A dual uniform polyhedron is the
dual of a
uniform polyhedron. Where a uniform polyhedron is
vertex-transitive, a dual uniform polyhedron is
face-transitive.
Enumeration
The face-transitive polyhedra comprise a 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 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:
regular 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 ...
,
regular octahedron,
regular dodecahedron, and
regular icosahedron. The regular octahedron is dual to the cube, and the regular icosahedron is dual to the regular dodecahedron. The regular tetrahedron is
self-dual, meaning its dual is the regular tetrahedron itself.
* 4 regular star
Kepler–Poinsot solids:
great dodecahedron,
small stellated dodecahedron,
great icosahedron, and
great stellated dodecahedron. The great dodecahedron is dual to the small stellated dodecahedron, and the great icosahedron is dual to the great stellated dodecahedron.
* 13 convex
Catalan solids, which are dual to the uniform convex
Archimedean solids
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 ...
.
* 53 star polyhedra, which are dual to the
uniform star polyhedra.
* The infinite series of
bipyramids, which are dual to the uniform
prisms, both convex and star.
* The infinite series of
trapezohedra, which are dual to the uniform
antiprisms, 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, each face of the dual polyhedron may be derived from the original polyhedron's corresponding
vertex figure by using the Dorman Luke construction.
Dorman Luke's construction proceeds as follows:
#Mark the points , , , of each edge connected to the vertex (in this case, the midpoints) such that .
#Draw the vertex figure .
#Draw the circumcircle of .
#Draw the line tangent to the circumcircle at each corner , , , .
#Mark the points , , , , where each two adjacent tangent lines meet.
The line segments , , , 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 .
In this example, 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 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.
See also
*
List of uniform polyhedra
Notes
References
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* {{cite book , first=Magnus , last=Wenninger , authorlink=Magnus Wenninger , title=Dual Models , publisher=Cambridge University Press , year=1983 , isbn=0-521-54325-8