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View the MidPoW (March 11 - 25, 2002):Tiling Triangles

Definition

When you fit individual tiles together with no gaps or overlaps to fill a flat space like

a ceiling, wall, or floor, you have a tiling. You can imagine that you can use a varietyof shapes to do this.

Here are some examples:

using one of the pentomino shapes

using rectangles

using triangles

Another word for a tiling is a tessellation. Read more here:What is a Tessellation?

A special kind of tiling or tessellation is rep-tiling. No, these rep-tilesaren't living

things, although some of you might be thinking of the figures in some of M.C.Escher's work. The "rep" in "rep-tile" stands for "replicating." Rep-tiles can be joined

together to make larger replicas of themselves. Learn more here:Reptilesby Andrew

Clarke.

Another special kind of tiling or tessellation is Penrose tilingnamed after the British

physicist and mathematician,Roger Penrose.The tiling is comprised of two rhombi,

http://mathforum.org/midpow/solutions/solution.ehtml?puzzle=143http://mathforum.org/midpow/solutions/solution.ehtml?puzzle=143http://mathforum.org/midpow/solutions/solution.ehtml?puzzle=143http://mathforum.org/sum95/suzanne/whattess.htmlhttp://mathforum.org/sum95/suzanne/whattess.htmlhttp://mathforum.org/sum95/suzanne/whattess.htmlhttp://www.geocities.com/alclarke0/PolyPages/Reptiles.htmhttp://www.geocities.com/alclarke0/PolyPages/Reptiles.htmhttp://www.geocities.com/alclarke0/PolyPages/Reptiles.htmhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Penrose.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Penrose.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Penrose.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Penrose.htmlhttp://www.geocities.com/alclarke0/PolyPages/Reptiles.htmhttp://mathforum.org/sum95/suzanne/whattess.htmlhttp://mathforum.org/midpow/solutions/solution.ehtml?puzzle=143

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one with angles of 36 and 144 degrees and one with angles of 72 and 108 degrees.

Learn more here:Aperiodic Tiling with Penrose Rhombsby Nancy Casey.

EXPLORING TESSELLATIONS WITH COMPUTER SOFTWARE

Once they have used Escher's tessellations to explore the transformations that can be used to

create tessellating art, students are given the opportunity to create their own artwork on acomputer with TesselMania! or Tessellation Exploration software. Within either

program, they select a polygon and a modifying rule or rules. Whenever a "bump" or "hole"

is added to a side, the program removes or adds the corresponding "hole" or "bump" from/to

the appropriate side according to the rule selected. Classic paint tools, including stamps, areavailable for adding interior interpreting features.

Depending on the program, the corresponding tessellation either accompanies thetessellating shape or appears at the press of a button.

http://www.cs.uidaho.edu/~casey931/puzzle/penrose/penrose.htmlhttp://www.cs.uidaho.edu/~casey931/puzzle/penrose/penrose.htmlhttp://www.cs.uidaho.edu/~casey931/puzzle/penrose/penrose.htmlhttp://www.cs.uidaho.edu/~casey931/puzzle/penrose/penrose.html

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Each student's original artwork can be printed on transfer material suitable for color

bubblejet printers, and then ironed onto a T-shirt. The results are truly outstanding!

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POLYGONS AND TESSELLATIONS

A simple connect-the-dots exercise is used to introduce the concept of apolygon. The 50-

sided polygon in the solution resembles a reptile. The creature is an adaptation of the one

that appears in Escher's lithographReptiles. In the graphic, several identical lizards

interlock in a jigsaw puzzle configuration ortessellation. A set of 15 large soft foam lizardsis available from the Imagination Project (telephone 888-477-6532 or 513-860-2711).

Reptiles - M. C. Escher

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If a third mirror is added to the hingedmirror assembly, the resulting equilateral

triangle prism can be used to generatetessellating art with reflectional symmetry.

(The configuration of mirrors is maintained

with an elastic band.) A generating trianglefor Escher's Lizard/Fish/Bat tessellation

(Tessellation 85) appears below.

Tessellation 85- M. C. Escher

(clickfor enlargement)

Students can create a generating triangle for a tessellation with reflectional symmetry by

drawing identical or distinct curves from the center of an equilateral triangle to its vertices.Two examples follow.

http://britton.disted.camosun.bc.ca/escher/lizard_fish_bat.jpghttp://britton.disted.camosun.bc.ca/escher/lizard_fish_bat.jpghttp://britton.disted.camosun.bc.ca/escher/lizard_fish_bat.jpghttp://britton.disted.camosun.bc.ca/escher/lizard_fish_bat.jpg

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Creating a Quilt Block with Tessellation Exploration

The following steps will allow you to reproduce many traditional quilt blocks as aTessellation

Explorationtile. Once you master the process, you can design your own patterns and then recreate

them with the software. You don't have to be artistically gifted to be creative with quilt blocks -

just careful!

