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