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Tessellations

Uncover the history of tessellations, from ancient Rome to modern art. Learn about regular and semi-regular tessellations, their rules, and examples of Escher's famous works. Discover classroom activities, NCTM standards, and real-world applications of tessellations.

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Tessellations

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  1. Tessellations Miranda Hodge December 11, 2003 MAT 3610

  2. What are Tessellations? • Tessellations are patterns that cover a plane with repeating figures so there is no overlapping or empty spaces.

  3. 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

  4. 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

  5. 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

  6. 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

  7. Examples of Escher’s Work Sun and Moon Horsemen

  8. 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

  9. 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

  10. 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

  11. Regular Tessellation cont. • Pentagons – No • Hexagons – Yes • Heptagons – No • Octagons – No • Any polygon with more than six sides doesn’t tessellate

  12. Semi-regular Tessellation • Uniform tessellations that contain two or more regular polygons • Same rules apply

  13. Semi-regular cont. • 3, 3, 3, 4, 4 • 8 Semi-regular tessellations

  14. Hyperbolic Tessellation • Infinitely many regular tessellations • {n,k} • n=number of sides • k=number of at each vertex • 1/n + 1/k = ½Euclidean • 1/n + 1/k < 1/2 Hyperbolic

  15. 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)

  16. 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

  17. 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

  18. Tessellations in the World • Uses for tessellations: • Tiling • Mosaics • Quilts • Tessellations are often used to solve problems in interior design and quilting

  19. 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

  20. Summary cont. • 8 Semi regular tessellations • Fun for geometry students!

  21. 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>.

  22. 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>.

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