1. Miranda Hodge December 11, 2003 MAT 3610
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
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

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
Fedrov – Russian Crystallographer

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
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

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
Give def of uniform
More than one way to tessellate hexagons and triangles

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

14. 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

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)
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.

16. Classroom Activities http://mathforum.org/pubs/boxer/tess.html
Boxer math tessellation tool
Teacher lesson plan
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 Nov < whattess.html>.
Bennett, D. “Tessellations Using Only Translations.” Teaching Mathematics with The Geometer’s Sketchpad. Emeryville, CA: Key Curriculum Press,
Boyd, Cindy J., et al. Geometry. New York: Glencoe McGraw-Hill,
“Escher Art Collection.” DaveMc’s Image Collection. 1 Dec <
“Geometry in Tessellations.” The Shodor Education Foundation, Inc Nov < tessgeom.html>.
Joyce, David E. “Hyperbolic Tessellations.” Clark University. Dec Nov < 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 <