Tiling by regular polygons

Tiling by regular polygons

Plane tilings by regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in "Harmonices Mundi".

Regular tilings

Following Grünbaum and Shephard (section 1.3), a tiling is said to be "regular" if the symmetry group of the tiling acts transitively on the "flags" of the tiling, where a flag is a triple consisting of a mutually incident vertex, edge and tile of the tiling. This means that for every pair of flags there is a symmetry operation mapping the first flag to the second. This is equivalent to the tiling being an edge-to-edge tiling by congruent regular polygons. There must be six equilateral triangles, four squares or three regular hexagons at a vertex, yielding the three "regular tessellations".

See also


* List of uniform tilings
* Wythoff symbol
* Tessellation
* Wallpaper group
* Regular polyhedron (the Platonic solids)
* Semiregular polyhedron (including the Archimedean solids)
* Hyperbolic geometry
* Penrose tiling

References

*
*
* D. M. Y. Sommerville, "An Introduction to the Geometry of n Dimensions." New York, E. P. Dutton, 1930. 196 pp. (Dover Publications edition, 1958) Chapter X: The Regular Polytopes

External links

Euclidean and general tiling links:

* cite web
author = Dutch, Steve
title = Uniform Tilings
url = http://www.uwgb.edu/dutchs/symmetry/uniftil.htm
accessdate = 2006-09-09
* cite web
author = Mitchell, K
title = Semi-Regular Tilings
url = http://people.hws.edu/mitchell/tilings/Part1.html
accessdate = 2006-09-09
*
** MathWorld | urlname=DemiregularTessellation | title=Demiregular tessellation

Hyperbolic tiling links:

* cite web
author = Eppstein, David
authorlink = David Eppstein
title = The Geometry Junkyard: Hyperbolic Tiling
url = http://www.ics.uci.edu/~eppstein/junkyard/hypertile.html
accessdate = 2006-09-09

* cite web
author = Hatch, Don
title = Hyperbolic Planar Tessellations
url = http://www.hadron.org/~hatch/HyperbolicTesselations/
accessdate = 2006-09-09

* cite web
author = Joyce, David
title = Hyperbolic Tessellations
url = http://aleph0.clarku.edu/~djoyce/poincare/poincare.html
accessdate = 2006-09-09


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