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Topic: Snub hexagonal tiling

	PolyGloss
		333 The t-truncate, a tiling of a simplex and its polytruncates.
		The dual of the xt{;4;3,8,2} is xt{;2;8,3,4}, a tiling of bioctagon prisms, 288 [248] to a vertex.
		This is a tiling of the simplex and its polytruncates.
	www.geocities.com /os2fan2/gloss.htm (16747 words)

	PS Wiki Encyclopedia (Site not responding. Last check: 2007-10-10)
		Following GrÃ¼nbaum 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.
		Such periodic tilings may be classified by the number of orbits of vertices, edges and tiles.
		Such tilings may also be known as uniform if they are vertex-transitive; there are eight families of such uniform tilings, each family having a real-valued parameter determining the overlap between sides of adjacent tiles or the ratio between the edge lengths of different tiles.
	70.84.119.226 /~puresear/PSWiki/index.php?title=Tilings_of_regular_polygons (683 words)

	The Geometry Junkyard: 3D Geometry
		Aperiodic space-filling tiles: John Conway describes a way of glueing two prisms together to form a shape that tiles space only aperiodically.
		Circumnavigating a cube and a tetrahedron, Henry Bottomley.
		John Conway and Charles Radin describe a three-dimensional generalization of the pinwheel tiling, the mathematics of which is messier due to the noncommutativity of three-dimensional rotations.
	www.ics.uci.edu /~eppstein/junkyard/3d.html (3839 words)

	[No title] (Site not responding. Last check: 2007-10-10)
		The hexagon- hexagon dihedral angle is larger than the hexagon-pentagon dihedral angle in the Euclidean limit (of small edge length) and I think that this always remains true as the edge length is increased (this is the shaky part of the argument).
		The tiling of the pentagon planes is regular, but the tiling of the hexagon planes is not regular -- the hexagon edges in pentagons planes are shorter than the hexagon edges perpendicular to the pentagon planes.
		One thing I'm not sure about: The hexagon- hexagon dihedral angle is larger than the hexagon-pentagon dihedral angle in the Euclidean limit (of small edge length) and I think that this always remains true as the edge length is increased (this is the shaky part of the argument).
	www.cecs.uci.edu /~eppstein/junkyard/buckyball.html (3314 words)

	[No title]
		Choose a vertex C, and let Ai, for i=1,...,m, be the incenter of the of the ith tile incident to C. Also, let Bi be the foot of the perpendicular from Ai to the arc separating the ith and (i+1)st tiles (the (m+1)st being identified as the first).
		Both values can be easily computed for the for the multiple tiling of the sphere by Schwarz triangle: D may be found from the area of (pqr) and the order g of the symmetry group generated by the reflection in its sides: 4πD=g(π/p+π/q+π/r−π), or, D=g/4(1/p+1/q+1/r−1).
		Note that in all the cases, the occurrence of α=π/2 implies that the cor‐ responding tile is a {2}, i.e., a digon, and may be discarded.
	www.math.technion.ac.il /kaleido/doc/uniform.txt (4091 words)

	The Geometry Junkyard: All Topics (Site not responding. Last check: 2007-10-10)
		This shape, constructed by inscribing circular arcs in a spiral tiling of squares, resembles but is not quite the same as a logarithmic spiral.
		The regular tiling by hexagons can be repeatedly subdivided and recombined into a tiling by hexagons 1/7 the size of the original, to form an interesting recursive structure.
		Snowflake reptile hexagonal substitution tiling (sometimes known as the Gosper Island) rediscovered by NASA and conjectured to perform visual processing in the human brain.
	kmh.ync.ac.kr /comScience/vision/junkyard/all.html (10040 words)

	Hyperbolic Planar Tesselations (Site not responding. Last check: 2007-10-10)
		We'll consider spherical, planar, and hyperbolic tilings all at once, using "Schwarz polygons" (a generalization of Schwarz triangles) to generate the symmetry groups, and using a generalized Coxeter-Dynkin symbol to name the resulting tesselations.
		The dual of the fundamental tiling is composed of "Schwarz polygons" denoted (p
		We can color each vertex of the uniform tiling (or Schwarz polygon of the dual) "even" or "odd" depending on whether it is generated as an even or odd number of reflections of a fixed initial vertex (or Schwarz polygon of the dual, respectively), i.e.
	www.hadron.org /%7Ehatch/HyperbolicTesselations (1397 words)

