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![Page 1: Tesselations (Tilings) Tessellation is defined by a covering of a infinite geometric plane figures of one type or a few types.](https://reader036.fdocuments.us/reader036/viewer/2022062713/56649cb95503460f94980013/html5/thumbnails/1.jpg)
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 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
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TesselationsA 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-4th grade Video
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K-16 Curriculum 6th – 8th 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 ReflectionsThese 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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5th-8th Grade Video
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Transformation
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Transformation
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Transformation
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Transformation
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Transformation
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Transformation
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Rotation
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Rotation
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Rotation
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Rotation
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Rotation
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Rotation
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Rotation
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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
12-16 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 TilingsSymmetric Tilings
1:p1 2:p2 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
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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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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 resourcesFor 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.
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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
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More web resourceshttp://goldennumber.net/quasicrystal.htmhttp://intendo.net/penrose/info.htmlhttp://quadibloc.com/math/penol.htmhttp://www.spsu.edu/math/tile/aperiodic/index.htmhttp://uwgb.edu/DutchS/symmetry/penrose.htmA Java applet to play with Penrose tiles:http://www.geocities.com/SiliconValley/Pines/1684/Penrose/htmlBob, a Penrose Tiling Generator and Explorerhttp://stephencollins.net/Web/Penrose/Default.aspx