To begin, download the compressed filesqgrids.zip.Unzip this collection of Tessellation

Explorationsquare grid files to a convenient location on your hard drive.

Step 1: Select a traditional quilt-block pattern that is based on a square grid. Suitable examples

will be found inQuilt Blocks GaloreandQuilt Block Collection(requiresAdobe Acrobat

Reader). For the best printed resource, consult Mary Ellen Hopkin'sThe It's Okay If You

Sit on My Quilt Bookwith over 350 color examples of geometric quilt blocks on a

gridded background. As an example, we will select the 4x4 quilt block called Cube

Lattice. You will require two copies of the block to complete the exercise.

Step 2: Cube Latticeis based on a 4x4 square grid. Using a marker, superimpose a drawing of

the 4x4 grid over one copy of the block. Notice the diagonal lines of the block. They

extend from corner to opposite corner of the corresponding squares on the grid.

Step 3: Launch Tessellation Exploration, then open the appropriate file in the directory of

square grids. For Cube Lattice, open the file . Zoom in to enlarge the view.

http://www.tomsnyder.com/products/product.asp?SKU=TESEXPhttp://www.tomsnyder.com/products/product.asp?SKU=TESEXPhttp://www.tomsnyder.com/products/product.asp?SKU=TESEXPhttp://www.tomsnyder.com/products/product.asp?SKU=TESEXPhttp://britton.disted.camosun.bc.ca/sqgrids.ziphttp://britton.disted.camosun.bc.ca/sqgrids.ziphttp://britton.disted.camosun.bc.ca/sqgrids.ziphttp://www.quilterscache.com/QuiltBlocksGalore.htmlhttp://www.quilterscache.com/QuiltBlocksGalore.htmlhttp://www.quilterscache.com/QuiltBlocksGalore.htmlhttp://britton.disted.camosun.bc.ca/quiltblocks.pdfhttp://britton.disted.camosun.bc.ca/quiltblocks.pdfhttp://britton.disted.camosun.bc.ca/quiltblocks.pdfhttp://get.adobe.com/reader/http://get.adobe.com/reader/http://get.adobe.com/reader/http://get.adobe.com/reader/http://www.amazon.com/exec/obidos/tg/detail/-/0929950054/qid=1116884173/sr=8-1/ref=sr_8_xs_ap_i1_xgl14/104-5816462-4354345?v=glance&s=books&n=507846http://www.amazon.com/exec/obidos/tg/detail/-/0929950054/qid=1116884173/sr=8-1/ref=sr_8_xs_ap_i1_xgl14/104-5816462-4354345?v=glance&s=books&n=507846http://www.amazon.com/exec/obidos/tg/detail/-/0929950054/qid=1116884173/sr=8-1/ref=sr_8_xs_ap_i1_xgl14/104-5816462-4354345?v=glance&s=books&n=507846http://www.amazon.com/exec/obidos/tg/detail/-/0929950054/qid=1116884173/sr=8-1/ref=sr_8_xs_ap_i1_xgl14/104-5816462-4354345?v=glance&s=books&n=507846http://www.amazon.com/exec/obidos/tg/detail/-/0929950054/qid=1116884173/sr=8-1/ref=sr_8_xs_ap_i1_xgl14/104-5816462-4354345?v=glance&s=books&n=507846http://www.amazon.com/exec/obidos/tg/detail/-/0929950054/qid=1116884173/sr=8-1/ref=sr_8_xs_ap_i1_xgl14/104-5816462-4354345?v=glance&s=books&n=507846http://get.adobe.com/reader/http://get.adobe.com/reader/http://britton.disted.camosun.bc.ca/quiltblocks.pdfhttp://www.quilterscache.com/QuiltBlocksGalore.htmlhttp://britton.disted.camosun.bc.ca/sqgrids.ziphttp://www.tomsnyder.com/products/product.asp?SKU=TESEXPhttp://www.tomsnyder.com/products/product.asp?SKU=TESEXP

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Step 4: Using the Linetool in the double-black color and the thinnest line width, add all

appropriate diagonal lines to the grid on the computer screen. Both cross hairs of the

cursor should line up with the horizontal and vertical grid lines when you depress the

mouse button to begin a diagonal line and again when you release it. Each diagonal line

should have exactly one pixel per row and column.

