Tesselations (Tilings)
Tessellation is defined by a covering of a infinite
geometric plane figures of one type or a few types.
Quick History
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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
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.
Nature
Science
Decoration
K-16 Curriculum
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K-5
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Shape recognition
Creating new shapes
Tilings
Polyominoes
K-4th grade
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Video
K-16 Curriculum
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6th – 8th grade
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Isometries of the Euclidean plane
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Transformation
Rotation
Reflections
Glide Reflections
Symmetry
Period vs Aperiodic
Periodic vs. Repeating Tilings
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Up and Down
Left to Right
Test for Period Tilings
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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
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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.
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What types of shapes can be rotated?
Four Kinds of Symmetry
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Reflections
Four Kinds of Symmetry
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Rotations
Four Kinds of Symmetry
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Glide Reflections
Four Kinds of Symmetry
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Slides
5th-8th Grade
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Video
Transformation
Transformation
Transformation
Transformation
Transformation
Transformation
Rotation
Rotation
Rotation
Rotation
Rotation
Rotation
Rotation
Rotation
K-16 Curriculum
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9-12
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Periodic vs Aperiodic Tilings
Formal Description of Wallpaper Tilings
Penrose Tilings
Science Connections
12-16
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Above with more detail
Wallpaper Tilings
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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
1:p1
10:p4
2:p2
11:p4m
3:pm
12:p4g
4:pg
13:p3
5:cm
6:pmm
14:p31m
7:pmg
15:p3m1
8:pgg
16:p6
9:cmm
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 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
York: W.H. Freeman and Co.
ciphers. New
Web Resources
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Wallpaper symmetries.
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Wall Paper Groups .
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http://www.kidsdomain.com/down/pc/tesselmaniap1.html
Kali Tiling Software
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http://www.geom.umn.edu/%7Eteach95/kt95/KTL.html
TesselMania Demo
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http://www.geom.umn.edu/software/tilings/TilingSoftware.html
Kaleideo Tile: Reflecting on Symmetry.
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http://www.xahlee.org/
Computer Software for Tiling.
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http://aleph0.clarku.edu/%7Edjoyce/wallpaper/index.html
http://www.geom.uiuc.edu/software/tilings/TilingSoftware.html
Symmetry
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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