In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most conveniently derived in this way.
The most familiar spherical polyhedron is the soccer ball, thought of as a spherical truncated icosahedron. The next most popular spherical polyhedron is the beach ball, thought of as a hosohedron.
Some "improper" polyhedra, such as hosohedra and their duals, dihedra, exist as spherical polyhedra, but their flat-faced analogs are degenerate. The example hexagonal beach ball, {2, 6}, is a hosohedron, and {6, 2} is its dual dihedron.
The first known man-made polyhedra are spherical polyhedra carved in stone. Many have been found in Scotland, and appear to date from the neolithic period (the New Stone Age).
During the 10th Century, the Islamic scholar Abū al-Wafā' Būzjānī (Abu'l Wafa) wrote the first serious study of spherical polyhedra.
Two hundred years ago, at the start of the 19th Century, Poinsot used spherical polyhedra to discover the four regular star polyhedra.
In the middle of the 20th Century, Coxeter used them to enumerate all but one of the uniform polyhedra, through the construction of kaleidoscopes (Wythoff construction).
All regular polyhedra, semiregular polyhedra, and their duals can be projected onto the sphere as tilings:
Schläfli symbol | {p,q} | t{p,q} | r{p,q} | t{q,p} | {q,p} | rr{p,q} | tr{p,q} | sr{p,q} |
---|---|---|---|---|---|---|---|---|
Vertex config. | pq | q.2p.2p | p.q.p.q | p.2q.2q | qp | q.4.p.4 | 4.2q.2p | 3.3.q.3.p |
Tetrahedral symmetry (3 3 2) | 33 | 3.6.6 | 3.3.3.3 | 3.6.6 | 33 | 3.4.3.4 | 4.6.6 | 3.3.3.3.3 |
V3.6.6 | V3.3.3.3 | V3.6.6 | V3.4.3.4 | V4.6.6 | V3.3.3.3.3 | |||
Octahedral symmetry (4 3 2) | 43 | 3.8.8 | 3.4.3.4 | 4.6.6 | 34 | 3.4.4.4 | 4.6.8 | 3.3.3.3.4 |
V3.8.8 | V3.4.3.4 | V4.6.6 | V3.4.4.4 | V4.6.8 | V3.3.3.3.4 | |||
Icosahedral symmetry (5 3 2) | 53 | 3.10.10 | 3.5.3.5 | 5.6.6 | 35 | 3.4.5.4 | 4.6.10 | 3.3.3.3.5 |
V3.10.10 | V3.5.3.5 | V5.6.6 | V3.4.5.4 | V4.6.10 | V3.3.3.3.5 | |||
Dihedral example (p=6) (2 2 6) | 62 | 2.12.12 | 2.6.2.6 | 6.4.4 | 26 | 2.4.6.4 | 4.4.12 | 3.3.3.6 |
n | 2 | 3 | 4 | 5 | 6 | 7 | ... |
---|---|---|---|---|---|---|---|
n-Prism (2 2 p) | ... | ||||||
n-Bipyramid (2 2 p) | ... | ||||||
n-Antiprism | ... | ||||||
n-Trapezohedron | ... |
Spherical tilings allow cases that polyhedra do not, namely hosohedra: figures as {2,n}, and dihedra: figures as {n,2}. Generally, regular hosohedra and regular dihedra are used.
