what is tessellation
TRANSCRIPT
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 Schläfli 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:
Shapetrianglesquarepentagonhexagonmore than six sides
Angle measure in degrees6090
108120
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. It is formed by regular polygons.2. 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. Hold down on one of the images and copy it to the clipboard.2. Open a paint program.
3. Paste the image.
4. 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/
BackgroundArt 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 polyominoesBeing inspired by the work of artist
MC Escher
Using a design you can create yourself based on
a square
Composition of Grey and Light
Brown
1. Copyrighted image reprinted with permission
citation
English Tilings, Holy Trinity, Leeds
2. Copyrighted image reprinted with permission
citation
Twisted Squares
3. Copyrighted image reprinted with
permissioncitation
Koch Snowflake
4. Copyrighted image reprinted with permission
citation
Designs in Tile - Commercial Bath
Installations
Chinese Pattern Fish by MC Escher Optical illusion
5. Copyrighted image reprinted with permission
citation
6. Copyrighted image reprinted with permission
citation
7. Copyrighted image reprinted with
permissioncitation
8. 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.htm
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)
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Graphing Calculator Scientific Calculator << a new window will open for these
Basically, a tessellation is a way to tile a floor (that goes on forever) with shapes so that there is no overlapping and no gaps. Remember the last puzzle you put together? Well, that was a tessellation! The shapes were just really weird.
Example:
We usually add a few more rules to make things interesting!
REGULAR TESSELLATIONS:
RULE #1 : The tessellation must tile a floor (that goes on forever) with no overlapping or gaps.
RULE #2 : The tiles must be regular polygons - and all the same.
RULE #3 : Each vertex must look the same.
What's a vertex?
where all the "corners" meet!
What can we tessellate using these rules?
Triangles? Yep!
Notice what happens at each vertex!
The interior angle of each equilateral triangle is 60 degrees.....
60 + 60 + 60 + 60 + 60 + 60 = 360 degrees
Squares? Yep!
What happens at each vertex?
90 + 90 + 90 + 90 = 360 degrees again!
So, we need to use regular polygons that add up to 360 degrees.
Will pentagons work?
The interior angle of a pentagon is 108 degrees. . .
108 + 108 + 108 = 324 degrees . . . Nope!
Hexagons?
120 + 120 + 120 = 360 degrees Yep!
Heptagons?
No way!! Now we are getting overlaps!
Octagons? Nope!
They'll overlap too. In fact, all polygons with more than six sides will overlap! So, the only regular polygons that tessellate are triangles, squares and hexagons!
SEMI-REGULAR TESSELLATIONS:
These tessellations are made by using two or more different regular polygons. The rules are still the same. Every vertex must have the exact same configuration.
Examples:
3, 6, 3, 6
3, 3, 3, 3, 6
These tessellations are both made up of hexagons and triangles, but their vertex configuration is different. That's why we've named them!
To name a tessellation, simply work your way around one vertex counting the number of sides of the polygons that form that vertex. The trick is to go around the vertex in order so that the smallest numbers possible appear first.
That's why we wouldn't call our 3, 3, 3, 3, 6 tessellation a 3, 3, 6, 3, 3!
Here's another tessellation made up of hexagons and triangles.
Can you see why this isn't an official semi-regular tessellation?
It breaks the vertex rule! Do you see where?
Here are some tessellations using squares and triangles:
3, 3, 3, 4, 4
3, 3, 4, 3, 4
Can you see why this one won't be a semi-regular tessellation?
MORE SEMI-REGULAR TESSELLATIONS
http://www.coolmath.com/lesson-tessellations-4.htm
TessellationFrom Wikipedia, the free encyclopedia
Jump to: navigation, search
A tessellation of pavement
A honeycomb is an example of a tessellated natural structure
A tessellation or tiling of the plane is a collection of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art of M. C. Escher. Tessellations are seen throughout art history, from ancient architecture to modern art.
In Latin, tessella is a small cubical piece of clay, stone or glass used to make mosaics.[1] The word "tessella" means "small square" (from "tessera", square, which in its turn is from the Greek word for "four"). It corresponds with the everyday term tiling which refers to applications of tessellations, often made of glazed clay.
