A noble
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 one which is
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 ...
(all faces the same) and
isogonal (all vertices the same). They were first studied in any depth by Hess and Bruckner in the late 19th century, and later by
Grünbaum.
Classes of noble polyhedra
Those sustaining two transitive properties simultaneously, either isogonal, isohedral or, isotoxal. The presence of regular polygons in the periphery ensues the three at the same time as exemplified in Platonic order. The presence of non-perfectly regular polygons expedites two of them. Thus, there are several main classes of noble polyhedra:
*
Regular polyhedra
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equival ...
.
*
Disphenoid
In geometry, a disphenoid () is a tetrahedron whose four faces are congruent acute-angled triangles. It can also be described as a tetrahedron in which every two edges that are opposite each other have equal lengths. Other names for the same sh ...
tetrahedra.
*
Crown polyhedra or Stephanoids. An infinite series of toroids.
* A variety of miscellaneous examples, e.g. the
stellated icosahedra D and
H, or their duals.
[Coxeter, ''Regular Polytopes'', 3rd ed. (1973), p. 117] It is not known whether there are finitely many, and if so how many might remain to be discovered.
If we allow some of Grünbaum's stranger constructions as polyhedra, then we have two more infinite series of toroids:
* Wreath polyhedra. These have triangular faces in coplanar pairs which share an edge.
* V-faced polyhedra. These have vertices in coincident pairs and degenerate faces.
Duality of noble polyhedra
We can distinguish between dual structural forms (topologies) on the one hand, and dual geometrical arrangements when reciprocated about a concentric sphere, on the other. Where the distinction is not made below, the term 'dual' covers both kinds.
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 noble polyhedron is also noble. Many are also self-dual:
* The nine regular polyhedra form dual pairs, with the tetrahedron being self-dual.
* The disphenoid tetrahedra are all topologically identical. Geometrically they come in dual pairs – one elongated, and one correspondingly squashed.
* A crown polyhedron is topologically self-dual. It does not seem to be known whether any geometrically self-dual examples exist.
* The wreath and V-faced polyhedra are dual to each other.
References
*
Grünbaum, B.; Polyhedra with hollow faces, ''Proc. NATO-ASI Conf. on polytopes: abstract, convex and computational, Toronto 1983,'' Ed. Bisztriczky, T. Et Al., Kluwer Academic (1994), pp. 43–70.
*
Grünbaum, B.
Are your polyhedra the same as my polyhedra?Discrete and Computational Geometry: The Goodman-Pollack Festschrift. B. Aronov, S. Basu, J. Pach, and Sharir, M., eds. Springer, New York 2003, pp. 461–488.
Polyhedra
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