Miranda Hodge December 11, 2003 MAT 3610 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.

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

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 Sumerians - Used mosaics for decoration Moorish - Islamic religion forbade artist to have people, animals, or real-world objects in their work Harmonice Mundi – Book written by Hohannes Kelper in 1619

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 Fedrov – Russian Crystallographer

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 Granada is located in the south of Spain They inspired Escher Escher Dutch graphic artist No formal mathematics training Inspired after second trip to Granada

Examples of Escher’s Work 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

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

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

Regular Tessellation cont. Pentagons – No Hexagons – Yes Heptagons – No Octagons – No 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 Give def of uniform More than one way to tessellate hexagons and triangles

Semi-regular cont. 3, 3, 3, 4, 4 8 Semi-regular tessellations Go through process of naming

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 This is because hyperbolic sum of angles in a triangle is less than 180

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) First a quasi-{5,4} tessellation. The pentagons are in red or yellow while the squares are in orange. It looks like a plaid disk.

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

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

Tessellations in the World Uses for tessellations: Tiling Mosaics Quilts 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 Any polygons with more than six sides do not tessellate

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

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>.

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>.