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Tessellations

Tessellations. What are Tessellations?. http://www.kscoufisdesign.com/tessellation1.swf. History of Tessellations. Many ancient cultures have used tessellations. Johannes Kepler conducted one of the first mathematical studies of tessellations.

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Tessellations

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  1. Tessellations

  2. What are Tessellations? • http://www.kscoufisdesign.com/tessellation1.swf

  3. History of Tessellations • Many ancient cultures have used tessellations. • Johannes Kepler conducted one of the first mathematical studies of tessellations. • E.S. Fedorov proved an aspect of tiling in 1891.

  4. They have been around a long time! Blue Mosque, Istanbul Iznik tile wall in Hagia Sophia in Istanbul. • http://www.sacred-destinations.com/turkey/istanbul-blue-mosque • http://www.planetware.com/picture/istanbul-hagia-sophia-ayasofya-tr-tr437.htm Islamic Art http://easyweb.easynet.co.uk/~iany/patterns/islamic.htm

  5. Islamic Floor Tiling Mideval Window Afganistan Mosiac

  6. Tessellations in Architecture Islamic Arch http://www.london-architecture.info/LO-001.htm http://www.london-architecture.info/LO-047.htm Islamic Minaret

  7. Everyday Tessellations! Floor Tiles Road pavement example

  8. Everyday stuff! - found in quilts, floor tiling, and wallpaper.

  9. More … Mud Flats Checker board

  10. Tessellations in Nature Snake skin spider Armadillo armor

  11. Tessellations in Sports

  12. MC Escher- Artist • Born June 17, 1898 in the Dutch province of Friesland. • He began studying tilings around 1935. • He created his most famous tessellation, “Reptiles” in 1943. • Escher drew his last tessellation in July of 1969. • He died at the age of 73 in March of 1972.

  13. Escher Art

  14. MC Escher official website • http://www.mcescher.com/

  15. A tessellation of a disk used to solve a finite element problem.-differential equations • These rectangular bricks are connected in a tessellation, which if considered an edge-to-edge tiling, topologically identical to a hexagonal tiling, with each hexagon flattened into a rectangle with the long edges divided into two edges by the neighboring bricks.

  16. Tessellations in Science • Basalticlava flows often display columnarjointing as a result of contraction forces causing cracks as the lava cools. The extensive crack networks that develop often produce hexagonal columns of lava. One example of such an array of columns is the Giant's Causeway in Ireland.

  17. Studying Science X-ray Crystallography • This picture depicts the transformation of eight points in an array to make a very small crystal lattice, which tessellates.

  18. This image suggests the relationship between tessellations, symmetry, and x-ray crystallograpy

  19. Hydrogen Peroxide, among other chemicals, has this tessellating, crystal structure.

  20. Basic Types of Tesselations Three main types of tessellations: * Regular * Semi-Regular * Demi-Regular

  21. Naming a Tessellation A tessellation is named by looking at a vertex point and finding how many polygons touch the vertex point. Conventions are named based on the type of polygons that touch the vertex point. The convention number represents the number of sides of each polygon. Examples: tessellation of squares- 4-4-4-4 tessellation of hexagons- 6-6-6

  22. A famous mathematician, Kepler studied tessellations and noted the regular tessellations or (tilings) of the plane. There are only three regular tessellations; one of triangles one of squares and one of hexagons

  23. Regular Tesselations • A regular tessellation is a pattern only using one regular polygon shape. • A regular polygon is any many sided shape that has sides of equal length and angles or equalmeasure.

  24. Semi-regular Tesselations • A semi-regular tessellation is a pattern consisting of more than one type of regular polygon. • The vertex arrangement is the same throughout the entire pattern

  25. SemiregularTesselation • tessellation has 3 triangles and two squares at each vertex, • this pattern can be folded to give unit cells of 2, 4, 6, or 8 squares per cell, and if wrapped around a cylinder, the pattern describes the way in which an overloaded cylinder buckles as it collapses.

  26. Demi-regular Tesselations • A demi-regular tessellation is a pattern of regular polygons in which there are two or three different polygon arrangements.

  27. How are Tesselations made? Tessellations are made through a group of techniques called transformations. Transformations are techniques used to take a shape and match it exactly to another. There are four types of transformations.

  28. Transformations Translations: when a shape is moved in any direction. Reflections: a mirror image that is made of the original shape. Rotations: the original shape is rotated around a central point. Glide reflections: the figure that results after reflection and translation.

  29. Other cool links on Tesselations Totally Tessellated! – created by 3 HS seniors http://library.thinkquest.org/16661/ Tessellations.org http://www.tessellations.org/ Cool Student made Tessellations http://math.slu.edu/escher/index.php/Tessellation_Art_Project_Student_Examples Note: Pictures in this presentation found online.

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