Difference between revisions of "User:Richard Pinch/sandbox-11"
(Start article: Honeycomb) |
(Start article: Schläfli symbol) |
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====References==== | ====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 "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 solid]]s correspond to: | ||
| + | * [[tetrahedron]]: $\{3,3\}$; | ||
| + | * [[cube]]: $\{4,3\}$; | ||
| + | * [[octahedron]]: $\{3,4\}$; | ||
| + | * [[dodecahedron]]: $\{5,3\}$; | ||
| + | * [[icosahedron]]: $\{3,5\}$. | ||
| + | There are three plane [[tessellation]]s: $\{3,6\}$, $\{4,4\}$, $\{6,3\}$. 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]]. | ||
| + | |||
| + | ====References==== | ||
| + | * H.S.M. Coxeter, "Regular complex polytopes" , Cambridge Univ. Press (1974) ISBN 0-521-20125-X {{ZBL|0732.51002}} | ||
Revision as of 18:32, 15 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:
- tetrahedron: $\{3,3\}$;
- cube: $\{4,3\}$;
- octahedron: $\{3,4\}$;
- dodecahedron: $\{5,3\}$;
- icosahedron: $\{3,5\}$.
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.
References
- H.S.M. Coxeter, "Regular complex polytopes" , Cambridge Univ. Press (1974) ISBN 0-521-20125-X Zbl 0732.51002
Richard Pinch/sandbox-11. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Richard_Pinch/sandbox-11&oldid=42532