Tessellations Miranda Hodge December 11 2003 MAT 3610

Tessellations Miranda Hodge December 11, 2003 MAT 3610

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

History of Tessellations ¢ The word tessellation comes from Latin word tessella Meaning “a square tablet” l The square tablets were used to make ancient Roman mosaics l ¢ 3 Did not call them tessellations

History cont. Sumerians used mosaics as early as 4000 B. C. ¢ Moorish artists 700 -1500 ¢ Used geometric designs for artwork l Decorated buildings l ¢ Harmonice Mundi (1619) l 4 Regular & Irregular

History cont. ¢ E. S. Fedorov (1891) l l ¢ 5 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

History cont. Alhambra Palace, Granada ¢ M. C. Escher ¢ Known as “The Father of Tessellations” l Created tessellations on woodworks l 1975 British Origami Society l • Popularity in the art world 6

Examples of Escher’s Work 7 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 ¢ 8

Regular Tessellation Uses only one type of regular polygon ¢ Rules: ¢ 1. the tessellation must tile an infinite floor with not gaps or overlapping l 2. the tiles must all be the same regular polygon l 3. each vertex must look the same l 9

Regular Tessellation cont. ¢ The interior angle must be a factor of 360° l ¢ 10 Where n is the number of sides What polygons will form a regular tessellation? l Triangles – Yes l Squares – Yes

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

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

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

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

Hyperbolic cont. ¢ ¢ Poincaré disk Regular Tessellation l ¢ Quasiregular Tessellation l l 15 {5, 4} built from two kinds of regular polygons so that two of each meet at each vertex, alternately Quasi-{5, 4)

Classroom Activities ¢ http: //mathforum. org/pubs/boxer/tess. html l l ¢ http: //www. shodor. org/interactivate/lessons/t essgeom. html l l ¢ 16 Boxer math tessellation tool Teacher lesson plan Teacher lessons plan Student worksheets Sketchpad Activities

NCTM Standards ¢ ¢ ¢ 17 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

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

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 ¢ l 19 Any polygons with more than six sides do not tessellate

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

Works Cited Alejandre, Suzanne. “What is a Tessellation? ” Math Forum 1994 -2003. 18 Nov. 2003. <http: //mathforum. org/sum 95/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 Mc. Graw-Hill, 1998. 523 -527. “Escher Art Collection. ” Dave. Mc’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. <http: //aleph 0. clarku. edu/~djoyce/poincare. html>. 21

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/tesspag 1. html>. 22