Step 5: Study the original quilt block (the copy without the superimposed square grid). With

the Penciltool and the thinnest line width, color double-white all pixels in horizontal

and vertical lines that are not part of the block pattern. Zoom in and out as needed. You

may use theErasertool where convenient, but the Penciltool works best for fixing

pixels adjacent to lines that are part of the block pattern.

Step 6: Color the patches of the block with the Paint Buckettool. For this example, we will use

the two colors of a standard color pair for the parallelograms, and double-black for the

squares.

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Step 7: Notice that the default setting for the corresponding tessellation is same coloring, but

you may elect to use contrasting coloring for a different effect.

SAME COLORING

CONTRASTING COLORING

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Tessellation

Atilingofregular polygons(in two dimensions),polyhedra(three dimensions), orpolytopes( dimensions) is called a

tessellation. Tessellations can be specified using aSchlfli symbol.

The breaking up of self-intersectingpolygons intosimple polygonsis also called tessellation (Woo et al. 1999), or

more properly,polygon tessellation.

There are exactly threeregular tessellationscomposed of regular polygons symmetrically tiling the plane.

http://mathworld.wolfram.com/Tiling.htmlhttp://mathworld.wolfram.com/Tiling.htmlhttp://mathworld.wolfram.com/Tiling.htmlhttp://mathworld.wolfram.com/RegularPolygon.htmlhttp://mathworld.wolfram.com/RegularPolygon.htmlhttp://mathworld.wolfram.com/RegularPolygon.htmlhttp://mathworld.wolfram.com/Polyhedron.htmlhttp://mathworld.wolfram.com/Polyhedron.htmlhttp://mathworld.wolfram.com/Polyhedron.htmlhttp://mathworld.wolfram.com/Polytope.htmlhttp://mathworld.wolfram.com/Polytope.htmlhttp://mathworld.wolfram.com/Polytope.htmlhttp://mathworld.wolfram.com/SchlaefliSymbol.htmlhttp://mathworld.wolfram.com/SchlaefliSymbol.htmlhttp://mathworld.wolfram.com/SchlaefliSymbol.htmlhttp://mathworld.wolfram.com/Intersection.htmlhttp://mathworld.wolfram.com/Intersection.htmlhttp://mathworld.wolfram.com/Intersection.htmlhttp://mathworld.wolfram.com/SimplePolygon.htmlhttp://mathworld.wolfram.com/SimplePolygon.htmlhttp://mathworld.wolfram.com/SimplePolygon.htmlhttp://mathworld.wolfram.com/PolygonTessellation.htmlhttp://mathworld.wolfram.com/PolygonTessellation.htmlhttp://mathworld.wolfram.com/PolygonTessellation.htmlhttp://mathworld.wolfram.com/RegularTessellation.htmlhttp://mathworld.wolfram.com/RegularTessellation.htmlhttp://mathworld.wolfram.com/RegularTessellation.htmlhttp://mathworld.wolfram.com/RegularTessellation.htmlhttp://mathworld.wolfram.com/PolygonTessellation.htmlhttp://mathworld.wolfram.com/SimplePolygon.htmlhttp://mathworld.wolfram.com/Intersection.htmlhttp://mathworld.wolfram.com/SchlaefliSymbol.htmlhttp://mathworld.wolfram.com/Polytope.htmlhttp://mathworld.wolfram.com/Polyhedron.htmlhttp://mathworld.wolfram.com/RegularPolygon.htmlhttp://mathworld.wolfram.com/Tiling.html

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Tessellations of the plane by two or moreconvex regularpolygonssuch that the samepolygonsin the same order

surround eachpolygon vertexare calledsemiregular tessellations,or sometimes Archimedean tessellations. In the

plane, there are eight such tessellations, illustrated above (Ghyka 1977, pp. 76-78; Williams 1979, pp. 37-41;

Steinhaus 1999, pp. 78-82; Wells 1991, pp. 226-227).

