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WHAT IS TESSELLATION? A tessellation is created when a shape is repeated and covers a plane without any gaps or overlaps. All of a regular polygon's angles and sides are congruent. If we tessellate the Euclidean plane with a regular polygon, the tessellation is a regular tessellation. Only three regular polygons can tessellate the Euclidean plane: triangles, squares, or hexagons. Since the regular polygons in a tessellation must fill the plane at each vertex, the polygon's interior angle measure must be an exact divisor of 360 degrees. This only works for the triangle, square, and hexagon and is the reason why only they can tessellate the Euclidean plane. Some tessellations are made with figures of animals such as birds. M.C. Escher is famous for his work with tessellations including ones with animals. Some links are included to show more about tessellations. http://wiki.answers.com/Q/What_is_a_tessellation

Tessellation

A tiling of regular polygons (in two dimensions), polyhedra (three dimensions), or polytopes ( dimensions) is called a tessellation. Tessellations can be specified using a SchlĂ¤fli symbol. The breaking up of self-intersecting polygons into simple polygons is also called tessellation (Woo et al. 1999), or more properly, polygon tessellation.

There are exactly three regular tessellations composed of regular polygons symmetrically tiling the plane.

Tessellations of the plane by two or more convex regular polygons such that the same polygons in the same order surround each polygon vertex are called semiregular 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).

There are 14 demiregular (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, a polyhedron which is capable of tessellating space is called a space-filling polyhedron. Examples include the cube, rhombic dodecahedron, and truncated octahedron. There is also a 16-sided space-filler and a convex polyhedron known as the Schmitt-Conway biprism which fills space only aperiodically. A tessellation of -dimensional polytopes is called a honeycomb. http://mathworld.wolfram.com/Tessellation.html

The Topic: Tessellations Easier - A tessellation is created when a shape is repeated over and over again. All the figures fit onto a flat surface exactly together without any gaps or overlaps. Harder - A tessellation is a repeating pattern composed of interlocking shapes (usually polygons) that can be extended infinitely. The tiling for a regular (or periodic) tessellation

is done with one repeated congruent regular polygon covering a plane in a repeating pattern without any openings or overlaps. Remember 'regular' means the sides of the polygon are all the same length, and 'congruent' means that the polygons fitted together are all the same size and shape. A semi-regular (or non-periodic) tessellation is formed by a regular arrangement of polygons, identically arranged at every vertex point. http://42explore.com/teslatn.htm

Definition A tessellation is created when a shape is repeated over and over again covering a plane without any gaps or overlaps. Another word for a tessellation is a tiling. Read more here: What is a Tiling? A dictionary* will tell you that the word "tessellate" means to form or arrange small squares in a checkered or mosaic pattern. The word "tessellate" is derived from the Ionic version of the Greek word "tesseres," which in English means "four." The first tilings were made from square tiles. A regular polygon has 3 or 4 or 5 or more sides and angles, all equal. A regular tessellation means a tessellation made up of congruent regular polygons. [Remember: Regular means that the sides and angles of the polygon are all equivalent (i.e., the polygon is both equiangular and equilateral). Congruent means that the polygons that you put together are all the same size and shape.] Only three regular polygons tessellate in the Euclidean plane: triangles, squares or hexagons. We can't show the entire plane, but imagine that these are pieces taken from planes that have been tiled. Here are examples of a tessellation of triangles a tessellation of squares a tessellation of hexagons When you look at these three samples you can easily notice that the squares are lined up with each other while the triangles and hexagons are not. Also, if you look at 6 triangles at a time, they form a hexagon, so the tiling of triangles and the tiling of hexagons are similar and they cannot be formed by directly lining shapes up under each other - a slide (or a glide!) is involved. You can work out the interior measure of the angles for each of these polygons: Shape

Angle measure in degrees

triangle square pentagon hexagon more than six sides

60 90 108 120 more than 120 degrees

Since the regular polygons in a tessellation must fill the plane at each vertex, the interior angle must be an exact divisor of 360 degrees. This works for the triangle, square, and hexagon, and you can show working tessellations for these figures. For all the others, the interior angles are not exact divisors of 360 degrees, and therefore those figures cannot tile the plane. Reinforce this idea with the Regular Tessellations interactive activity: Teacher Lesson Plan || Student Page

Naming Conventions A tessellation of squares is named "4.4.4.4". Here's how: choose a vertex, and then look at one of the polygons that touches that vertex. How many sides does it have? Since it's a square, it has four sides, and that's where the first "4" comes from. Now keep going around the vertex in either direction, finding the number of sides of the polygons until you get back to the polygon you started with. How many polygons did you count? There are four polygons, and each has four sides.