Wallpaper groups
Tilings with translational symmetry can be categorized by wallpaper group, of which 17 exist. All seventeen of these patterns are known to exist in the Alhambra palace in Granada, Spain. Of the three regular tilings two are in the category p6m and one is in p4m.
[edit] Tessellations and color
If this parallelogram pattern is colored before tiling it over a plane, seven colors are required to ensure each complete parallelogram has a consistent color that is distinct from that of adjacent areas. (This tiling can be compared to the surface of a torus.) Coloring after tiling, only four colors are needed.
When discussing a tiling that is displayed in colors, to avoid ambiguity one needs to specify whether the colors are part of the tiling or just part of its illustration. See also symmetry.
The four color theorem states that for every tessellation of a normal Euclidean plane, with a set of four available colors, each tile can be colored in one color such that no tiles of equal color meet at a curve of positive length. Note that the coloring guaranteed by the four-color theorem will not in general respect the symmetries of the tessellation. To produce a coloring which does, as many as seven colors may be needed, as in the picture at right.
[edit] Tessellations with quadrilaterals
Copies of an arbitrary quadrilateral can form a tessellation with 2-fold rotational centers at the midpoints of all sides, and translational symmetry whose basis vectors are the diagonal of the quadrilateral or, equivalently, one of these and the sum or difference of the two. For an asymmetric quadrilateral this tiling belongs to wallpaper group p2. As fundamental domain we have the quadrilateral. Equivalently, we can construct a parallelogram subtended by a minimal set of translation vectors, starting from a rotational center. We can divide this by one diagonal, and take one half (a triangle) as fundamental domain. Such a triangle has the same area as the quadrilateral and can be constructed from it by cutting and pasting.
[edit] Regular and semi-regular tessellations
Hexagonal tessellation of a floor
A regular tessellation is a highly symmetric tessellation made up of congruent regular polygons. Only three regular tessellations exist: those made up of equilateral triangles, squares, or hexagons. A semiregular tessellation uses a variety of regular polygons; there are eight of these. The arrangement of polygons at every vertex point is identical. An edge-to-edge tessellation is even less regular: the only requirement is that adjacent tiles only share full sides, i.e. no tile shares a partial side with any other tile. Other types of tessellations exist, depending on types of figures and types of pattern. There are regular versus irregular, periodic versus aperiodic, symmetric versus asymmetric, and fractal tessellations, as well as other classifications.
Penrose tilings using two different polygons are the most famous example of tessellations that create aperiodic patterns. They belong to a general class of aperiodic tilings that can be constructed out of self-replicating sets of polygons by using recursion.
A monohedral tiling is a tessellation in which all tiles are congruent. Spiral monohedral tilings include the Voderberg tiling discovered by Hans Voderberg in 1936, whose unit tile is a nonconvex enneagon; and the Hirschhorn tiling discovered by Michael Hirschhorn in the 1970s, whose unit tile is an irregular pentagon.
[edit] Self-dual tessellations
Tilings and honeycombs can also be self-dual. All n-dimensional hypercubic honeycombs with Schlafli symbols {4,3n−2,4}, are self-dual.
[edit] Tessellations and computer models
A tessellation of a disk used to solve a finite element problem.
These rectangular bricks are connected in a tessellation which, considered as an edge-to-edge tiling, is topologically identical to a hexagonal tiling; each hexagon is flattened into a rectangle whose long edges are divided in two by the neighboring bricks.
This basketweave tiling is topologically identical to the Cairo pentagonal tiling, with one side of each rectangle counted as two edges, divided by a vertex on the two neighboring rectangles.
In the subject of computer graphics, tessellation techniques are often used to manage datasets of polygons and divide them into suitable structures for rendering. Normally, at least for real-time rendering, the data is tessellated into triangles, which is sometimes referred to as triangulation. Tessellation is a staple feature of DirectX 11 and OpenGL.[2][3]
In computer-aided design the constructed design is represented by a boundary representation topological model, where analytical 3D surfaces and curves, limited to faces and edges constitute a continuous boundary of a 3D body. Arbitrary 3D bodies are often too complicated to analyze directly. So they are approximated (tessellated) with a mesh of small, easy-to-analyze pieces of 3D volume — usually either irregular tetrahedrons, or irregular hexahedrons. The mesh is used for finite element analysis.
