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Topic: Tessellation

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	Math Forum: What Is a Tessellation?
		The word "tessellate" is derived from the Ionic version of the Greek word "tesseres," which in English means "four." The first tilings were made from square tiles.
		Since the regular polygons in a tessellation must fill the plane at each vertex, the interior angle must be an exact divisor of 360 degrees.
		Since a mosaic extends over a given area without leaving any region uncovered, the geometric meaning of the word tessellate is "to cover the plane with a pattern in such a way as to leave no region uncovered." By extension, space or hyperspace may also be tessellated.
	mathforum.org /sum95/suzanne/whattess.html (753 words)

	Cool math Lessons - Geometry - What are Tessellations?
		Basically, a tessellation is a way to tile a floor (that goes on forever) with shapes so that there is no overlapping and no gaps.
		These tessellations are both made up of hexagons and triangles, but their vertex configuration is different.
		To name a tessellation, simply work your way around one vertex counting the number of sides of the polygons that form that vertex.
	www.coolmath.com /tesspag1.htm (320 words)

	NationMaster - Encyclopedia: Tessellation
		A tessellation of a plane (or of space) is any subdivision of the plane (or space) into regions or "cells" that border each other exactly, with no gaps in between.
		The four color theorem states that for every tessellation of the plane, with a set of four available colors, each tile can be colored in one color such that no tiles of equal color meet at a curve of positive length.
		Copies of an arbitrary quadrilateral can form a tessellation with 2-fold rotational centers at the midpoints of all sides, and translational symmetry with as minimal set of translation vectors a pair according to the diagonals of the quadrilateral, or equivalently, one of these and the sum or difference of the two.
	www.nationmaster.com /encyclopedia/Tessellation (566 words)

	Tessellation Summary
		A tessellation of a plane (or of space) is any subdivision of the plane (or space) into regions or "cells" that border each other exactly, with no gaps in between.
		The four color theorem states that for every tessellation of the plane, with a set of four available colors, each tile can be colored in one color such that no tiles of equal color meet at a curve of positive length.
		Copies of an arbitrary quadrilateral can form a tessellation with 2-fold rotational centers at the midpoints of all sides, and translational symmetry with as minimal set of translation vectors a pair according to the diagonals of the quadrilateral, or equivalently, one of these and the sum or difference of the two.
	www.bookrags.com /Tessellation (3766 words)

	Tessellations - Escher and how to make your own
		Tessellations here mean designs featuring animals, birds, etc, which can fill the page, without over-lapping, to form a pattern.
		You can see a tessellation in the top right corner as you move around the site.
		On these pages, you will find information about all aspects of tessellations, from their history and development to complete galleries of examples by David, Seth and the Master, M.C. Escher, the pioneer of the art.
	www.tessellations.org (241 words)

	Hyperbolic Tessellations
		A regular tessellation, or tiling, is a covering of the plane by regular polygons so that the same number of polygons meet at each vertex.
		For a dual tessellation you reverse the roles of the faces and the vertices.
		The dual of a {5,4} tessellation is a {4,5} tessellation, that is, a tiling by squares, five squares meeting at each vertex.
	aleph0.clarku.edu /~djoyce/poincare/poincare.html (784 words)

	Hyperbolic Tessellation Fractals
		In two dimensions, tessellation is the regular tiling of a surface using polygons.
		The first number in the symbol is the number of sides of the polygon and second is the number of polygons surrounding each vertex.
		Notice that as the tessellation approaches the border of the disk, the image repeats in finer and finer detail, approaching infinity.
	www.hiddendimension.com /TessellationFractals.html (200 words)

	Tessellation Design: Tessellations Part 1
		One definition of a tessellation is: "the careful juxtaposition of elements into a coherent pattern"* I've named my practice accordingly because that juxtaposition is what I strive for in my work.
		In mathematics, tessellation refers to the study of "tiling" or how regular shapes can be placed to fill an infinite space with no gaps and no overlapping shapes.
		Regular tessellations are those which consist of a single shape.
	www.karenvagts.com /kvtessellations1.html (309 words)

	Tantalizing Tesselations!!
		Throughout the activities in this unit, the students are asked to respond to the concept of tessellations.
		As they finish tessellating, the group raises their hands to indicate they are finished.
		Explain to students that when they are done they are to write the steps they used for tessellating in their journals.
	mathcentral.uregina.ca /RR/database/RR.09.96/archamb1.html (3245 words)

