Tessellations
Miranda Hodge December 11, 2003 MAT 3610
What are Tessellations?
Tessellations are patterns that cover a plane with repeating figures so there is no overlapping or empty spaces.
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History of Tessellations
The word tessellation comes from Latin word tessella
Meaning “a square tablet” The square tablets were used to make ancient Roman mosaics
Did not call them tessellations
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History cont.
Sumerians used mosaics as early as 4000 B.C. Moorish artists 700-1500
Used geometric designs for artwork Decorated buildings
Harmonice Mundi (1619)
Regular & Irregular
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History cont.
E.S. Fedorov (1891)
Found methods for repeating tilings over a plane “Unofficial” beginning of the mathematical study of tessellations
Many discoveries have be made about tessellations since Fedorov’s work
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History cont.
Alhambra Palace, Granada M.C. Escher
Known as “The Father of Tessellations” Created tessellations on woodworks 1975 British Origami Society
• Popularity in the art world
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Examples of Escher’s Work
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Sun and Moon
Horsemen
Tessellation Basics
Formed by translating, rotating, and reflecting polygons The sum of the measures of the angles of the polygons surrounding at a vertex is 360° Regular Tessellation Semi-regular Tessellation Hyperbolic Tessellation
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Regular Tessellation
Uses only one type of regular polygon Rules:
1. the tessellation must tile an infinite floor with not gaps or overlapping 2. the tiles must all be the same regular polygon 3. each vertex must look the same
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Regular Tessellation cont.
The interior angle must be a factor of 360°
Where n is the number of sides
180(n 2) n
What polygons will form a regular tessellation?
Triangles – Yes Squares – Yes
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Regular Tessellation cont.
Pentagons – No Hexagons – Yes Heptagons – No Octagons – No Any polygon with more than six sides doesn’t tessellate
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Semi-regular Tessellation
Uniform tessellations that contain two or more regular polygons Same rules apply
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Semi-regular cont.
3, 3, 3, 4, 4
8 Semi-regular tessellations
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Hyperbolic Tessellation
Infinitely many regular tessellations {n,k}
n=number of sides k=number of at each vertex
1/n + 1/k = ½ Euclidean 1/n + 1/k < 1/2 Hyperbolic
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Hyperbolic cont.
Poincaré disk Regular Tessellation
{5,4}
Quasiregular Tessellation
built from two kinds of regular polygons so that two of each meet at each vertex, alternately Quasi-{5,4)
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Classroom Activities
http://mathforum.org/pubs/boxer/tess.html
Boxer math tessellation tool Teacher lesson plan
http://www.shodor.org/interactivate/lessons/t essgeom.html
Teacher lessons plan Student worksheets
Sketchpad Activities
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NCTM Standards
Apply transpositions and symmetry to analyze mathematical situations Analyze characteristics and properties of twoand three-dimensional geometric shapes and develop mathematical arguments about geometric relationships Apply appropriate techniques, tools, and formulas to determine measurement
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Tessellations in the World
Uses for tessellations:
Tiling Mosaics Quilts
Tessellations are often used to solve problems in interior design and quilting
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Summary of Tessellations
Patterns that cover a plane with repeating figures so there is no overlapping or empty spaces. Found throughout history MC Escher Triangles, Squares, and Hexagons tessellate
Any polygons with more than six sides do not tessellate
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Summary cont.
8 Semi regular tessellations Fun for geometry students!
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Works Cited
Alejandre, Suzanne. “What is a Tessellation?” Math Forum 1994-2003. 18 Nov. 2003.. Bennett, D. “Tessellations Using Only Translations.” Teaching Mathematics with The Geometer’s Sketchpad. Emeryville, CA: Key Curriculum Press, 2002. 18-19. Boyd, Cindy J., et al. Geometry. New York: Glencoe McGraw-Hill, 1998. 523-527. “Escher Art Collection.” DaveMc’s Image Collection. 1 Dec. 2003. < http://www.cs.unc.edu/~davemc/Pic/Escher/>. “Geometry in Tessellations.” The Shodor Education Foundation, Inc. 19972003. 18 Nov. 2003. < http://www.shodor.org/interactivate/lessons/ tessgeom.html>. Joyce, David E. “Hyperbolic Tessellations.” Clark University. Dec. 1998. 18 Nov.2003. .
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Works Cited cont.
Seymour, Dale and Jill Britton. Introduction to Tessellations. Palo Alto: Dale Seymour Publications, 1989. “Tessellations by Karen.” Coolmath.com. 18 Nov. 2003. .
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