Hosohedron
Set of regularn-gonal hosohedra | |
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![]() Exampleregularhexagonal hosohedron on a sphere | |
Type | regularpolyhedronorspherical tiling |
Faces | ndigons |
Edges | n |
Vertices | 2 |
Euler char. | 2 |
Vertex configuration | 2n |
Wythoff symbol | n| 2 2 |
Schläfli symbol | {2,n} |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() |
Symmetry group | Dnh [2,n] (*22n) order4n |
Rotation group | Dn [2,n]+ (22n) order2n |
Dual polyhedron | regularn-gonaldihedron |
![](https://upload.wikimedia.org/wikipedia/commons/thumb/1/1e/BeachBall.jpg/220px-BeachBall.jpg)
Inspherical geometry,ann-gonalhosohedronis atessellationofluneson aspherical surface,such that each lune shares the same twopolar oppositevertices.
Aregularn-gonal hosohedron hasSchläfli symbol{2,n},with eachspherical lunehavinginternal angle2π/nradians(360/ndegrees).[1][2]
Hosohedra as regular polyhedra
[edit]For a regular polyhedron whose Schläfli symbol is {m,n}, the number of polygonal faces is:
ThePlatonic solidsknown to antiquity are the only integer solutions form≥ 3 andn≥ 3. The restrictionm≥ 3 enforces that the polygonal faces must have at least three sides.
When considering polyhedra as aspherical tiling,this restriction may be relaxed, sincedigons(2-gons) can be represented asspherical lunes,having non-zeroarea.
Allowingm= 2 makes
and admits a new infinite class of regular polyhedra, which are the hosohedra. On a spherical surface, the polyhedron {2,n} is represented asnabutting lunes, with interior angles of2π/n.All these spherical lunes share two common vertices.
Space | Spherical | Euclidean | |||||
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Tiling name |
Henagonal hosohedron |
Digonal hosohedron |
Trigonal hosohedron |
Square hosohedron |
Pentagonal hosohedron |
... | Apeirogonal hosohedron |
Tiling image |
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... | ![]() |
Schläfli symbol |
{2,1} | {2,2} | {2,3} | {2,4} | {2,5} | ... | {2,∞} |
Coxeter diagram |
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... | ![]() ![]() ![]() ![]() ![]() |
Faces and edges |
1 | 2 | 3 | 4 | 5 | ... | ∞ |
Vertices | 2 | 2 | 2 | 2 | 2 | ... | 2 |
Vertex config. |
2 | 2.2 | 23 | 24 | 25 | ... | 2∞ |
Kaleidoscopic symmetry
[edit]Thedigonalspherical lunefaces of a-hosohedron,,represent the fundamental domains ofdihedral symmetry in three dimensions:the cyclic symmetry,,,order.The reflection domains can be shown by alternately colored lunes as mirror images.
Bisecting each lune into two spherical triangles creates an-gonalbipyramid,which represents thedihedral symmetry,order.
Symmetry (order) | Schönflies notation | |||||||
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Orbifold notation | ||||||||
Coxeter diagram | ![]() ![]() ![]() |
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-gonal hosohedron | Schläfli symbol | |||||||
Alternately colored fundamental domains | ![]() |
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Relationship with the Steinmetz solid
[edit]The tetragonal hosohedron is topologically equivalent to thebicylinder Steinmetz solid,the intersection of two cylinders at right-angles.[3]
Derivative polyhedra
[edit]Thedualof the n-gonal hosohedron {2,n} is then-gonaldihedron,{n,2}. The polyhedron {2,2} 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 atruncatedvariation. The truncatedn-gonal hosohedron is the n-gonalprism.
Apeirogonal hosohedron
[edit]In the limit, the hosohedron becomes anapeirogonal hosohedronas a 2-dimensional tessellation:
Hosotopes
[edit]Multidimensionalanalogues in general are calledhosotopes.A regular hosotope withSchläfli symbol{2,p,...,q} has two vertices, each with avertex figure{p,...,q}.
Thetwo-dimensional hosotope,{2}, is adigon.
Etymology
[edit]The term “hosohedron” appears to derive from the Greek ὅσος (hosos) “as many”, the idea being that a hosohedron can have “as manyfaces as desired”.[4]It was introduced by Vito Caravelli in the eighteenth century.[5]
See also
[edit]![](https://upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png)
References
[edit]- ^Coxeter,Regular polytopes,p. 12
- ^Abstract Regular polytopes, p. 161
- ^Weisstein, Eric W."Steinmetz Solid".MathWorld.
- ^Steven Schwartzman (1 January 1994).The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English.MAA. pp.108–109.ISBN978-0-88385-511-9.
- ^Coxeter, H.S.M. (1974).Regular Complex Polytopes.London: Cambridge University Press. p. 20.ISBN0-521-20125-X.
The hosohedron {2,p} (in a slightly distorted form) was named by Vito Caravelli (1724–1800)…
- McMullen, Peter;Schulte, Egon (December 2002),Abstract Regular Polytopes(1st ed.),Cambridge University Press,ISBN0-521-81496-0
- Coxeter, H.S.M,Regular Polytopes(third edition), Dover Publications Inc.,ISBN0-486-61480-8