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Difference between revisions of "User:Richard Pinch/sandbox-11"

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A symbol encoding classes of polygons, polyhedra, polytopes and tessellations.
 
A symbol encoding classes of polygons, polyhedra, polytopes and tessellations.
  
The symbol $\{p\}$ denotes a regular $p$-gon; the symbol $\{p,q\}$ a polyhedron with faces which are regular $p$-gons, $q$ of which meet at each vertex.  The [[Platonic solid]]s correspond to:
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The symbol $\{p\}$ denotes a regular $p$-gon; the symbol $\{p,q\}$ a polyhedron with faces which are regular $p$-gons, $q$ of which meet at each vertex.  The [[Platonic solids]] correspond to:
 
* [[tetrahedron]]: $\{3,3\}$;
 
* [[tetrahedron]]: $\{3,3\}$;
 
* [[cube]]: $\{4,3\}$;
 
* [[cube]]: $\{4,3\}$;
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The symbol $\{p,q,r\}$ denotes a [[polytope]] in four dimensions or a [[honeycomb]].
 
The symbol $\{p,q,r\}$ denotes a [[polytope]] in four dimensions or a [[honeycomb]].
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The plane symbol may be extended to $\left\lbrace\frac{p}{q}\right\rbrace$ (where $p,q$ are coprime) denoting a $p$-gram or star polygon: a figure inscribed in a regular $p$-gon by joining every $q$-th vertex. 
  
 
====References====
 
====References====
 
* H.S.M. Coxeter,  "Regular complex polytopes" , Cambridge Univ. Press  (1974) ISBN 0-521-20125-X  {{ZBL|0732.51002}}
 
* H.S.M. Coxeter,  "Regular complex polytopes" , Cambridge Univ. Press  (1974) ISBN 0-521-20125-X  {{ZBL|0732.51002}}
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=Glide=
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''glide reflection''
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An indirect (orientation-reversing) Euclidean isometry.  In the plane, given a line $\ell$, a glide with axis $\ell$ is the composite of a translation in the direction of $\ell$ and [[reflection]] in $\ell$ as mirror.  In space, given a plane $\Pi$, a glide is the composite of a translation parallel to $\Pi$ and reflection in $\Pi$.
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The indirect isometries of the Euclidean plane are all glide reflections (including reflections as a special case).
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The indirect isometries of Euclidean space are either glide reflections or [[rotatory reflection]]s (including reflections as a special case).
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====References====
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* H. S. M. Coxeter, "The Beauty of Geometry: Twelve Essays" Dover (1999) ISBN 0486409198  {{ZBL|0941.51001}}
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* E.G. Rees, "Notes on Geometry" Springer (1983)) ISBN 3-540-12053-X {{ZBL|0498.51001}}
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=Rotatory reflection=
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''rotatory inversion''
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An indirect (orientation-reversing) isometry of Euclidean space.  Given a plane $\Pi$ and a line $\ell$ perpendicular to $\Pi$, a rotatory reflection is the composite of a rotation with $\ell$ as axis and reflection in $\Pi$.
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A '''rotatory inversion''': given a line $\ell$ and a point $P$ on $\ell$, the composite of a rotation with $\ell$ as axis and [[central inversion]] (or reflection) in the point $P$. 
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Every rotatory reflection can be expressed as a rotatory inversion, and conversely.
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 +
The indirect isometries of Euclidean space are either rotatory reflections or [[glide reflection]]s (including reflections as a special case).
 +
 +
====References====
 +
* H. S. M. Coxeter, "The Beauty of Geometry: Twelve Essays" Dover (1999) ISBN 0486409198  {{ZBL|0941.51001}}
 +
* E.G. Rees, "Notes on Geometry" Springer (1983)) ISBN 3-540-12053-X {{ZBL|0498.51001}}
 +
 +
 +
=Central inversion=
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''central symmetry'', ''reflection in a point''
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An isometry of a [[Euclidean space]] with respect to a centre $O$.  The image of point $A$ is that point $A'$ on the line $\overline{OA}$ such that $A'O = OA$.  In the Euclidean plane, this is a rotation by a half-turn about the point $O$.
 +
 +
 +
====References====
 +
* H. S. M. Coxeter, "The Beauty of Geometry: Twelve Essays" Dover (1999) ISBN 0486409198  {{ZBL|0941.51001}}
 +
* E.G. Rees, "Notes on Geometry" Springer (1983)) ISBN 3-540-12053-X {{ZBL|0498.51001}}

Latest revision as of 22:35, 17 December 2017

Honeycomb

"A symmetrical subdivision of a three-dimensional manifold into a number of polyhedral cells all alike, each rotation that is a symmetry of a cell being also a symmetry of the entire configuration."