	Gamasutra - Features - "Tessellation of 4x4 Bezier Patches for the Intel P3" [03.17.00]
		There are three basic steps in the tessellation process: taking samples of the parametric surface, connecting these samples into triangles, and generating the tessellated surface vertices.
		The simplest method is to compute the surface position and normal in object space (the space where the control points are defined), and then feed these vertices to the triangle transform and lighting routines.
		A more complex, but faster, method is to transform the control points to screen space and tessellate using the transformed control points.
	www.gamasutra.com /features/20000317/barad_01.htm (747 words)

	Tessellation
		Examples of tessellation pieces will be given, along with over head examples of tessellation art samples, and the students will have to discover which type of tessellation each piece is according to the list they created from exploration.
		Each student should be able to look at a tessellation and determine what shape this art form is steaming from, what angles are in it, what vertices are being used, and what tessellation type is being expressed.
		The student does a self check of their tessellation to see if it is an actual, authentic tessellation and then they are ready for a conference with me. I conference with the students to make sure they are on the right track and have performed the correct procedures to create the tessellation types.
	www.d23.org /tip/Frameworks/Tessellation.html (2401 words)

	Penn College Outreach for K-12: Pennsylvania Governor's Institute for Mathematics
		All images in the tessellation are identical and the tessellation fills the specified amount of space.
		The design does not tessellate or the original shape is not adjusted at all to create a unique fundamental region (For example, the student uses squares as the fundamental region).
		Is the fundamental region of your tessellation a polygon?
	www.pct.edu /k12/gov-math/2003/6-8-tantalizing-tessellations.html (1776 words)

	Earth tessellation II
		The stress field derived from the fracture tessellation is consistent with a uniformly and simultaneously expanding Gondwana, as predicted by the hypothesis of Anderson (1982).
		The scale of the fracture tessellation was a function of the strength of the Gondwanan lithosphere and the geometry of a spherical shell.
		This is consistent with the hypothesis of Anderson (1982) that the geoid anomaly marks the paleoposition of Gondwana, and that Gondwana spread outward from the anomaly as it broke apart.
	www.mantleplumes.org /EarthTess2.html (3628 words)

	tessellation
		Congruent means that the polygons that you put together are all the same size and shape.] Only three regular polygons tessellate in the Euclidean plane: triangles, squares or hexagons.We can't show the entire plane, but imagine that these are pieces taken from planes that have been tiled.
		Since the regular polygons in a tessellation must fill the plane at each vertex, the interior angle must be an exact divisor of 360 degrees.
		For all the others, the interior angles are not exact divisors of 360 degrees, and therefore those figures cannot tile the plane.
	www.metu.edu.tr /~e123640/project/regulartessellation1.htm (283 words)

	Totally Tessellated: Tessellations
		Tessellations are different from patterns in that patterns usually do not have distinct closed shapes.
		On the left is a true tessellation; on the right is not a tessellation but a pattern.
		There is a class of techniques used to create tessellations of shapes that represent actual objects; these tessellations are not necessarily composed of polygons.
	library.thinkquest.org /16661/background/tessellations.html (188 words)

	Tessellation.html
		The word tessellation comes from the Latin Tessella, which was a small Square stone or tile used in ancient Roman mosaics.
		A plane tessellation is a pattern made up of one or more shapes, completely covering a surface without any gaps or overlaps.
		When the finished tesselations are put on display, they will be viewed as a very difficult and complex endevor, at least to those who do not understand how they were created, Let them marvel.
	www.taospaint.com /Tessellation.html (1069 words)

	Toolbox
		This tool allows you to build tessellations and other designs by attaching the corners of various shapes to one another.
		To make a tessellation, you need to create an arrangement of repeating shapes which leaves no spaces or overlaps between its pieces.
		Begin building your tessellation by clicking on one of the "models" in the list on the right and then dragging it out into the main area.
	www.boxermath.com /plp/modules/online/workshop/toolbox/mosaictool.html (257 words)