For a tessellation of regular congruent hexagons, if you choose a vertex and count the sides of the polygons that touch it, you'll see that there are three polygons and each has six sides, so this tessellation is called "6.6.6":

A tessellation of triangles has six polygons surrounding a vertex, and each of them has three sides: "3.3.3.3.3.3".

Semi-regular Tessellations You can also use a variety of regular polygons to make semi-regular tessellations. A semiregular tessellation has two properties which are: 1. 2.

It is formed by regular polygons. The arrangement of polygons at every vertex point is identical.

Here are the eight semi-regular tessellations:

Interestingly there are other combinations that seem like they should tile the plane because the arrangements of the regular polygons fill the space around a point. For example:

If you try tiling the plane with these units of tessellation you will find that they cannot be extended infinitely. Fun is to try this yourself.

1.

2. 3. 4.

Hold down on one of the images and copy it to the clipboard. Open a paint program. Paste the image. Now continue to paste and position and see if you can tessellate it.

There are an infinite number of tessellations that can be made of patterns that do not have the same combination of angles at every vertex point. There are also tessellations made of polygons that do not share common edges and vertices. You can learn more by following the links listed in Other Tessellation Links and Related Sites. Michael South has contributed some thoughts to the discussion. *Steven Schwartzman's The Words of Mathematics (1994, The Mathematical Association of America) says: tessellate (verb), tessellation (noun): from Latin tessera "a square tablet" or "a die used for gambling." Latin tessera may have been borrowed from Greek tessares, meaning "four," since a square tile has four sides. The diminutive of tessera was tessella, a small, square piece of stone or a cubical tile used in mosaics. Since a mosaic extends over a given area without leaving any region uncovered, the geometric meaning of the word tessellate is "to cover the plane with a pattern in such a way as to leave no region uncovered." By extension, space or hyperspace may also be tessellated. http://mathforum.org/sum95/suzanne/whattess.html

What is Tessellate!? This activity allows the user to generate a polygon that will repeat without overlapping across a plane. Starting from a rectangle, triangle or hexagon, the user bends the lines of the polygon, creating a new polygon. The user can choose several different colors to enhance the pattern, and can observe the different effects that colors have on tessellations. Starting from a simple polygon such as a triangle,

then selecting colors for the shapes, the result is a plane covered with repetitions of the shape with no overlapping or gaps in between the shapes.

Or starting from hexagon

and bending the lines to form an irregular polygon

then selecting colors for the shapes, renders this image:

Tessellations occur naturally in the world, and are frequently used in designs for works of art and architecture. They can assist students in conceptualizing infinity, learning about the different types of symmetry, and making observations about how colors and shapes affect perception. http://www.shodor.org/interactivate/activities/Tessellate/

Background Art and mathematics can be combined in designs that are a fascinating mix of detail and beauty by creating tessellations.

To make a tessellation you need to create a pattern of repeating shapes which leaves no spaces or overlaps between its pieces. Tessellations are made by reflecting (flipping), translating (sliding) and rotating (turning) the two-dimensional shape or shapes that you choose to use. Your choice of colours for each of the shapes adds further beauty to your design. Here are some examples of tessellations. Look at each of one carefully and discuss with a partner the shapes that you can see. Using regular polygons and circles

Using polyominoes

Composition of Grey and Light Brown

English Tilings, Holy Trinity, Leeds

Being inspired by Using a design you can the work of artist create yourself based on a MC Escher square Twisted Squares Koch Snowflake

3. Copyrighted image reprinted with permission citation 2. Copyrighted image reprinted with permission reprinted with permission citation citation

1. Copyrighted image Designs in Tile Commercial Bath Installations

Chinese Pattern

4. Copyrighted image reprinted with permission citation

Fish by MC Escher

Optical illusion

8. Copyrighted image reprinted with permission citation

5. Copyrighted image reprinted with permission 6. Copyrighted image reprinted with permission citation citation

7. Copyrighted image reprinted with permission citation

Tessellations in History Geometric and artistic shapes and patterns, including tessellations, have been used by people for thousands of years to create beautiful tiles. Visit the following website and look at the range of tiles created by people from many lands and during different times in history. Tessellations in history

Discuss with your partner the shapes and colours that have been used in each set of tiles. Discuss which ones you like best and why? Find other examples of tessellations within your school, home and community environment. http://www.cap.nsw.edu.au/bb_site_intro/stage2_Modules/tesselations/tesselations.h tm tessellate (verb), tessellation (noun): from Latin tessera "a square tablet" or "a die used for gambling." Latin tessera may have been borrowed from Greek tessares, meaning "four," since a square tile has four sides. The diminutive of tessera was

tessella, a small, square piece of stone or a cubical tile used in mosaics. Since a mosaic extends over a given area without leaving any region uncovered, the geometric meaning of the word tessellate is "to cover the plane with a pattern in such a way as to leave no region uncovered." By extension, space or hyperspace may also be tessellated. http://mathforum.org/sum95/suzanne/whattess.html

