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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.
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
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.
Nature
Science
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
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 ReflectionsThese 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
Transformation
Transformation
Transformation
Transformation
Transformation
Transformation
Rotation
Rotation
Rotation
Rotation
Rotation
Rotation
Rotation
Rotation
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!
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
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 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.
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 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