In
spherical geometry, 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 of
lunes on a
spherical surface, such that each lune shares the same two
polar opposite
A polar opposite is the diametrically opposite point of a circle or sphere. It is mathematically known as an antipodal point, or antipode when referring to the Earth. It is also an idiom often used to describe people and ideas that are opposites.
...
vertices.
A
regular -gonal hosohedron has
Schläfli symbol with each
spherical lune having
internal angle radians ( degrees).
Hosohedra as regular polyhedra
For a regular polyhedron whose Schläfli symbol is , the number of polygonal faces is :
:
The
Platonic solids 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, this restriction may be relaxed, since
digons (2-gons) can be represented as
spherical lunes, 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 s ...
.
Allowing ''m'' = 2 makes
:
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
digonal
spherical lune faces of a
-hosohedron,
, represent the fundamental domains of
dihedral symmetry in three dimensions: the cyclic symmetry
,