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

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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

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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.

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Nature

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Science

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Decoration

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

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K-4 th grade Video Video

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

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Periodic vs. Repeating Tilings Up and Down Left to Right

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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?

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

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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?

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Four Kinds of Symmetry Reflections

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Four Kinds of Symmetry Rotations

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Four Kinds of Symmetry Glide Reflections

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Four Kinds of Symmetry Slides

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5 th -8 th Grade Video Video

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Transformation

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Rotation

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

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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!

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17 Wallpaper Tilings Symmetric Tilings 1:p12:p23:pm4:pg5:cm6:pmm7:pmg8:pgg9:cmm 10:p411:p4m12:p4g13:p314:p31m15:p3m116:p617:p6m

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

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The kite and dart can be found in the pentagram

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The seven vertex neighborhoods of kites and darts

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

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The cartwheel pattern surrounding Batman

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Alterations to the shape of the tiles to force aperiodicity

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

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Penrose rhombs

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The seven vertex neighborhoods of Penrose rhombs

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Decagons in a Penrose pattern

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A tiling of rhombs

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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.

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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

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More web resources A Java applet to play with Penrose tiles: html Bob, a Penrose Tiling Generator and Explorer

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