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 Tessellations are thousands of years old and can be found all over the world in a variety of cultures and in a variety of forms. They can be traced all the way back to the Sumerian civilization (about 4000 B.C.) in which the walls of homes and temples were decorated by designs of tessellations constructed from slabs of hardened clay. Not only did these tessellations provide decoration but they also became part of the structure of the buildings. Since then, tessellations have been found in many of the artistic elements of wide-ranging cultures including the Egyptians, Moors, Romans, Persians, Greek, Byzatine, Arabic, Japanese, and Chinese The 17 official names were given in 1952 by International Union of Crystallography

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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Decoration

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 In order to say that the tiling is periodic, as opposed to merely repeating, a mathematician would want to be sure the repetition happens in a regular way. If you were to walk from left to right across the the tiling shown on the right, you would keep seeing the thin tile again and again, but the repetition isn't at all regular, since the distance between occurences of the thin tiles keeps increasing. On the other hand, if you were to cross the tiling from top to bottom, instead of left to right, the pattern would repeat regularly. So we also need to be sure that a tiling is as regular as possible in all directions, before declaring it is periodic. In general, it a hard to say what "as regularly as possible" means in a precise mathematical way. But for a plane tiling, it basically means that the tiling repeats in two independent directions, since the plane is 2-dimensional.

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? So how can we tell if a tiling is periodic? One way is to construct a lattice. A lattice is a grid consisting of two sets of evenly spaced parallel lines. In the image on the left the lattice is the black lines (the blue lines represent the cooridinate axes in the plane). By the way it is made, you can see that a lattice repeats regularly in two directions. The parallelograms formed by the lattice are called period parallelograms. A tiling is periodic when we can lay a lattice over the tiling in such a way so that the period parallelograms contain idential pieces of the tiling. We call these pieces fundamental domains for the 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

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K-16 Curriculum 9-12 12-16 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!

17 Wallpaper Tilings Symmetric Tilings 3:pm 4:pg 5:cm 6:pmm 7:pmg 8:pgg 9:cmm 10:p4 11:p4m 12:p4g 13:p3 14:p31m 15:p3m1 16:p6 17:p6m For wallpaper groups the full notation begins with either p or c, for a primitive cell or a face-centred cell; these are explained below. This is followed by a digit, n, indicating the highest order of rotational symmetry: 1-fold (none), 2-fold, 3-fold, 4-fold, or 6-fold. The next two symbols indicate symmetries relative to one translation axis of the pattern, referred to as the "main" one; if there is a mirror perpendicular to a translation axis we choose that axis as the main one (or if there are two, one of them). The symbols are either m, g, or 1, for mirror, glide reflection, or none. The axis of the mirror or glide reflection is perpendicular to the main axis for the first letter, and either parallel or tilted 180°/n (when n > 2) for the second letter. Many groups include other symmetries implied by the given ones. The short notation drops digits or an m that can be deduced, so long as that leaves no confusion with another group. p2 (p211): Primitive cell, 2-fold rotation symmetry, no mirrors or glide reflections. p4g (p4gm): Primitive cell, 4-fold rotation, glide reflection perpendicular to main axis, mirror axis at 45°. cmm (c2mm): Centred cell, 2-fold rotation, mirror axes both perpendicular and parallel to main axis. p31m (p31m): Primitive cell, 3-fold rotation, mirror axis at 60°.

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

Penrose rhombs

Penrose rhombs

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 (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. Wall Paper Groups . 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