Tesselations (Tilings)Tessellation is defined by a covering of a infinite geometric plane figures of one type or a few types.
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
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.
K-16 Curriculum • K-5 • Shape recognition • Creating new shapes • Tilings • Polyominoes
K-4th grade • Video
K-16 Curriculum • 6th – 8th grade • Isometries of the Euclidean plane • Transformation • Rotation • Reflections • Glide Reflections • Symmetry • Period vs Aperiodic
Periodic vs. Repeating Tilings • Up and Down • Left to Right
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?
Fundamental Domain • The pieces that are repeated in a periodic tiling is called fundamental domains. Can there be more than one fundamental domains?
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?
Four Kinds of Symmetry • Reflections
Four Kinds of Symmetry • Rotations
Four Kinds of Symmetry • Glide Reflections
Four Kinds of Symmetry • Slides
5th-8th Grade • Video
K-16 Curriculum • 9-12 • Periodic vs Aperiodic Tilings • Formal Description of Wallpaper Tilings • Penrose Tilings • Science Connections • 12-16 • Above with more detail
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!
Kites and darts are formed from rhombuses with degree measures of 72° and 108°
The kite and dart can be found in the pentagram
The infinite sun pattern The infinite star pattern The two Penrose patterns with perfect symmetry
The kites and darts can be changed into other shapes as well, as Penrose showed by making an illustration of non-periodic tiling chickens
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.
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
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