spherical geometry 300px, A sphere with a spherical triangle on it. Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2-dimensional surface and other terms like "ball" or "solid sp ...

, an

-gonal In mathematics, a polygonal number is a number represented as dots or pebbles arranged in the shape of a regular polygon. The dots are thought of as alphas (units). These are one type of 2-dimensional figurate numbers. Definition and examples ...

hosohedron is a

tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of g ...

of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular -gonal hosohedron has

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mor ...

with each

spherical lune In spherical geometry, a spherical lune (or biangle) is an area on a sphere bounded by two half great circles which meet at antipodal points. It is an example of a digon, θ, with dihedral angle θ. The word "lune" derives from ''luna'', the ...

having

internal angle In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) i ...

radian The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...

s ( degrees).

Hosohedra as regular polyhedra

For a regular polyhedron whose Schläfli symbol is , the number of polygonal faces is : :

N_2=\frac.

The

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 edge ...

s known to antiquity are the only integer solutions for ''m'' ≥ 3 and ''n'' ≥ 3. The restriction ''m'' ≥ 3 enforces that the polygonal faces must have at least three sides. When considering polyhedra as a

spherical tiling 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 ...

, this restriction may be relaxed, since

digon In geometry, a digon 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 vis ...

s (2-gons) can be represented as

s, having non-zero

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open ...

. Allowing ''m'' = 2 makes :

N_2=\frac=n,

and admits a new infinite class of regular polyhedra, which are the hosohedra. On a spherical surface, the polyhedron is represented as ''n'' abutting lunes, with interior angles of . All these spherical lunes share two common vertices.

Kaleidoscopic symmetry

The

2n

digonal

faces of a

2n

-hosohedron,

\

, represent the fundamental domains of

dihedral symmetry in three dimensions In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group that as an abstract group is a dihedral group Dih''n'' (for ''n'' ≥ 2). Types Ther ...

: the cyclic symmetry

C_

/math>, (*nn), order 2n . The reflection domains can be shown by alternately colored lunes as mirror images.

Bisecting each lune into two spherical triangles creates an n -gonal

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 n ...

, which represents the

dihedral symmetry In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, g ...

D_

, order

4n

Relationship with the Steinmetz solid

The tetragonal hosohedron is topologically equivalent to the bicylinder Steinmetz solid, the intersection of two cylinders at right-angles.

Derivative polyhedra

The dual of the n-gonal hosohedron is the ''n''-gonal

dihedron 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 fla ...

, . The polyhedron is self-dual, and is both a hosohedron and a dihedron. A hosohedron may be modified in the same manner as the other polyhedra to produce a truncated variation. The truncated ''n''-gonal hosohedron is the n-gonal prism.

Apeirogonal hosohedron

In the limit, the hosohedron becomes an

apeirogonal hosohedron In geometry, an apeirogonal hosohedron or infinite hosohedronConway (2008), p. 263 is a tiling of the plane consisting of two vertices at infinity. It may be considered an improper regular tiling of the Euclidean plane, with Schläfli symbol ...

as a 2-dimensional tessellation: : Apeirogonal hosohedron

Hosotopes

Multidimensional analogues in general are called hosotopes. A regular hosotope with

has two vertices, each with a

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 l ...

. The two-dimensional hosotope, , is a

Etymology

The term “hosohedron” appears to derive from the Greek ὅσος (''hosos'') “as many”, the idea being that a hosohedron can have “as many faces as desired”. It was introduced by Vito Caravelli in the eighteenth century.

References

* * Coxeter, H.S.M, ''Regular Polytopes'' (third edition), Dover Publications Inc.,

External links

* {{Tessellation Polyhedra Tessellation Regular polyhedra