Space | Spherical | Euclidean | |||||
---|---|---|---|---|---|---|---|
Tiling name | Henagonal hosohedron | Digonal hosohedron | Trigonal hosohedron | Square hosohedron | Pentagonal hosohedron | ... | Apeirogonal hosohedron |
Tiling image | ... | ||||||
Schläfli symbol | {2,1} | {2,2} | {2,3} | {2,4} | {2,5} | ... | {2,∞} |
Coxeter diagram | ... | ||||||
Faces and edges | 1 | 2 | 3 | 4 | 5 | ... | ∞ |
Vertices | 2 | 2 | 2 | 2 | 2 | ... | 2 |
Vertex config. | 2 | 2.2 | 23 | 24 | 25 | ... | 2∞ |
Space | Spherical | Euclidean | |||||
---|---|---|---|---|---|---|---|
Tiling name | Monogonal dihedron | Digonal dihedron | Trigonal dihedron | Square dihedron | Pentagonal dihedron | ... | Apeirogonal dihedron |
Tiling image | ... | ||||||
Schläfli symbol | {1,2} | {2,2} | {3,2} | {4,2} | {5,2} | ... | {∞,2} |
Coxeter diagram | ... | ||||||
Faces | 2 {1} | 2 {2} | 2 {3} | 2 {4} | 2 {5} | ... | 2 {∞} |
Edges and vertices | 1 | 2 | 3 | 4 | 5 | ... | ∞ |
Vertex config. | 1.1 | 2.2 | 3.3 | 4.4 | 5.5 | ... | ∞.∞ |
Spherical polyhedra having at least one inversive symmetry are related to projective polyhedra [1] (tessellations of the real projective plane) – just as the sphere has a 2-to-1 covering map of the projective plane, projective polyhedra correspond under 2-fold cover to spherical polyhedra that are symmetric under reflection through the origin.
The best-known examples of projective polyhedra are the regular projective polyhedra, the quotients of the centrally symmetric Platonic solids, as well as two infinite classes of even dihedra and hosohedra: [2]
In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra.
In geometry, an octahedron is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.
In geometry, a polyhedron is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
In geometry, a 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells. The 4-polytopes were discovered by the Swiss mathematician Ludwig Schläfli before 1853.
In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in n dimensions to form a new figure. Starting with an original figure, the process extends specific elements such as its edges or face planes, usually in a symmetrical way, until they meet each other again to form the closed boundary of a new figure. The new figure is a stellation of the original. The word stellation comes from the Latin stellātus, "starred", which in turn comes from Latin stella, "star". Stellation is the reciprocal or dual process to faceting.
In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron, great rhombicosidodecahedron, omnitruncated dodecahedron or omnitruncated icosahedron is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces.
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 equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.
In spherical geometry, an n-gonalhosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices.
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent.
In geometry, the complete or final stellation of the icosahedron is the outermost stellation of the icosahedron, and is "complete" and "final" because it includes all of the cells in the icosahedron's stellation diagram. That is, every three intersecting face planes of the icosahedral core intersect either on a vertex of this polyhedron or inside of it. It was studied by Max Brückner after the discovery of Kepler–Poinsot polyhedron. It can be viewed as an irregular, simple, and star polyhedron.
In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids.
In geometry, a digon, or a 2-gon, is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visualised in elliptic space. It may also be viewed as a representation of a graph with two vertices, see "Generalized polygon".
A dihedron is a type of polyhedron, made of two polygon faces which share the same set of n edges. In three-dimensional Euclidean space, it is degenerate if its faces are flat, while in three-dimensional spherical space, a dihedron with flat faces can be thought of as a lens, an example of which is the fundamental domain of a lens space L(p,q). Dihedra have also been called bihedra, flat polyhedra, or doubly covered polygons.
In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality.
In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon in which every n – 1 consecutive sides belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a regular polyhedron is a skew polygon such that every two consecutive sides belongs to one of the faces. Petrie polygons are named for mathematician John Flinders Petrie.
In mathematics, a regular 4-polytope or regular polychoron is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.
In geometry, the regular skew polyhedra are generalizations to the set of regular polyhedra which include the possibility of nonplanar faces or vertex figures. Coxeter looked at skew vertex figures which created new 4-dimensional regular polyhedra, and much later Branko Grünbaum looked at regular skew faces.
In geometry, a (globally) projective polyhedron is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations of the sphere – and toroidal polyhedra – tessellations of the toroids.