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Tessellations
Miranda HodgeDecember 11, 2003
MAT 3610
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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 tessellaMeaning “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 700-1500Used 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
Sun and Moon Horsemen
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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
What polygons will form a regular tessellation? Triangles – Yes
Squares – Yes
180(2)nn−
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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 sidesk=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/tessgeom.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 two- and 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:TilingMosaicsQuilts
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.<http://mathforum.org/sum95/suzanne/
whattess.html>.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. 1997-2003. 18 Nov. 2003. < http://www.shodor.org/interactivate/lessons/tessgeom.html>.
Joyce, David E. “Hyperbolic Tessellations.” Clark University. Dec. 1998. 18 Nov.2003. <http://aleph0.clarku.edu/~djoyce/poincare/poincare.html>.
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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. <http://www.coolmath.com/tesspag1.html>.