“Hosohedron”的意思、由来-开放百科全书

词条

Hosohedron

释义

Hosohedra as regular polyhedra
Kaleidoscopic symmetry
Relationship with the Steinmetz solid
Derivative polyhedra
Apeirogonal hosohedron
Hosotopes
Etymology
See also
References
External links

{{Infobox polyhedron
| name =Set of regular n-gonal hosohedra
| image =Hexagonal Hosohedron.svg
| caption =Example hexagonal hosohedron on a sphere
| type =Regular polyhedron or spherical tiling
| euler = 2
| faces =n digons
| edges =n
| vertices =2
| vertex_config =2ⁿ
| schläfli ={2,n}
| wythoff =n {{!}} 2 2
| coxeter ={{CDD|node_1|2x|node|n|node}}
| symmetry =D_nh, [2,n], (*22n), order 4n
| rotsymmetry =D_n, [2,n]⁺, (22n), order 2n
| surface_area =
| volume =
| angle =
| dual =dihedron
| properties =
| vertex_figure =
| net =}}

In geometry, an n-gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices.

A regular n-gonal hosohedron has Schläfli symbol {2, n}, with each spherical lune having internal angle {{sfrac|2{{pi}}|n}} radians ({{sfrac|360|n}} degrees).^[1]^[2]

Hosohedra as regular polyhedra

For a regular polyhedron whose Schläfli symbol is {m, n}, the number of polygonal faces may be found by:

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. Allowing m = 2 admits a new infinite class of regular polyhedra, which are the hosohedra. On a spherical surface, the polyhedron {2, n} is represented as n abutting lunes, with interior angles of {{sfrac|2{{pi}}|n}}. All these lunes share two common vertices.

A regular trigonal hosohedron, {2,3}, represented as a tessellation of 3 spherical lunes on a sphere.	A regular tetragonal hosohedron, represented as a tessellation of 4 spherical lunes on a sphere.

Kaleidoscopic symmetry

The digonal (lune) faces of a 2n-hosohedron, {2,2n}, represents the fundamental domains of dihedral symmetry in three dimensions: C_nv, [n], (*nn), order 2n. The reflection domains can be shown as alternately colored lunes as mirror images. Bisecting the lunes into two spherical triangles creates bipyramids and define dihedral symmetry D_nh, order 4n.

Symmetry	C_1v,	C_2v, [2]	C_3v, [3]	C_4v, [4]	C_5v, [5]	C_6v, [6]
Hosohedron	{2,2	{2,4	{2,6	{2,8	{2,10	{2,12
Fundamental domains

Relationship with the Steinmetz solid

The tetragonal hosohedron is topologically equivalent to the bicylinder Steinmetz solid, the intersection of two cylinders at right-angles.^[3]

Derivative polyhedra

The dual of the n-gonal hosohedron {2, n} is the n-gonal dihedron, {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 a truncated variation. The truncated n-gonal hosohedron is the n-gonal prism.

Apeirogonal hosohedron

In the limit the hosohedron becomes an apeirogonal hosohedron as a 2-dimensional tessellation:

Hosotopes

Multidimensional analogues in general are called hosotopes. A regular hosotope with Schläfli symbol {2,p,...,q} has two vertices, each with a vertex figure {p,...,q}.

The two-dimensional hosotope, {2}, is a digon.

Etymology

The term “hosohedron” was coined by H.S.M. Coxeter, and possibly derives from the Greek ὅσος (hosos) “as many”, the idea being that a hosohedron can have “as many faces as desired”.^[4]

References

1. ^Coxeter, Regular polytopes, p. 12
2. ^Abstract Regular polytopes, p. 161
3. ^{{mathworld|urlname=SteinmetzSolid|title=Steinmetz Solid}}
4. ^{{cite book|author=Steven Schwartzman|title=The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English|url=https://books.google.com/books?id=SRw4PevE4zUC&pg=PA109|date=1 January 1994|publisher=MAA|isbn=978-0-88385-511-9|pages=108–109}}

{{citation | last1 = McMullen | first1 = Peter | author1-link = Peter McMullen | first2 = Egon | last2 = Schulte | title = Abstract Regular Polytopes | edition = 1st | publisher = Cambridge University Press | isbn = 0-521-81496-0 |date=December 2002 }}
Coxeter, H.S.M; Regular Polytopes (third edition). Dover Publications Inc. {{ISBN|0-486-61480-8}}

External links

{{mathworld | urlname = Hosohedron | title = Hosohedron}}

3 : Polyhedra|Tessellation|Regular polyhedra

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。