A noble
polyhedron
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...
is one which is
isohedral (all faces the same) and
isogonal (all vertices the same). They were first studied in any depth by
Edmund Hess and
Max Brückner in the late 19th century, and later by
Branko Grünbaum
Branko Grünbaum (; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descent[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 eq ...](_blank)
, that is, the five
Platonic solids
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 edge ...
and the four
Kepler–Poinsot polyhedra.
*
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 ...
tetrahedra.
*
Crown polyhedra, also known as stephanoid polyhedra.
* A variety of miscellaneous examples, e.g. the
stellated icosahedra D and
H, or their duals.
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 (besides the crown polyhedra mentioned above):
* 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 of a noble polyhedron is also noble. Many are also self-dual:
* The five 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.
Generating other noble polyhedra
In 2008, Robert Webb discovered a new noble polyhedron, a
faceting
Stella octangula as a faceting of the cube
In geometry, faceting (also spelled facetting) is the process of removing parts of a polygon, polyhedron or polytope, without creating any new Vertex (geometry), vertices.
New edges of a faceted po ...
of the
snub cube.
This was the first new class of noble polyhedra (with chiral octahedral symmetry) to be discovered since Brückner's work over a century before. In 2020, Ulrich Mikloweit generated 52 noble polyhedra by extending isohedral facetings of
uniform polyhedra
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive—there is an isometry mapping any vertex onto any other. It follows that all vertices are congruent. Uniform polyhedra may be regular (if also fac ...
, of which 24 were already described by Brückner and 19 were entirely new.
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?{{Webarchive, url=https://web.archive.org/web/20160803160413/http://www.math.washington.edu/~grunbaum/Your%20polyhedra-my%20polyhedra.pdf , date=2016-08-03 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.
External links
List of noble polyhedraat Polytope Wiki
Polyhedra