http://mathworld.wolfram.com/Polygon.htmlhttp://mathworld.wolfram.com/Polygon.htmlhttp://mathworld.wolfram.com/Polygon.htmlhttp://mathworld.wolfram.com/Polygon.htmlhttp://mathworld.wolfram.com/Polygon.htmlhttp://mathworld.wolfram.com/Polygon.htmlhttp://mathworld.wolfram.com/PolygonVertex.htmlhttp://mathworld.wolfram.com/PolygonVertex.htmlhttp://mathworld.wolfram.com/PolygonVertex.htmlhttp://mathworld.wolfram.com/SemiregularTessellation.htmlhttp://mathworld.wolfram.com/SemiregularTessellation.htmlhttp://mathworld.wolfram.com/SemiregularTessellation.htmlhttp://mathworld.wolfram.com/SemiregularTessellation.htmlhttp://mathworld.wolfram.com/PolygonVertex.htmlhttp://mathworld.wolfram.com/Polygon.htmlhttp://mathworld.wolfram.com/Polygon.html

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There are 14demiregular(or polymorph) tessellations which are orderly compositions of the three regular and eight

semiregular tessellations (Critchlow 1970, pp. 62-67; Ghyka 1977, pp. 78-80; Williams 1979, p. 43; Steinhaus 1999,

pp. 79 and 81-82).

In three dimensions, apolyhedronwhich is capable of tessellating space is called aspace-filling polyhedron.

Examples include thecube,rhombic dodecahedron,andtruncated octahedron.There is also a 16-sided space-filler

and a convexpolyhedronknown as theSchmitt-Conway biprismwhich fills space only aperiodically.

A tessellation of -dimensional polytopes is called ahoneycomb.

SEE ALSO:Archimedean Solid,Cairo Tessellation,Cell,Demiregular Tessellation,Dual Tessellation,Hexagonal

Grid,Hinged Tessellation,Honeycomb,Honeycomb Conjecture,Kepler's Monsters,Regular Tessellation,Schlfli

Symbol,Semiregular Polyhedron,Semiregular Tessellation,Space-Filling Polyhedron,Spiral Similarity,Square

Grid,Symmetry,Tiling,Triangular Grid,Triangular Symmetry Group,Triangulation,Wallpaper Groups

REFERENCES:

