Tessellations

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>.