Mesh of a surface is usually generated per individual faces and edges (approximated to polylines) so that original limit vertices are included into mesh. To ensure approximation of the original surface suits needs of the further processing 3 basic parameters are usually defined for the surface mesh generator:
Maximum allowed distance between planar approximation polygon and the surface (aka "sag"). This parameter ensures that mesh is similar enough to the original analytical surface (or the polyline is similar to the original curve).
Maximum allowed size of the approximation polygon (for triangulations it can be maximum allowed length of triangle sides). This parameter ensures enough detail for further analysis.
Maximum allowed angle between two adjacent approximation polygons (on the same face). This parameter ensures that even very small humps or hollows that can have significant effect to analysis will not disappear in mesh.
Algorithm generating mesh is driven by the parameters (example: CATIA V5 tessellation library). Some computer analyses require adaptive mesh. Mesh is being locally enhanced (using stronger parameters) in areas where it is needed during the analysis.
Some geodesic domes are designed by tessellating the sphere with triangles that are as close to equilateral triangles as possible.
[edit] Tessellations in nature
Basaltic lava flows often display columnar jointing as a result of contraction forces causing cracks as the lava cools. The extensive crack networks that develop often produce hexagonal columns of lava. One example of such an array of columns is the Giant's Causeway in Northern Ireland.
The Tessellated pavement in Tasmania is a rare sedimentary rock formation where the rock has fractured into rectangular blocks.
[edit] Number of sides of a polygon versus number of sides at a vertex
For an infinite tiling, let a be the average number of sides of a polygon, and b the average number of sides meeting at a vertex. Then (a − 2)(b − 2) = 4. For example, we have the
combinations , for the tilings in the article Tilings of regular polygons.
A continuation of a side in a straight line beyond a vertex is counted as a separate side. For example, the bricks in the picture are considered hexagons, and we have combination (6, 3).
Similarly, for the basketweave tiling often found on bathroom floors, we have .
For a tiling which repeats itself, one can take the averages over the repeating part. In the general case the averages are taken as the limits for a region expanding to the whole plane. In cases like an infinite row of tiles, or tiles getting smaller and smaller outwardly, the outside is not negligible and should also be counted as a tile while taking the limit. In extreme cases the limits may not exist, or depend on how the region is expanded to infinity.
For finite tessellations and polyhedra we have
where F is the number of faces and V the number of vertices, and χ is the Euler characteristic (for the plane and for a polyhedron without holes: 2), and, again, in the plane the outside counts as a face.
The formula follows observing that the number of sides of a face, summed over all faces, gives twice the total number of sides in the entire tessellation, which can be expressed in terms of the number of faces and the number of vertices. Similarly the number of sides at a vertex, summed over all vertices, also gives twice the total number of sides. From the two results the formula readily follows.
In most cases the number of sides of a face is the same as the number of vertices of a face, and the number of sides meeting at a vertex is the same as the number of faces meeting at a vertex. However, in a case like two square faces touching at a corner, the number of sides of the outer face is 8, so if the number of vertices is counted the common corner has to be counted twice. Similarly the number of sides meeting at that corner is 4, so if the number of faces at that corner is counted the face meeting the corner twice has to be counted twice.
A tile with a hole, filled with one or more other tiles, is not permissible, because the network of all sides inside and outside is disconnected. However it is allowed with a cut so that the tile with the hole touches itself. For counting the number of sides of this tile, the cut should be counted twice.
For the Platonic solids we get round numbers, because we take the average over equal numbers: for (a − 2)(b − 2) we get 1, 2, and 3.
