Concave Hexagons
Inhaltsverzeichnis
Reference
Paul Gailiunas: Concave Hexagons. In: Bridges 2009. Pages 243–250
DOI
Abstract
The tilings (n.3.n.3) exist in the spherical, Euclidean or hyperbolic plane, depending on whether n is less than, equal to, or greater than 6. In all cases the dual tiling consists of rhombi, which can be taken in pairs to form "regular" concave hexagons. In the case of the spherical examples the tilings can be illustrated by colouring the faces of rhombic polyhedra. In the Euclidean plane "regular" concave hexagons allow tilings that cannot be constructed from the dual (6.3.6.3) tiling, some of which allow analogous tilings of non-"regular" concave hexagons. Some Escher-like designs are derived from such tilings.
Some of the possibilities in the hyperbolic plane are briefly considered.
Extended Abstract
Bibtex
Used References
[1] P.Gailiunas, A Polyhedral Byway. Bridges Proceedings 2001, pp.115-122.
[2] S.T.Coffin, The Puzzling World of Polyhedral Dissections, OUP, 1990, pp.126−7.
[3] P.Gailiunas, Spiral Tilings. Bridges Proceedings 2000, pp.133-140, available online at http://www.mi.sanu.ac.yu/vismath/gal/index.html
[4] D.Schattschneider, Visions of Symmetry, W.H.Freeman and Company, 1990.
[5] P.Gailiunas, Some Monohedral Tilings Derived from Regular Polygons, Bridges Donostia Proceedings 2007, p.11.
[6] http://www.bridgesmathart.org/art-exhibits/bridges2007/gailiunas.html
[7] D.Dunham, 168 Butterflies on a Polyhedron of Genus 3, Bridges Proceedings 2002, pp.197-204.
[8] Grünbaum and Shephard, Tilings and Patterns, W.H.Freeman and Company, 1987, p.289.
[9] http://www.mi.sanu.ac.yu/vismath/gailiunas2008/index.html
Links
Full Text
http://archive.bridgesmathart.org/2009/bridges2009-243.pdf