A regular honeycomb is described by a Schläfli symbol $\{p,q,r\}$ denoting polyhedral cells that are Platonic solids $\{p,q\}$, such that every face $\{p\}$ belongs to just two cells, and every edge to $r$ cells.

References

  • H.S.M. Coxeter "Twisted honeycombs", Conference Board of the Mathematical Sciences. Regional Conference Series in Mathematics. No.4. American Mathematical Society (1970) ISBN 0-8218-1653-5 Zbl 0217.46502
  • H.S.M. Coxeter, "Regular complex polytopes" , Cambridge Univ. Press (1974) ISBN 0-521-20125-X Zbl 0732.51002

Schläfli symbol

A symbol encoding classes of polygons, polyhedra, polytopes and tessellations.

The symbol $\{p\}$ denotes a regular $p$-gon; the symbol $\{p,q\}$ a polyhedron with faces which are regular $p$-gons, $q$ of which meet at each vertex. The Platonic solids correspond to:

There are three plane tessellations: $\{3,6\}$, $\{4,4\}$, $\{6,3\}$. The dual solid or tessellation to $\{p,q\}$ is $\{q,p\}$.

The symbol $\{p,q,r\}$ denotes a polytope in four dimensions or a honeycomb.

The plane symbol may be extended to $\left\lbrace\frac{p}{q}\right\rbrace$ (where $p,q$ are coprime) denoting a $p$-gram or star polygon: a figure inscribed in a regular $p$-gon by joining every $q$-th vertex.

References

  • H.S.M. Coxeter, "Regular complex polytopes" , Cambridge Univ. Press (1974) ISBN 0-521-20125-X Zbl 0732.51002

Glide

glide reflection

An indirect (orientation-reversing) Euclidean isometry. In the plane, given a line $\ell$, a glide with axis $\ell$ is the composite of a translation in the direction of $\ell$ and reflection in $\ell$ as mirror. In space, given a plane $\Pi$, a glide is the composite of a translation parallel to $\Pi$ and reflection in $\Pi$.

The indirect isometries of the Euclidean plane are all glide reflections (including reflections as a special case). The indirect isometries of Euclidean space are either glide reflections or rotatory reflections (including reflections as a special case).


References

  • H. S. M. Coxeter, "The Beauty of Geometry: Twelve Essays" Dover (1999) ISBN 0486409198 Zbl 0941.51001
  • E.G. Rees, "Notes on Geometry" Springer (1983)) ISBN 3-540-12053-X Zbl 0498.51001

Rotatory reflection

rotatory inversion

An indirect (orientation-reversing) isometry of Euclidean space. Given a plane $\Pi$ and a line $\ell$ perpendicular to $\Pi$, a rotatory reflection is the composite of a rotation with $\ell$ as axis and reflection in $\Pi$.

A rotatory inversion: given a line $\ell$ and a point $P$ on $\ell$, the composite of a rotation with $\ell$ as axis and central inversion (or reflection) in the point $P$.

Every rotatory reflection can be expressed as a rotatory inversion, and conversely.

The indirect isometries of Euclidean space are either rotatory reflections or glide reflections (including reflections as a special case).

References

  • H. S. M. Coxeter, "The Beauty of Geometry: Twelve Essays" Dover (1999) ISBN 0486409198 Zbl 0941.51001
  • E.G. Rees, "Notes on Geometry" Springer (1983)) ISBN 3-540-12053-X Zbl 0498.51001


Central inversion

central symmetry, reflection in a point

An isometry of a Euclidean space with respect to a centre $O$. The image of point $A$ is that point $A'$ on the line $\overline{OA}$ such that $A'O = OA$. In the Euclidean plane, this is a rotation by a half-turn about the point $O$.


References

  • H. S. M. Coxeter, "The Beauty of Geometry: Twelve Essays" Dover (1999) ISBN 0486409198 Zbl 0941.51001
  • E.G. Rees, "Notes on Geometry" Springer (1983)) ISBN 3-540-12053-X Zbl 0498.51001
How to Cite This Entry:
Richard Pinch/sandbox-11. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-11&oldid=42536