A dual uniform polyhedron is the
dual
Dual or Duals may refer to:
Paired/two things
* Dual (mathematics), a notion of paired concepts that mirror one another
** Dual (category theory), a formalization of mathematical duality
*** see more cases in :Duality theories
* Dual (grammatical ...
of 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 ...
. Where a uniform polyhedron is
vertex-transitive
In geometry, a polytope (e.g. a polygon or polyhedron) or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of fa ...
, a dual uniform polyhedron is
face-transitive
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 congrue ...
.
Enumeration
The face-transitive polyhedra comprise the set of 9 regular polyhedra, two finite sets comprising 66 non-regular polyhedra, and two infinite sets:
* 5 regular 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 e ...
s, which are dual to each other (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 ...
is its own dual).
* 4 regular star
Kepler-Poinsot solids, which are dual to each other.
* 13 convex
Catalan solid
In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. There are 13 Catalan solids. They are named for the Belgian mathematician Eugène Catalan, who first described them in 1865.
The Catalan s ...
s, which are dual to the uniform convex
Archimedean solids
In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed o ...
.
* 53 star polyhedra, which are dual to the
uniform star polyhedra
In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting. Each polyhedron can contain either star polygon faces, star polygon vertex figures, ...
.
* The infinite series of
bipyramid
A (symmetric) -gonal bipyramid or dipyramid is a polyhedron formed by joining an -gonal pyramid and its mirror image base-to-base. An -gonal bipyramid has triangle faces, edges, and vertices.
The "-gonal" in the name of a bipyramid does ...
s, which are dual to the uniform
prisms, both convex and star.
* The infinite series of
trapezohedra
In geometry, an trapezohedron, -trapezohedron, -antidipyramid, -antibipyramid, or -deltohedron is the dual polyhedron of an antiprism. The faces of an are congruent and symmetrically staggered; they are called ''twisted kites''. With a hig ...
, which are dual to the uniform
antiprism
In geometry, an antiprism or is a polyhedron composed of two parallel direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway notation .
Antiprisms are a subclass ...
s, both convex and star.
The full set are described by Wenninger, together with instructions for constructing models, in his book ''Dual Models''.
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 ...
, 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 of a polyhedron. Mark a point somewhere along each connected edge. Draw lines ...
by using the Dorman Luke construction.
[, p. 117; , p. 30.]
As an example, the illustration below shows the vertex figure (red) of the
cuboctahedron
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it ...
being used to derive the corresponding face (blue) of the
rhombic dodecahedron
In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.
Properties
The rhombic dodecahed ...
.
Dorman Luke's construction proceeds as follows:
#Mark the points ''A'', ''B'', ''C'', ''D'' of each edge connected to the vertex ''V'' (in this case, the midpoints) such that ''VA'' = ''VB'' = ''VC'' = ''VD''.
#Draw the vertex figure ''ABCD''.
#Draw the circumcircle of ''ABCD''.
#Draw the line tangent to the circumcircle at each corner ''A'', ''B'', ''C'', ''D''.
#Mark the points ''E'', ''F'', ''G'', ''H'' where each two adjacent tangent lines meet.
The line segments ''EF'', ''FG'', ''GH'', ''HE'' are already drawn, as parts of the tangent lines. The polygon ''EFGH'' is the face of the dual polyhedron that corresponds to the original vertex ''V''.
In this example, the size of the vertex figure was chosen so that its circumcircle lies on the
intersphere
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 con ...
of the cuboctahedron, which also becomes the intersphere of the dual rhombic dodecahedron. Dorman Luke's construction can only be used when a polyhedron has such an intersphere, so that the vertex figure has a circumcircle. For instance, it can be applied to the
uniform polyhedra
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 f ...
.
See also
*
List of uniform polyhedra
In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive ( transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are c ...
Notes
References
*.
*.
*
* {{cite book , first=Magnus , last=Wenninger , authorlink=Magnus Wenninger , title=Dual Models , publisher=Cambridge University Press , year=1983 , isbn=0-521-54325-8