What are Tessellations? (page 1 of 4) Your browser does not support the IFRAME tag. Your browser does Your browser does not not support the support the IFRAME IFRAME tag. tag. Graphing Calculator Scientific Calculator

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Tessellation

A tiling of regular polygons (in two dimensions), polyhedra (three dimensions), or polytopes ( dimensions) is called a tessellation. Tessellations can be specified using a SchlĂ¤fli symbol. The breaking up of self-intersecting polygons into simple polygons is also called tessellation (Woo et al. 1999), or more properly, polygon tessellation.

There are exactly three regular tessellations composed of regular polygons symmetrically tiling the plane.

Tessellations of the plane by two or more convex regular polygons such that the same polygons in the same order surround each polygon vertex are called semiregular 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).

There are 14 demiregular (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, a polyhedron which is capable of tessellating space is called a space-filling polyhedron. Examples include the cube, rhombic dodecahedron, and truncated octahedron. There is also a 16-sided space-filler and a convex polyhedron known as the Schmitt-Conway biprism which fills space only aperiodically. A tessellation of -dimensional polytopes is called a honeycomb. http://mathworld.wolfram.com/Tessellation.html

The Topic: Tessellations Easier - A tessellation is created when a shape is repeated over and over again. All the figures fit onto a flat surface exactly together without any gaps or overlaps. Harder - A tessellation is a repeating pattern composed of interlocking shapes (usually polygons) that can be extended infinitely. The tiling for a regular (or periodic) tessellation

is done with one repeated congruent regular polygon covering a plane in a repeating pattern without any openings or overlaps. Remember 'regular' means the sides of the polygon are all the same length, and 'congruent' means that the polygons fitted together are all the same size and shape. A semi-regular (or non-periodic) tessellation is formed by a regular arrangement of polygons, identically arranged at every vertex point. http://42explore.com/teslatn.htm

Definition A tessellation is created when a shape is repeated over and over again covering a plane without any gaps or overlaps. Another word for a tessellation is a tiling. Read more here: What is a Tiling? A dictionary* will tell you that the word "tessellate" means to form or arrange small squares in a checkered or mosaic pattern. The word "tessellate" is derived from the Ionic version of the Greek word "tesseres," which in English means "four." The first tilings were made from square tiles. A regular polygon has 3 or 4 or 5 or more sides and angles, all equal. A regular tessellation means a tessellation made up of congruent regular polygons. [Remember: Regular means that the sides and angles of the polygon are all equivalent (i.e., the polygon is both equiangular and equilateral). Congruent means that the polygons that you put together are all the same size and shape.] Only three regular polygons tessellate in the Euclidean plane: triangles, squares or hexagons. We can't show the entire plane, but imagine that these are pieces taken from planes that have been tiled. Here are examples of a tessellation of triangles a tessellation of squares a tessellation of hexagons When you look at these three samples you can easily notice that the squares are lined up with each other while the triangles and hexagons are not. Also, if you look at 6 triangles at a time, they form a hexagon, so the tiling of triangles and the tiling of hexagons are similar and they cannot be formed by directly lining shapes up under each other - a slide (or a glide!) is involved. You can work out the interior measure of the angles for each of these polygons: Shape

Angle measure in degrees

triangle square pentagon hexagon more than six sides

60 90 108 120 more than 120 degrees

Since the regular polygons in a tessellation must fill the plane at each vertex, the interior angle must be an exact divisor of 360 degrees. This works for the triangle, square, and hexagon, and you can show working tessellations for these figures. For all the others, the interior angles are not exact divisors of 360 degrees, and therefore those figures cannot tile the plane. Reinforce this idea with the Regular Tessellations interactive activity: Teacher Lesson Plan || Student Page

Naming Conventions A tessellation of squares is named "4.4.4.4". Here's how: choose a vertex, and then look at one of the polygons that touches that vertex. How many sides does it have? Since it's a square, it has four sides, and that's where the first "4" comes from. Now keep going around the vertex in either direction, finding the number of sides of the polygons until you get back to the polygon you started with. How many polygons did you count? There are four polygons, and each has four sides.

For a tessellation of regular congruent hexagons, if you choose a vertex and count the sides of the polygons that touch it, you'll see that there are three polygons and each has six sides, so this tessellation is called "6.6.6":

A tessellation of triangles has six polygons surrounding a vertex, and each of them has three sides: "3.3.3.3.3.3".

Semi-regular Tessellations You can also use a variety of regular polygons to make semi-regular tessellations. A semiregular tessellation has two properties which are: 1. 2.