From the formula for a finite polyhedron we see that in the case that while expanding to an infinite polyhedron the number of holes (each contributing −2 to the Euler characteristic) grows proportionally with the number of faces and the number of vertices, the limit of (a − 2)(b − 2) is larger than 4. For example, consider one layer of cubes, extending in two directions, with one of every 2 × 2 cubes removed. This has combination (4, 5), with (a − 2)(b − 2) = 6 = 4(1 + 2 / 10)(1 + 2 / 8), corresponding to having 10 faces and 8 vertices per hole.
Note that the result does not depend on the edges being line segments and the faces being parts of planes: mathematical rigor to deal with pathological cases aside, they can also be curves and curved surfaces.
[edit] Tessellations of other spaces
An example tessellation A torus can be tiled by a
of the surface of a sphere by a truncated icosidodecahedron.
repeating matrix of isogonal quadrilaterals.
M.C.Escher, Circle Limit III (1959)
As well as tessellating the 2-dimensional Euclidean plane, it is also possible to tessellate other n-dimensional spaces by filling them with n-dimensional polytopes. Tessellations of other spaces are often referred to as honeycombs. Examples of tessellations of other spaces include:
Tessellations of n-dimensional Euclidean space. For example, filling 3-dimensional Euclidean space with cubes to create a cubic honeycomb.
Tessellations of n-dimensional elliptic space, either the n -sphere (spherical tiling, spherical polyhedron) or n-dimensional real projective space (elliptic tiling, projective polyhedron).
For example, projecting the edges of a regular dodecahedron onto its circumsphere creates a tessellation of the 2-dimensional sphere with regular spherical pentagons, while taking the quotient by the antipodal map yields the hemi-dodecahedron, a tiling of the projective plane.
Tessellations of n-dimensional hyperbolic space. For example, M. C. Escher's Circle Limit III depicts a tessellation of the hyperbolic plane using the Poincaré disk model with congruent fish-like shapes. The hyperbolic plane admits a tessellation with regular p-gons meeting in q's
whenever ; Circle Limit III may be understood as a tiling of octagons meeting in threes, with all sides replaced with jagged lines and each octagon then cut into four fish.
See (Magnus 1974) for further non-Euclidean examples.
There are also abstract polyhedra which do not correspond to a tessellation of a manifold because they are not locally spherical (locally Euclidean, like a manifold), such as the 11-cell and the 57-cell. These can be seen as tilings of more general spaces.
http://en.wikipedia.org/wiki/Tessellation
TessellationsPatterns covering the plane by fitting together replicas of the same basic shape have been created by Nature and Man either by accident or design. Examples range from the simple hexagonal pattern of the bees' honeycomb or a tiled floor to the intricate decorations used by the Moors in thirteenth century Spain or the elaborate mathematical, but artistic, mosaics created by Maurits Escher this century. These patterns are called tessellations.
What is a tessellation?
In geometrical terminology a tessellation is the pattern resulting from the arrangement of regular polygons to cover a plane without any interstices (gaps) or overlapping. The patterns are usually repeating. There are three types of tessellation.
Regular Tessellations
Regular tessellations are made up entirely of congruent regular polygons all meeting vertex to vertex. There are only three regular tessellations which use a network of equilateral triangles, squares and hexagons.
Those using triangles and hexagons-
Semi-regular Tessellations
Semi-regular tessellations are made up with two or more types of regular polygon which are fitted together in such a way that the same polygons in the same cyclic order surround every vertex. There are eight semi-regular tessellations which comprise different combinations of equilateral triangles, squares, hexagons, octagons and dodecagons.
Those using triangles and hexagons-
Non-regular Tessellations
Non-regular tessellations are those in which there is no restriction on the order of the polygons around vertices. There is an infinite number of such tessellations.
Taking account of the above mathematical definitions it will be readily appreciated that most patterns made up with one or more polyiamonds are not strictly tessellations because the component polyiamonds are not regular polygons. The patterns might more accurately be called mosaics or tiling patterns. Regular tessellations in the mathematical sense are possible, however, with the moniamond, the triangular tetriamond and the hexagonal hexiamond. Semi-regular tesselations are possible with combinations of the moniamond and the hexagonal hexiamond. Nevertheless I will apply the term tessellation (as other authors have) to describe the patterns resulting from the arrangement of one or more polyiamonds to cover the plane without any interstices or overlapping.