http://mathworld.wolfram.com/DemiregularTessellation.htmlhttp://mathworld.wolfram.com/DemiregularTessellation.htmlhttp://mathworld.wolfram.com/DemiregularTessellation.htmlhttp://mathworld.wolfram.com/Polyhedron.htmlhttp://mathworld.wolfram.com/Polyhedron.htmlhttp://mathworld.wolfram.com/Polyhedron.htmlhttp://mathworld.wolfram.com/Space-FillingPolyhedron.htmlhttp://mathworld.wolfram.com/Space-FillingPolyhedron.htmlhttp://mathworld.wolfram.com/Space-FillingPolyhedron.htmlhttp://mathworld.wolfram.com/Cube.htmlhttp://mathworld.wolfram.com/Cube.htmlhttp://mathworld.wolfram.com/Cube.htmlhttp://mathworld.wolfram.com/RhombicDodecahedron.htmlhttp://mathworld.wolfram.com/RhombicDodecahedron.htmlhttp://mathworld.wolfram.com/RhombicDodecahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/Polyhedron.htmlhttp://mathworld.wolfram.com/Polyhedron.htmlhttp://mathworld.wolfram.com/Polyhedron.htmlhttp://mathworld.wolfram.com/Schmitt-ConwayBiprism.htmlhttp://mathworld.wolfram.com/Schmitt-ConwayBiprism.htmlhttp://mathworld.wolfram.com/Schmitt-ConwayBiprism.htmlhttp://mathworld.wolfram.com/Honeycomb.htmlhttp://mathworld.wolfram.com/Honeycomb.htmlhttp://mathworld.wolfram.com/Honeycomb.htmlhttp://mathworld.wolfram.com/ArchimedeanSolid.htmlhttp://mathworld.wolfram.com/ArchimedeanSolid.htmlhttp://mathworld.wolfram.com/ArchimedeanSolid.htmlhttp://mathworld.wolfram.com/CairoTessellation.htmlhttp://mathworld.wolfram.com/CairoTessellation.htmlhttp://mathworld.wolfram.com/CairoTessellation.htmlhttp://mathworld.wolfram.com/Cell.htmlhttp://mathworld.wolfram.com/Cell.htmlhttp://mathworld.wolfram.com/Cell.htmlhttp://mathworld.wolfram.com/DemiregularTessellation.htmlhttp://mathworld.wolfram.com/DemiregularTessellation.htmlhttp://mathworld.wolfram.com/DemiregularTessellation.htmlhttp://mathworld.wolfram.com/DualTessellation.htmlhttp://mathworld.wolfram.com/DualTessellation.htmlhttp://mathworld.wolfram.com/DualTessellation.htmlhttp://mathworld.wolfram.com/HexagonalGrid.htmlhttp://mathworld.wolfram.com/HexagonalGrid.htmlhttp://mathworld.wolfram.com/HexagonalGrid.htmlhttp://mathworld.wolfram.com/HexagonalGrid.htmlhttp://mathworld.wolfram.com/HingedTessellation.htmlhttp://mathworld.wolfram.com/HingedTessellation.htmlhttp://mathworld.wolfram.com/HingedTessellation.htmlhttp://mathworld.wolfram.com/Honeycomb.htmlhttp://mathworld.wolfram.com/Honeycomb.htmlhttp://mathworld.wolfram.com/Honeycomb.htmlhttp://mathworld.wolfram.com/HoneycombConjecture.htmlhttp://mathworld.wolfram.com/HoneycombConjecture.htmlhttp://mathworld.wolfram.com/KeplersMonsters.htmlhttp://mathworld.wolfram.com/KeplersMonsters.htmlhttp://mathworld.wolfram.com/KeplersMonsters.htmlhttp://mathworld.wolfram.com/RegularTessellation.htmlhttp://mathworld.wolfram.com/RegularTessellation.htmlhttp://mathworld.wolfram.com/RegularTessellation.htmlhttp://mathworld.wolfram.com/SchlaefliSymbol.htmlhttp://mathworld.wolfram.com/SchlaefliSymbol.htmlhttp://mathworld.wolfram.com/SchlaefliSymbol.htmlhttp://mathworld.wolfram.com/SchlaefliSymbol.htmlhttp://mathworld.wolfram.com/SemiregularPolyhedron.htmlhttp://mathworld.wolfram.com/SemiregularPolyhedron.htmlhttp://mathworld.wolfram.com/SemiregularTessellation.htmlhttp://mathworld.wolfram.com/SemiregularTessellation.htmlhttp://mathworld.wolfram.com/SemiregularTessellation.htmlhttp://mathworld.wolfram.com/Space-FillingPolyhedron.htmlhttp://mathworld.wolfram.com/Space-FillingPolyhedron.htmlhttp://mathworld.wolfram.com/Space-FillingPolyhedron.htmlhttp://mathworld.wolfram.com/SpiralSimilarity.htmlhttp://mathworld.wolfram.com/SpiralSimilarity.htmlhttp://mathworld.wolfram.com/SpiralSimilarity.htmlhttp://mathworld.wolfram.com/SquareGrid.htmlhttp://mathworld.wolfram.com/SquareGrid.htmlhttp://mathworld.wolfram.com/SquareGrid.htmlhttp://mathworld.wolfram.com/Symmetry.htmlhttp://mathworld.wolfram.com/Symmetry.htmlhttp://mathworld.wolfram.com/Symmetry.htmlhttp://mathworld.wolfram.com/Tiling.htmlhttp://mathworld.wolfram.com/Tiling.htmlhttp://mathworld.wolfram.com/Tiling.htmlhttp://mathworld.wolfram.com/TriangularGrid.htmlhttp://mathworld.wolfram.com/TriangularGrid.htmlhttp://mathworld.wolfram.com/TriangularGrid.htmlhttp://mathworld.wolfram.com/TriangularSymmetryGroup.htmlhttp://mathworld.wolfram.com/TriangularSymmetryGroup.htmlhttp://mathworld.wolfram.com/TriangularSymmetryGroup.htmlhttp://mathworld.wolfram.com/Triangulation.htmlhttp://mathworld.wolfram.com/Triangulation.htmlhttp://mathworld.wolfram.com/Triangulation.htmlhttp://mathworld.wolfram.com/WallpaperGroups.htmlhttp://mathworld.wolfram.com/WallpaperGroups.htmlhttp://mathworld.wolfram.com/WallpaperGroups.htmlhttp://mathworld.wolfram.com/WallpaperGroups.htmlhttp://mathworld.wolfram.com/Triangulation.htmlhttp://mathworld.wolfram.com/TriangularSymmetryGroup.htmlhttp://mathworld.wolfram.com/TriangularGrid.htmlhttp://mathworld.wolfram.com/Tiling.htmlhttp://mathworld.wolfram.com/Symmetry.htmlhttp://mathworld.wolfram.com/SquareGrid.htmlhttp://mathworld.wolfram.com/SquareGrid.htmlhttp://mathworld.wolfram.com/SpiralSimilarity.htmlhttp://mathworld.wolfram.com/Space-FillingPolyhedron.htmlhttp://mathworld.wolfram.com/SemiregularTessellation.htmlhttp://mathworld.wolfram.com/SemiregularPolyhedron.htmlhttp://mathworld.wolfram.com/SchlaefliSymbol.htmlhttp://mathworld.wolfram.com/SchlaefliSymbol.htmlhttp://mathworld.wolfram.com/RegularTessellation.htmlhttp://mathworld.wolfram.com/KeplersMonsters.htmlhttp://mathworld.wolfram.com/HoneycombConjecture.htmlhttp://mathworld.wolfram.com/Honeycomb.htmlhttp://mathworld.wolfram.com/HingedTessellation.htmlhttp://mathworld.wolfram.com/HexagonalGrid.htmlhttp://mathworld.wolfram.com/HexagonalGrid.htmlhttp://mathworld.wolfram.com/DualTessellation.htmlhttp://mathworld.wolfram.com/DemiregularTessellation.htmlhttp://mathworld.wolfram.com/Cell.htmlhttp://mathworld.wolfram.com/CairoTessellation.htmlhttp://mathworld.wolfram.com/ArchimedeanSolid.htmlhttp://mathworld.wolfram.com/Honeycomb.htmlhttp://mathworld.wolfram.com/Schmitt-ConwayBiprism.htmlhttp://mathworld.wolfram.com/Polyhedron.htmlhttp://mathworld.wolfram.com/TruncatedOctahedron.htmlhttp://mathworld.wolfram.com/RhombicDodecahedron.htmlhttp://mathworld.wolfram.com/Cube.htmlhttp://mathworld.wolfram.com/Space-FillingPolyhedron.htmlhttp://mathworld.wolfram.com/Polyhedron.htmlhttp://mathworld.wolfram.com/DemiregularTessellation.html

