SafekipediaIn spherical geometry, an n‑gonal hosohedron is a special way of covering a sphere with shapes called lunes. Imagine slicing a sphere with thin wedges, like slices of pizza, but each slice connects two points directly across the sphere from each other. These points are known as polar opposite vertices, and each wedge or lune meets at these points.
A regular n‑gonal hosohedron follows a precise pattern and is described using something called a Schläfli symbol, written as {2, n}. This symbol tells us exactly how the shape is built. Each spherical lune in this pattern has a specific size, measured by its internal angle. That angle is calculated as 2π/n radians, or simply 360/n degrees, which means the size of each wedge changes depending on the number of sides, n.
Hosohedra are interesting because they show how flat shapes can cover a curved surface in a very orderly way. They help mathematicians understand the relationships between different geometric shapes and how they fit together on spheres.
Hosohedra as regular polyhedra
Further information: List of regular polytopes and compounds § Spherical 2
In geometry, a hosohedron is a special kind of shape on a sphere. It is made up of sections called lunes, which are like slices of the sphere between two points. Each lune has the same two points at the ends.
When we look at these shapes as patterns on a sphere, we can include shapes with just two sides, called digons. This lets us create a whole new group of regular polyhedra called hosohedra. On a sphere, a hosohedron with n sides looks like n lunes all meeting together, and each lune has an angle of 360 divided by n degrees. All these lunes share the same two points on the sphere.
The main shapes known from ancient times, called Platonic solids, need each face to have at least three sides. But on a sphere, we can relax this rule and include these interesting two-sided shapes.
A regular trigonal hosohedron, {2,3}, represented as a tessellation of 3 spherical lunes on a sphere. | A regular tetragonal hosohedron, {2,4}, represented as a tessellation of 4 spherical lunes on a sphere. |
Kaleidoscopic symmetry
The many flat sections called "lunes" on a special shape called a hosohedron show a type of symmetry called dihedral symmetry. These lunes can be colored in alternating ways to show how they reflect and repeat.
If you split each lune into two smaller pieces, you get a shape called a bipyramid. This shows an even larger symmetry group.
| Symmetry (order 2 n {\displaystyle 2n} ) | Schönflies notation | C n v {\displaystyle C_{nv}} | C 1 v {\displaystyle C_{1v}} | C 2 v {\displaystyle C_{2v}} | C 3 v {\displaystyle C_{3v}} | C 4 v {\displaystyle C_{4v}} | C 5 v {\displaystyle C_{5v}} | C 6 v {\displaystyle C_{6v}} |
|---|---|---|---|---|---|---|---|---|
| Orbifold notation | ( ∗ n n ) {\displaystyle (*nn)} | ( ∗ 11 ) {\displaystyle (*11)} | ( ∗ 22 ) {\displaystyle (*22)} | ( ∗ 33 ) {\displaystyle (*33)} | ( ∗ 44 ) {\displaystyle (*44)} | ( ∗ 55 ) {\displaystyle (*55)} | ( ∗ 66 ) {\displaystyle (*66)} | |
| Coxeter diagram |   | |  |  |  |  |  | |
| [ n ] {\displaystyle [n]} | [ ] {\displaystyle [\,\,]} | [ 2 ] {\displaystyle } | [ 3 ] {\displaystyle } | [ 4 ] {\displaystyle } | [ 5 ] {\displaystyle } | [ 6 ] {\displaystyle } | ||
| 2 n {\displaystyle 2n} -gonal hosohedron | Schläfli symbol | { 2 , 2 n } {\displaystyle \{2,2n\}} | { 2 , 2 } {\displaystyle \{2,2\}} | { 2 , 4 } {\displaystyle \{2,4\}} | { 2 , 6 } {\displaystyle \{2,6\}} | { 2 , 8 } {\displaystyle \{2,8\}} | { 2 , 10 } {\displaystyle \{2,10\}} | { 2 , 12 } {\displaystyle \{2,12\}} |
| Alternately colored fundamental domains |  |  |  |  |  |  | ||
Relationship with the Steinmetz solid
The tetragonal hosohedron is the same shape as the bicylinder Steinmetz solid. This solid is formed when two cylinders intersect at right angles to each other.
Derivative polyhedra
The dual of the n-gonal hosohedron {2, n} is the n-gonal dihedron, {n, 2}. The polyhedron {2,2} is special because it is self-dual, meaning it is both a hosohedron and a dihedron.
Like other polyhedra, a hosohedron can be changed to create a new shape. When we truncate, or cut off, the n-gonal hosohedron, we get an n-gonal prism.
Main article: Dual
Main articles: Truncated , Prism
Apeirogonal hosohedron
In the limit, the hosohedron becomes an apeirogonal hosohedron as a 2-dimensional tessellation. This means it stretches out in a way that covers an infinite flat surface with its shapes.
Hosotopes
Further information: List of regular polytopes and compounds § Spherical 3
Multidimensional versions of these shapes are called hosotopes. A regular hosotope with a special symbol called a Schläfli symbol {2, p, ..., q} has just two points, and each point has a pattern called a vertex figure {p,...,q}.
In two dimensions, the hosotope {2} is known as a digon. This means it is a simple shape with two sides on a flat or curved surface.
Etymology
The word "hosohedron" comes from an ancient Greek word meaning "as many." This name reflects that a hosohedron can have many faces, depending on how it is made. The term was first used by a person named Vito Caravelli a long time ago in the 1700s.
This article is a child-friendly adaptation of the Wikipedia article on Hosohedron, available under CC BY-SA 4.0.
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