	Tessellation - Life and Art of M.C Escher
		History of mosaic tessellations, mathematics and science, historical image gallery and fascinating demonstration pages for the work of M. Escher.
		Tessellations by Marjorie Rice: See her love of nature, drawing, the patterns found in nature and their mathematical relationships.
		Tessellating Animation Japanese artist Makoto Nakamura has taken tessellation a step beyond the plane with his awesome animations.
	www.princetonol.com /groups/iad/lessons/middle/tessell.htm (582 words)

	reptiles (Site not responding. Last check: )
		Different types of tessellations are produced by adding more specific conditions of regularity and piecing together to this general principal.
		Escher's tessellations can be made by cutting out a piece from one of the sides of a shape and rotating it 90º about a centre of rotation which is located on one of the vertices of the shape.
		Once this piece has been rotated through 90º it is stuck to another side of the shape, if the original shape which tessellates the plane is either a triangle, a quadrilateral or a pentagon.
	descartes.cnice.mecd.es /ingles/maths_workshop/Escher's_tessellations/reptiles.htm (597 words)

	8. Voronoi Tessellation & Percolation
		The application of a non parametric percolation to the tessellation cells exceeding this noise level leads directly to a source list which is free of any assumptions about the source geometry.
		The algorithm used here for the construction of this tessellation is a two-dimensional adaption of the recipe described by Tanemura, Ogawa, and Ogita [2] for three dimensions.
		We use open boundaries on a plane, finite-area surface, where the constructibility of the out most cells is guaranteed by restricting the tessellation to the central region of the area actually covered by the photon distribution.
	ledas-cxc.star.le.ac.uk /udocs/docs/swdocs/detect/html/node16.html (3205 words)

	Math Forum: Tessellation Tutorials by Suzanne Alejandre
		The Geometer's Sketchpad and Tessellations by Cathi Sanders
		Tessellations from Hawaii - using the Geometer's Sketchpad method
		The unique art of tessellations can now be created by you.
	mathforum.org /sum95/suzanne/tess.intro.html (120 words)

	ArtLex on Tessellations
		Often a repeating geometric pattern, many of which may also be referred to as tiling.
		The study of tessellations can integrate many disciplines across any full curriculum -- in art, math, language arts, social studies, etc.
		It was developed for a 1998 exhibition in the Netherlands, where, the site says, visitors stood for two hours to watch what can now be seen online: a vivid unraveling of how Escher fooled the eye by compressing multiple viewpoints into one.
	www.artlex.com /ArtLex/t/tesselation.html (553 words)

	tessellation (Site not responding. Last check: )
		Tessellation is an arrangement or repeated use of geometric figures to completely cover a plane surface without overlapping and no gaps.
		Basically, a tessellation is a way to tile a floor (that goes on forever) with shapes so that there is no overlapping and no gaps.
		There are two basic types of repeating patterns that used one or more tiles in a repeative way: the three "regular" and eight "semi-regular" tessellations.
	www.metu.edu.tr /~e123640/project/wha_is_a_tessellation.htm (142 words)

	Computer Tools and Algorithms for Origami Tessellation design
		Origami tessellations are a genre of origami that started in the late 1960s with models such as Momatani’s stretch wall and later works by Shuzo Fujimoto in his book Seizo soru origami asobi no shotai (Creative invitation to paper play).
		Origami tessellations are generated from a tiling of the plane.
		The conceptually simplest algorithm to generate the light pattern is to consider each pixel in the folded representation and count the number of polygons that it lies in.
	www.sanger.ac.uk /Users/agb/Origami/Tess/Bateman.html (1358 words)

	Activity 10
		A semi-regular tessellation is a tessellation that is composed of two or more regular polygons with the same code at each vertex.
		The dual of a tessellation is formed by connect the centers of the shapes in a tessellation so that these segments do not pass through a vertex of the tessellation.
		A demi-regular tessellation is a tessellation composed of regular polygons with more than two different vertex codes arranged in a repeating pattern.
	homepage.mac.com /efithian/Geometry/Activity-10.html (658 words)

	Earth tessellation
		The tessellation of polygons is mathematically precise and intolerant, fixed by projection of a single pentagon.
		Initial opening of the Central Atlantic may have moved Gondwana and its fracture tessellation off the outbreak point of CAMP, displacing the younger parts of the hotspot tessellation with respect to the Cape Verde (or Fernando) hotspot.
		The overall congruence of the hotspots to the tessellation implies, however, that Gondwana was largely stagnant during Late Jurassic-Early Cretaceous magmatic outbreaks, consistent with a loop in the Gondwanan apparent polar wander path (DeWitt et al.
	www.mantleplumes.org /EarthTess.html (1243 words)

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