Tesselations (Tilings) Tessellation is defined by a covering of a infinite geometric plane figures of one type or a few

# Tesselations (Tilings) Tessellation is defined by a covering of a infinite geometric plane figures of one type or a few

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## Tesselations (Tilings) Tessellation is defined by a covering of a infinite geometric plane figures of one type or a few

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1. Tesselations (Tilings)Tessellation is defined by a covering of a infinite geometric plane figures of one type or a few types.

2. Quick History • Sumerian civilization (about 4000 B.C.) • The word was founded  in 1660. The Latin root tessellare means to pave. -Stone paved streets in the 1600’s. •  17 Wallpaper Tilings (Periodic)-1952 • Penrose Tilings (Aperiodic)-Roger Penrose -1974

3. Tesselations A tiling is just a way of covering a flat surface with smaller shapes or tiles that fit together nicely, without gaps or overlaps. Tilings come in many varieties, both man-made ones, and ones in nature.

4. Nature

5. Science

6. Decoration

7. K-16 Curriculum • K-5 • Shape recognition • Creating new shapes • Tilings • Polyominoes

9. K-16 Curriculum • 6th – 8th grade • Isometries of the Euclidean plane • Transformation • Rotation • Reflections • Glide Reflections • Symmetry • Period vs Aperiodic

10. Periodic vs. Repeating Tilings • Up and Down • Left to Right

11. Test for Period Tilings • Construct a lattice • By the way it is made, you can see that a lattice repeats regularly in two directions. • A tiling is periodic when we can lay a lattice over the tiling in such a way so that each parallelograms contains identical pieces of the tiling. Where would we see a periodic tiling?

12. Fundamental Domain • The pieces that are repeated in a periodic tiling is called fundamental domains. Can there be more than one fundamental domains?

13. Four Kinds of Symmetry • Slides • Rotations • Reflections • Glide Reflections These different ways of moving things in the plane are called isometries. What types of shapes can be rotated?

14. Four Kinds of Symmetry • Reflections

15. Four Kinds of Symmetry • Rotations

16. Four Kinds of Symmetry • Glide Reflections

17. Four Kinds of Symmetry • Slides

19. Transformation

20. Transformation

21. Transformation

22. Transformation

23. Transformation

24. Transformation

25. Rotation

26. Rotation

27. Rotation

28. Rotation

29. Rotation

30. Rotation

31. Rotation

32. Rotation

33. K-16 Curriculum • 9-12 • Periodic vs Aperiodic Tilings • Formal Description of Wallpaper Tilings • Penrose Tilings • Science Connections • 12-16 • Above with more detail

34. Wallpaper Tilings • Some of the most fascinating tilings are the so-called wallpaper tilings. These tilings are so symmetric that they can be built up by starting with a single tile by following simple sets of rules. But perhaps the most interesting thing about the wallpaper tilings is that there are exactly seventeen of them!

35. 17 Wallpaper TilingsSymmetric Tilings

36. Kites and darts are formed from rhombuses with degree measures of 72° and 108°

37. The kite and dart can be found in the pentagram

38. The seven vertex neighborhoods of kites and darts

39. The infinite sun pattern The infinite star pattern The two Penrose patterns with perfect symmetry

40. The cartwheel pattern surrounding Batman

41. Alterations to the shape of the tiles to force aperiodicity

42. The kites and darts can be changed into other shapes as well, as Penrose showed by making an illustration of non-periodic tiling chickens

43. Penrose rhombs

44. Penrose rhombs

45. The seven vertex neighborhoods of Penrose rhombs

46. Decagons in a Penrose pattern

47. A tiling of rhombs

48. Print resources For all practical purposes: introduction to contemporary mathematics (3rd ed.). (1994). New York: W.H. Freeman and Co. Gardner, M. (1989) Penrose tiles to trapdoor ciphers. New York: W.H. Freeman and Co.

49. Web Resources • Wallpaper symmetries. • http://aleph0.clarku.edu/%7Edjoyce/wallpaper/index.html • Wall Paper Groups . • http://www.xahlee.org/ • Computer Software for Tiling. • http://www.geom.umn.edu/software/tilings/TilingSoftware.html • Kaleideo Tile: Reflecting on Symmetry. • http://www.geom.umn.edu/%7Eteach95/kt95/KTL.html • TesselMania Demo • http://www.kidsdomain.com/down/pc/tesselmaniap1.html • Kali Tiling Software • http://www.geom.uiuc.edu/software/tilings/TilingSoftware.html • Symmetry • http://www.scienceu.com/geometry/articles/tiling/symmetry/p2.html

50. More web resources http://goldennumber.net/quasicrystal.htm http://intendo.net/penrose/info.html http://quadibloc.com/math/penol.htm http://www.spsu.edu/math/tile/aperiodic/index.htm http://uwgb.edu/DutchS/symmetry/penrose.htm A Java applet to play with Penrose tiles: http://www.geocities.com/SiliconValley/Pines/1684/Penrose/html Bob, a Penrose Tiling Generator and Explorer http://stephencollins.net/Web/Penrose/Default.aspx