It is formed by regular polygons. The arrangement of polygons at every vertex point is identical.

Here are the eight semi-regular tessellations:

Interestingly there are other combinations that seem like they should tile the plane because the arrangements of the regular polygons fill the space around a point. For example:

If you try tiling the plane with these units of tessellation you will find that they cannot be extended infinitely. Fun is to try this yourself.

1.

2. 3. 4.

Hold down on one of the images and copy it to the clipboard. Open a paint program. Paste the image. Now continue to paste and position and see if you can tessellate it.

There are an infinite number of tessellations that can be made of patterns that do not have the same combination of angles at every vertex point. There are also tessellations made of polygons that do not share common edges and vertices. You can learn more by following the links listed in Other Tessellation Links and Related Sites. Michael South has contributed some thoughts to the discussion. *Steven Schwartzman's The Words of Mathematics (1994, The Mathematical Association of America) says: tessellate (verb), tessellation (noun): from Latin tessera "a square tablet" or "a die used for gambling." Latin tessera may have been borrowed from Greek tessares, meaning "four," since a square tile has four sides. The diminutive of tessera was tessella, a small, square piece of stone or a cubical tile used in mosaics. Since a mosaic extends over a given area without leaving any region uncovered, the geometric meaning of the word tessellate is "to cover the plane with a pattern in such a way as to leave no region uncovered." By extension, space or hyperspace may also be tessellated. http://mathforum.org/sum95/suzanne/whattess.html

What is Tessellate!? This activity allows the user to generate a polygon that will repeat without overlapping across a plane. Starting from a rectangle, triangle or hexagon, the user bends the lines of the polygon, creating a new polygon. The user can choose several different colors to enhance the pattern, and can observe the different effects that colors have on tessellations. Starting from a simple polygon such as a triangle,

then selecting colors for the shapes, the result is a plane covered with repetitions of the shape with no overlapping or gaps in between the shapes.

Or starting from hexagon

and bending the lines to form an irregular polygon

then selecting colors for the shapes, renders this image:

Tessellations occur naturally in the world, and are frequently used in designs for works of art and architecture. They can assist students in conceptualizing infinity, learning about the different types of symmetry, and making observations about how colors and shapes affect perception. http://www.shodor.org/interactivate/activities/Tessellate/

Background Art and mathematics can be combined in designs that are a fascinating mix of detail and beauty by creating tessellations.

To make a tessellation you need to create a pattern of repeating shapes which leaves no spaces or overlaps between its pieces. Tessellations are made by reflecting (flipping), translating (sliding) and rotating (turning) the two-dimensional shape or shapes that you choose to use. Your choice of colours for each of the shapes adds further beauty to your design. Here are some examples of tessellations. Look at each of one carefully and discuss with a partner the shapes that you can see. Using regular polygons and circles

Using polyominoes

Composition of Grey and Light Brown

English Tilings, Holy Trinity, Leeds

Being inspired by Using a design you can the work of artist create yourself based on a MC Escher square Twisted Squares Koch Snowflake

3. Copyrighted image reprinted with permission citation 2. Copyrighted image reprinted with permission reprinted with permission citation citation

1. Copyrighted image Designs in Tile Commercial Bath Installations

Chinese Pattern

4. Copyrighted image reprinted with permission citation

Fish by MC Escher

Optical illusion

8. Copyrighted image reprinted with permission citation

5. Copyrighted image reprinted with permission 6. Copyrighted image reprinted with permission citation citation

7. Copyrighted image reprinted with permission citation

Tessellations in History Geometric and artistic shapes and patterns, including tessellations, have been used by people for thousands of years to create beautiful tiles. Visit the following website and look at the range of tiles created by people from many lands and during different times in history. Tessellations in history

Discuss with your partner the shapes and colours that have been used in each set of tiles. Discuss which ones you like best and why? Find other examples of tessellations within your school, home and community environment. http://www.cap.nsw.edu.au/bb_site_intro/stage2_Modules/tesselations/tesselations.h tm tessellate (verb), tessellation (noun): from Latin tessera "a square tablet" or "a die used for gambling." Latin tessera may have been borrowed from Greek tessares, meaning "four," since a square tile has four sides. The diminutive of tessera was

tessella, a small, square piece of stone or a cubical tile used in mosaics. Since a mosaic extends over a given area without leaving any region uncovered, the geometric meaning of the word tessellate is "to cover the plane with a pattern in such a way as to leave no region uncovered." By extension, space or hyperspace may also be tessellated. http://mathforum.org/sum95/suzanne/whattess.html

What are Tessellations? (page 1 of 4) Your browser does not support the IFRAME tag. Your browser does Your browser does not not support the support the IFRAME IFRAME tag. tag. Graphing Calculator Scientific Calculator

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