The following definitions and descriptions refer to tessellations of polyiamonds. Examples are restricted , with some noteable exceptions, to tessellations of individual polyiamonds.
Tessellations can be created by performing one or more of three basic operations, translation, rotation and reflection, on a polyiamond (see Figure).
Translation - sliding the polyiamond along the plane. The translation operation can be applied to all polyiamonds.
Rotation - rotating the polyiamond in the plane. The rotation operation can be applied to all polyiamonds which do not possess circular symmetry, for example the hexagonal hexiamond, which remains unchanged following rotation through 60o or multiples thereof.
Reflection - reflecting the polyiamond in the plane, as if being viewed in a mirror. The reflection operation is limited to polyiamonds which are enantiomorphic. An enantiomorphic polyiamond is one which cannot be superimposed on its reflection, its mirror image.
I propose the following classification of polyiamond tessellations which is based on the operations performed on the polyiamond being tessellated..
Simple tessellations are those in which only the translation operation is used.
Complex tessellations are those in which one or both of the rotation and reflection operations is used with the translation operation.
A single or multiple of a polyiamond may be combined to form a figure which is capable of tessellating the plane using only the translation operation. This figure will be called the unit cell.
A particular unit cell may be filled by multiples of different polyiamonds. Gardner described how five pairs of heptiamonds could be used to fill the same unit cell tessellation pattern. You will be able to find many other examples in the illustrations later.
Tessellations may be further classified according to how the unit cells containing one or more polyiamonds are arranged. If the unit cells are arranged such that a regular repeating pattern is produced the tessellation is termed periodic. If the arrangement produces an irregular or random pattern the tessellation is termed aperiodic. Another arrangement which produces a tessellation with a centre of circular symmetry is termed radial - such tessellations, with the exception of special cases, are complex and will comprise two three or six unit cells each containing an infinite number of poyiamonds.
All tesselations which are regular belong to a set of seventeen different symmetry groups which exhaust all the ways in which patterns can be repeated endlessly in two dimensions.
The reader should realise that polyiamonds of odd order cannot provide simple tessellations. Every polyiamond of odd order is by definition unbalanced. The rotation and reflection operations must be used in order to provide balanced unit cells for tessellation.
All of the polyiamonds of order eight or less, with the exception of one of the heptiamonds will tessellate the plane. The exception is the V-shaped heptiamond. Gardner (6th book p.248) posed the problem of identifying this
heptiamond and reproduced an impossibilty proof of Gregory. However, in combination with other heptiamonds or other polyiamonds, tesselations using this V-shaped heptiamond can be achieved.
http://www.mathpuzzle.com/Tessel.htm
What Is a Tessellation?
About This Project || What is a Tessellation? || Tessellation Tutorials || Tessellation Links
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:
Shapetrianglesquarepentagonhexagonmore than six sides
Angle measure in degrees6090
108120
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. It is formed by regular polygons.2. 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. Hold down on one of the images and copy it to the clipboard.2. Open a paint program.
3. Paste the image.
4. 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
Types of TessellationBy Mary Anne Thygesen, eHow Contributor
updated: September 11, 2009
There are many types and shapes that make up tessellations. Tessellations are tilings of a surface that have no gaps or overlaps, and have rotational symmetry.
Shapes that Make Regular Tessellations
1.
Equilateral triangle, square, hexagon.
Equilateral triangles, squares and hexagons are the shapes that make regular tessellations. These are common shapes for floor tiles and quilting templates.
Shapes that Make Semi Regular Tessellations
2.
Semi-regular tessellation shapes.
The shapes that make up semi-regular tessellations are composed of two or three regular polygons. In the image, from left to right, top to bottom: 6.3.3.3.3., 3.3.3.4.4., 3.3.4.3.4., 3.3.6.6., 3.6.3.6., 3.4.4.6. and 3.4.6.4. Notice the numbers correspond to the sides and angles of the individual shapes.