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http://www.amazon.com/exec/obidos/ASIN/0486253570/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0486253570/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0486253570/ref=nosim/weisstein-20http://library.thinkquest.org/16661/http://library.thinkquest.org/16661/http://library.thinkquest.org/16661/http://www.amazon.com/exec/obidos/ASIN/0769000835/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0769000835/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0769000835/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0500340331/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0500340331/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0500340331/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0906212200/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0906212200/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0906212200/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0226282473/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0226282473/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0226282473/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0716719258/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0716719258/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0486235424/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0486235424/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0486235424/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0486201635/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0486201635/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0486201635/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/B0006BMVAU/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/B0006BMVAU/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/B0006BMVAU/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0933174659/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0933174659/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0933174659/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0716720647/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0716720647/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0716720647/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/082181933X/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/082181933X/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/082181933X/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0965640582/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0965640582/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0965640582/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0486409147/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0486409147/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0486409147/ref=nosim/weisstein-20http://www.vicher.cz/puzzle/telesa/telesa.htmhttp://www.vicher.cz/puzzle/telesa/telesa.htmhttp://www.vicher.cz/puzzle/telesa/telesa.htmhttp://www.ericweisstein.com/encyclopedias/books/Tilings.htmlhttp://www.ericweisstein.com/encyclopedias/books/Tilings.htmlhttp://www.ericweisstein.com/encyclopedias/books/Tilings.htmlhttp://www.amazon.com/exec/obidos/ASIN/0140118136/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0140118136/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0140118136/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0140118136/ref=nosim/weisstein-20http://www.ericweisstein.com/encyclopedias/books/Tilings.htmlhttp://www.vicher.cz/puzzle/telesa/telesa.htmhttp://www.amazon.com/exec/obidos/ASIN/0486409147/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0965640582/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/082181933X/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0716720647/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0933174659/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/B0006BMVAU/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0486201635/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0486235424/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0716719258/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0226282473/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0906212200/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0500340331/ref=nosim/weisstein-20http://www.amazon.com/exec/obidos/ASIN/0769000835/ref=nosim/weisstein-20http://library.thinkquest.org/16661/http://www.amazon.com/exec/obidos/ASIN/0486253570/ref=nosim/weisstein-20

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Williams, R.The Geometrical Foundation of Natural Structure: A Source Book of Design. New York: Dover, pp. 35-43, 1979.

Woo, M.; Neider, J.; Davis, T.; and Shreiner, D. Ch. 11 inOpenGL 1.2 Programming Guide, 3rd ed.: The Official Guide to Learning OpenGL,

Version 1.2.Reading, MA: Addison-Wesley, 1999.

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