Presentation on theme: "Tesselations (Tilings) Tessellation is defined by a covering of a infinite geometric plane figures of one type or a few types."— Presentation transcript:
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 The Latin root tessellare means to pave. -Stone paved streets in the 1600’s. 17 Wallpaper Tilings (Periodic)-1952 Penrose Tilings (Aperiodic)-Roger Penrose
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 6 th – 8 th 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
5 th -8 th Grade Video Video
K-16 Curriculum 9-12 Periodic vs Aperiodic Tilings Formal Description of Wallpaper Tilings Penrose Tilings Science Connections 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 seven vertex neighborhoods of kites and darts
The infinite sun pattern The infinite star pattern The two Penrose patterns with perfect symmetry
The cartwheel pattern surrounding Batman
Alterations to the shape of the tiles to force aperiodicity
The kites and darts can be changed into other shapes as well, as Penrose showed by making an illustration of non-periodic tiling chickens
The seven vertex neighborhoods of Penrose rhombs
Decagons in a Penrose pattern
A tiling of rhombs
Print resources For all practical purposes: introduction to contemporary mathematics (3 rd 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. Wallpaper symmetries Wall Paper Groups. Wall Paper Groups Computer Software for Tiling. Computer Software for Tiling Kaleideo Tile: Reflecting on Symmetry. Kaleideo Tile: Reflecting on Symmetry TesselMania Demo TesselMania Demo Kali Tiling Software Kali Tiling Software Symmetry Symmetry
More web resources A Java applet to play with Penrose tiles: html Bob, a Penrose Tiling Generator and Explorer