Shapes that Make Demi-Regular Tessellations
3.
Demi-regular tessellation shapes.
The shapes that construct demi-regular tessellations are more complex. Demi-regular tessellation shapes have two vertexes.
Shapes that Make Nonregular Tessellations
4.
Quadrilles
Nonregular tessellation shapes are shapes where the interior angles add up to 360 degrees. M.C. Escher made these shapes famous in his works.
Famous Ties5. Alhambra, a 14th century Moorish castle located in Granada, Spain, provided inspiration to M.C.
Escher.
Warning6. Not all tilings are tessellations in the mathematical sense. Many tilings are missing rotational
symmetry. A good way to check for rotational symmetry is to put your finger at the vertex, the point where all corners come together, and rotate the image to see if it looks the same all the way around.
Read more: Types of Tessellation | eHow.com http://www.ehow.com/facts_5375263_types-tessellation.html#ixzz0zzf5AAAfhttp://www.ehow.com/facts_5375263_types-tessellation.html
History of Tessellations Tessellations have been around for centuries and are still quite prevalent
today. However the study of tessellations in mathematics has a relatively
short history. In 1619, Johannes Kepler did one of the first documented
studies of tessellations when he wrote about the regular and semiregular
tessellation, which are coverings of a plane with regular polygons. Some two
hundred years later in 1891, the Russian crystallographer E. S. Fedorov
proved that every tiling of the plane is constructed in accordance to one of
seventeen different groups of isometries. Fedorov's work marked the
unofficial beginning of the mathematical study of tessellations. Other
prominent contributors include Shubnikov and Belov (1951); and Heinrich
Heesch and Otto Kienzle (1963).1
However, the most famous contributor was the Dutch artist, M. C. Escher
(1898-1972). M.C. Escher was a man studied and greatly appreciated by
respected mathematicians, scientists and crystallographers yet he had no
formal training in science or mathematics. He was a humble man who
considered himself neither an artist nor a mathematician.2 He is most
famous for his so-called impossible structures, such as Ascending and
Descending, Relativity, his Transformation Prints, such as Metamorphosis I,
Metamorphosis II and Metamorphosis III, Sky & Water I or Reptiles. During
his lifetime, M.C. Escher made 448 lithographs, woodcuts and wood
engravings and over 2000 drawings and sketches.3 (click here see examples
of his work.)
http://jwilson.coe.uga.edu/EMT668/EMAT6680.2003.fall/Thomas/PROJECT/
History.htm
History of Tessellations
Tessellations are visible in the world around us. You can see the polygons tessellating in mosaic titled floors and beehives.
Tiling can be seen in crystallization as well as quiltmaking. Tessellation is defined by a covering of a infinite geometric plane figures of one type or a few types. The word was founded in 1660. The Latin root tessellare means to pave. We can see how this word might have been derived if we look look at the stone paved streets of the 1600.
Tessellations are formed by translating, rotating, and reflecting polygons in a plane.
In our pages, you will see many samples of tessellating and you can view the work of a famous mathematician and architect. M.C.Escher,
and our student Galleria to see works created by the design team and other students in our geometry classes. As we create our designs using the software, the concepts of symmetry, transformations, and isometries are evident in everything that we do at our site.
http://library.thinkquest.org/CR0212121/history/historypage.htm
Shapes that tessellate
Index --- grids --- squares (examples) --- triangles (examples) --- Escher-style (examples) --- for teachers
Triangles, squares and hexagons are the only regular shapes which tessellate by themselves. You can have other tessellations of regular shapes if you use more than one type of shape. You can
even tessellate pentagons, but they won't be regular ones. Tessellations can be used for tile patterns or in patchwork quilts!
Single regular shapes TrianglesSquaresHexagons
Large grid of trianglesLarge grid of squaresLarge grid of hexagons
Multiple regular shapes Squares, trianglesHexagons, trianglesHexagons, squares, trianglesOctagons, squaresDodecagons, trianglesDodecagons, hexagons, squares
Large grid of squares and trianglesLarge grid of hexagons and trianglesLarge grid of hexagons, squares and trianglesLarge grid of octagons and squaresLarge grid of dodecagons and trianglesLarge grid of dodecagons, hexagons and squares
Other shapes Irregular pentagonsWaffle patternFish patterns
Large grid of irregular pentagonsLarge grid of waffle patternLarge grid of fish patterns
Triangles - these make pretty tessellations. Two triangles make a diamond. Six triangles make a hexagon.