	Multidimensional Glossary
		In the plane, this changes the checkerboard tiling into another checkerboard tiling rotated 45° to the original; in 3-space this produces the uniform honeycomb of regular tetrahedra and octahedra; and in 4-space this produces the regular honeycomb of hexadecachora.
		The possible cells (cell sets) for the ico regiment are the 24 octahedra of the icositetrahedron, the 48 tetrahemihexahedra inscribable in the 24 octahedra, the twelve equatorial cuboctahedra of the icositetrachoron, the twelve octahemioctahedra inscribable in those cuboctahedra, and the twelve hexahemioctahedra inscribable in those cuboctahedra.
		Because each central hexagon of the ico is common to the three octahemioctahedra or hexahemioctahedra that pass through it, no ico polychoron can use all twelve of either cell that are available; a face must belong to exactly two (not three) cells in any polychoron.
	members.aol.com /Polycell/glossary.html (16079 words)

	Hyperbolic Planar Tesselations
		tiling) is uniform if its faces are regular and its symmetry group (including reflections) is transitive on the vertices.
		The fundamental tiling with one of its edge types (i.e.
		A tiling is not uniquely determined by its vertex configuration!
	www.plunk.org /~hatch/HyperbolicTesselations (1397 words)

	Hyperbolic Planar Tesselations
		cube (234) (2,3,3,3,4,3) = (3,3,3,4,3) snub cuboctahedron (235) (4,6,10) truncated icosidodecahedron (235) (2,6,5,6) = (6,5,6) truncated icosahedron (23 5) (5,4,3,4) rhombicosidodecahedron (2 35) (3,10,2,10) = (3,10,10) truncated dodecahedron (23 5) (2,3)
		regular hexagon tiling (236) (2,3,3,3,6,3) = (3,3,3,6,3) snub hexagon tiling (244) (4,8,8) truncated square tiling (244) (2,8,4,8) = (8,4,8) truncated square tiling (24 4) (4,4,4,4) = 4
		square tiling (2 4 4) (same as (24 4)) (244) (2,3,4,3,4,3) = (3,4,3,4,3) snub square tiling (22inf) (4,4,inf) infinity-gonal prism (22inf) (2,4,inf,4) = (4,inf,4) infinity-gonal prism (22 inf) (same as (22inf)) (2 2inf) (2,inf,2,inf) = (inf,inf) infinity-gonal dihedron (22 inf) (2,2)
	www.hadron.org /~hatch/HyperbolicTesselations (1397 words)

	The Geometry Junkyard: All Topics (Site not responding. Last check: 2007-10-10)
		Equilateral pentagons that tile the plane, Livio Zucca.
		Ghost diagrams, Paul Harrison's software for finding tilings with Wang-tile-like hexagonal tiles, specified by matching rules on their edges.
		Marcin Malinowski still has some Escher-like tiling patterns on his home page, although his old Escheresque wallpaper group page seems to be irretrievably gone.
	www.cecs.uci.edu /~eppstein/junkyard/all.html (9742 words)

	The Geometry Junkyard: All Topics
		This page also includes background material on tiling and aperiodicity as well as some of the theory of Penrose tilings.
		A closely related problem of smoothing a triangular mesh by moving points one at a time to optimize the angles of incident triangles can be solved in linear time by LP-type algorithms [Matousek, Sharir, and Welzl, SCG 1992].
		Doug Zare nicely summarizes the shapes that can arise on intersecting a simplex with a hyperplane: if there are p points on the hyperplane, m on one side, and n on the other side, the shape is (a projective transformation of) a p-iterated cone over the product of m-1 and n-1 dimensional simplices.
	www.math.ntnu.edu.tw /~jcchuan/all.html (7425 words)

	[No title]
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	www.math.technion.ac.il /~rl/docs/uniform.ps (1652 words)

	The Geometry Junkyard: Polyhedra and Polytopes (Site not responding. Last check: 2007-10-10)
		The International Bone-Roller's Guild ponders the isohedra: polyhedra that can act as fair dice, because all faces are symmetric to each other.
		Companion site to a middle school text by Jill Britton, with links to many other web sites involving symmetry or tiling.
		See also the section on resistors in the rec.puzzles faq.
	www1.ics.uci.edu /~eppstein/junkyard/polytope.html (2177 words)

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