeirogon () or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes.
In some literature, the term "apeirogon" may refer only to the ...
:
*
Square tiling
In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of meaning it has 4 squares around every vertex.
Conway called it a quadrille.
The internal angle of th ...
:
*
Cubic honeycomb
The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation (or honeycomb) in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a re ...
:
* In general, all regular ''n''-dimensional Euclidean
hypercubic honeycombs: .
The self-dual (infinite) regular
hyperbolic
Hyperbolic is an adjective describing something that resembles or pertains to a hyperbola (a curve), to hyperbole (an overstatement or exaggeration), or to hyperbolic geometry.
The following phenomena are described as ''hyperbolic'' because they ...
honeycombs are:
* Compact hyperbolic tilings:
,
, ... .
* Paracompact hyperbolic tiling:
* Compact hyperbolic honeycombs:
,
, and
* Paracompact hyperbolic honeycombs:
,
,
, and
See also
*
Conway polyhedron notation
In geometry, Conway polyhedron notation, invented by John Horton Conway and promoted by George W. Hart, is used to describe polyhedra based on a seed polyhedron modified by various prefix operations.
Conway and Hart extended the idea of using op ...
*
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 self-dual.
The dual of an isogonal (vertex-transitive) polygon is an isotoxal (ed ...
*
Self-dual graph
In the mathematical discipline of graph theory, the dual graph of a plane graph is a graph that has a vertex for each face of . The dual graph has an edge for each pair of faces in that are separated from each other by an edge, and a self-loop ...
*
Self-dual polygon
In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
References
Notes
Bibliography
*.
*.
*.
*.
*.
* .
*.
External links
*
*
*
{{DEFAULTSORT:Dual Polyhedron
Polyhedra
Polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on th ...
Polytopes