Design a triangle tessellation online More examples of triangle tessellations Large grid of triangles to print or save
Design a triangle mosaic online - which you can use to make tessellations
Squares - rather obvious! By colouring them, you can build up more complicated patterns. Click here to make a square tessellation online. Make an Escher-style tessellation online
Design a square tessellation online Design an Escher-style tessellation online Optical illusions based on tessellations Large grid of triangles to print or save
Design mosaics online - which you can use to make tessellations
Hexagons - Bees make natural tessellations of hexagons in their honeycombs.
Hexagons can be made with triangle grids. Design a triangle tessellation online Design a triangle mosaic online
Large grid of hexagons to print or save
Squares and triangles - There are two ways to mix squares and triangles in the same pattern.
This is the boring way. You can make it more interesting with colouring.
The other grid looks strange, as we don't recognise squares as easily when they are tilted.
Large grids of triangles and squares to print or save - both types
Hexagons and triangles - There are two regular ways to mix hexagons and triangles in the same pattern. You can play around with the triangles grid to find other ways to mix hexagons and triangles.
This packs the triangles closely, with a few triangles between.
Here, the hexagons are further apart, with a line of triangles between them.
These tessellations can be made with triangle grids. Design a triangle tessellation online Design a triangle mosaic online
Large grids of triangles and hexagons to print or save - both types
Hexagons, squares and triangles
Large grid of hexagons, squares and triangles to print or save
Large grid of octagons and squares to print or save
Dodecagons (12 sides) and triangles - Since the sides of the shapes must be the same length, so they can fit together, you end up with the dodecagons being much larger than the triangles.
Large grid of dodecagons and triangles to print or save
Dodecagons, hexagons and squares
Large grid of dodecagons, hexagons and squares to print or save
Pentagons - Regular pentagons won't make a tessellation, but rather squashed ones will.
Large grid of irregular pentagons to print or save
Penrose tiles - these are based on five-fold symmetry, and they never repeat! (external site.)
Waffle - You don't have to restrict yourself to regular shapes. Here is a waffle pattern, and a suggested colour scheme.
Large waffle grid to print or save
Fish - These patterns are based on a simple fish design, made of a tilted square with a triangle for a tail.
It's surprising how many ways you can fit these together.
Large grid of fish to print or save
http://gwydir.demon.co.uk/jo/tess/grids.htm
2D
What is a 2d shape?A two-dimensional shape is a geometric figure having length and width, but no thickness or depth. Examples of 2-d shapes are drawings of squares, circles, or other objects. All two-dimensional figures can be drawn on a flat surface.
http://wiki.answers.com/Q/What_is_a_2d_shape
Regular 2-D Shapes - Polygons
Move the mouse over the shapes to discover their properties.
Triangle Square
Pentagon Hexagon
Heptagon Octagon
Nonagon Decagon
Hendecagon Dodecagon
These shapes are known as regular polygons. A polygon is a many sided shape with straight sides.
To be a regular polygon all the sides and angles must be the same.
http://www.mathsisfun.com/shape.html
POLYHEDRAL
Dictionary entry overview: What does polyhedral mean?
• POLYHEDRAL (adjective) The adjective POLYHEDRAL has 1 sense:
1. of or relating to or resembling a polyhedron
Familiarity information: POLYHEDRAL used as an adjective is very rare.
Dictionary entry details
• POLYHEDRAL (adjective)
Sense 1 polyhedral [BACK TO TOP]
Meaning:
Of or relating to or resembling a polyhedron
Classified under:
Relational adjectives (pertainyms)
Pertainym:
polyhedron (a solid figure bounded by plane polygons or faces)
http://www.audioenglish.net/dictionary/polyhedral.htm