TESSELLATION

Definition

A tessellation is created when a shape is repeated over and over again covering a

plane without any gaps or overlaps.

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 the 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 was a small cubical piece of clay, stone or glass used to make

mosaics. 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 tessellation, often made of glazed clay.

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.

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.

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. (To see why, we compare this tiling to the surface of a

Torus.) If we tile before coloring, only four colors are needed.

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. (To see why, we compare this tiling to the surface of a

Torus.) If we tile before coloring, only four colors are needed.

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 with as minimal set of

translation vectors a pair according to the diagonals 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.

Regular and irregular 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 tesselations, 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. The Voderberg tiling

discovered by Hans Voderberg in 1936, which is the earliest known spiral tiling. The unit

tile is a bent enneagon. The Hirschhorn tiling discovered by Michael Hirschhorn in the

1970s. The unit tile is an irregular pentagon.

Tessellations and computer graphics

When A tessellation of a disk used to solve a finite element problem.

These rectangular bricks are connected in a tessellation, which if

considered an edge-to-edge tiling, topologically identical to a

hexagonal tiling, with each hexagon flattened into a rectangle with

the long edges divided into two edges 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. In computer-

aided design, arbitrary 3D shapes are often too complicated to analyze directly. So they

are divided (tessellated) into a mesh of small, easy-to-analyze pieces -- usually either

irregular tetrahedrons, or irregular hexahedrons. The mesh is used for finite element

analysis Some geodesic domes are designed by tessellating the sphere with triangles

that are as close to equilateral triangles as possible.

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 Ireland.

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 (3, 6), (3 \tfrac{1}{3},5), (3 \tfrac{3}{4},4 \tfrac{2}{7}), (4, 4), (6, 3),

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 bathroom floor tiling we have (5, 3 1/3).

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

( a - 2 ) ( b - 2 ) = 4 ( 1 - \frac{\chi}{F} ) ( 1 - \frac{\chi}{V} )

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 number of sides, 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 faces, gives also twice the 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.

Tessellations of other spaces

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 - for example, projecting the edges of a

dodecahedron onto its circumsphere creates a tessellation of the 2-dimensional sphere

with regular spherical pentagons.

* Tessellations of n-dimensional hyperbolic space - for example, M. C. Escher's Circle

Limit III depicts a tessellation of the hyperbolic plane with congruent fish-like shapes.

The hyperbolic plane admits a tessellation with regular p-gons meeting in q's whenever \

tfrac{1}{p}+\tfrac{1}{q} < \tfrac{1}{2}; 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.

History

In every civilization and culture, colored tilings and patterns appear among the earliest

decorations.... In particular, 2-color patterns arose -- early and frequently through a

device known as 'counterchange'.... An early paper with remarkable counterchange

designs formed by diagonally divided squares -- one-half black, one-half white -- was

published by Truchet (1704).

Regular Polygon Tessellations I

Objective: To understand which regular polygons will tile by themselves, which

won't,and why.

Materials: For teaching students, hands-on manipulatives such as Pattern Blocks

or Tessel-Gons are recommended.

A polygon is a many-sided shape. A regular polygon is one in which all of the sides and

angles are equal. Some examples are shown below.

These are referred to as, respectively, (regular) triangle, square, pentagon, hexagon,

heptagon, and octagon. A vertex is a point at which three or more tiles in a tessellation

meet. Two tiles cannot meet in a point, but would have to meet in line. First, try

tessellating with hexagons. This works, as shown below, with three hexagons meeting

at each vertex.

Since the interior angles get larger as the number of sides in a polygon gets larger, no

regular polygons with more than six sides can tessellate by themselves. (Hexagons

already have the minimum possible number of tiles meeting at each vertex, three.) If

you have a Tessel-gons set, you can try tessellating with octagons to illustrate this

point. Next, try tessellating with squares. This is also possible, as shown below.

Here four tiles meet at each vertex. Since there are no integers between three and four,

pentagons must not tessellate. This is shown below. (Foam rubber regular pentagons

can be purchased from Tessellations for $0.25 each, if you want to be able to

demonstrate this hands on.)

Finally, try tessellating with triangles. This is also possible, as shown below, where the

number of tiles meeting at each vertex is six. Since there are no regular polygons with

less than three sides, the only regular polygons which will tile be themselves are

triangles, squares, and hexagons.

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

color in 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.

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 of the polygon are all the same length. 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

triangle

square

pentagon

hexagon

more than six sides

   

Angle measure in degrees

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:

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:

   

A regular tiling of polygons (in two dimensions), polyhedra (three dimensions), or

polytopes (n 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

here 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 n-dimensional polytopes is called a honeycomb.

Polygon Tessellation

The breaking up of self-intersecting polygons into simple polygons (illustrated above) is

also called tessellation (Woo et al. 1999).

Symmetry in Tessellations

Objective: To understand the different types of mathematical symmetry found in

tessellations.

Materials: For teaching students, hands-on manipulatives are recommended. Any of the

Puzzellations puzzles can be used to illustrate these basic symmetries.

Three types of mathematical symmetry are commonly found in tessellations. These are

translational symmetry, rotational symmetry, and glide reflection symmetry. Recall when

reading this lesson that tessellations extend to infinity; the diagrams shown below are

finite portions of infinite tessellations.

1.Translationa lSymmetry

A tessellation possesses translational symmetry if it can be translated by some vector

and remain unchanged. Any tessellation with this property has inifinitely many different

translation vectors due to the infinite extent of tessellations. The tessellation below has

translational symmetry; two possible vectors are shown. Find additional vectors.

2. Rotational Symmetry

A tessellation possesses rotational symmetry if it can be rotated by some angle about

some point and remain unchanged. A tessellation which can be rotated by 1/n of a full

revolution and remain unchanged is said to posses n-fold rotational symmetry. In the

example below, point A is a point of 3-fold rotational symmetry, while point B is a point

of 2-fold rotational symmetry. Try to identify a point of 6-fold rotational symmetry.

3.GlideReflectionSymmetry

A tessellation possesses glide reflection symmetry if it can be translated by some vector

and then reflected about that vector and remain unchanged. A special case of glide

rereflection symmetry is simple reflection or mirror symmetry, where the vector has a

value of zero. The example below illustrates glide reflection. Try to find some lines of

simple reflection symmetry for the first tessellation above. Does the second tessellation

above posses glide reflection symmetry?

Using any of the Puzzellations puzzles, pattern blocks, etc., try to construct other

tessellations which exhibit the symmetries discussed here.

Tessellation

A Tessellation (or Tiling) is when you cover a surface with a pattern of flat shapes so

that there are no overlaps or gaps

Examples:

   

Rectangles Octagons and Squares Pentagons and Quadrilaterals

Regular Tessellations

A regular tessellation is a pattern made by repeating a regular polygon. There are only

3 regular tessellations:

   

Triangles Squares Hexagons3.3.3.3.3.3 4.4.4.4 6.6.6

Look at a Vertex ...

A vertex is just a "corner point".

What shapes meet here?

Three hexagons meet at this vertex,and a hexagon has 6 sides.

So this is called a "6.6.6" tessellation.

 

For a regular tessellation, the pattern is identical at each vertex!

Semi-regular Tessellations

A semi-regular tessellation is made of two or more regular polygons. The pattern at each vertex must be the same! There are only 8 semi-regular tessellations:

   

3.3.3.3.6   3.3.3.4.4   3.3.4.3.4         

   

3.4.6.4   3.6.3.6   3.12.12         

     

4.6.12   4.8.8             

Other Tessellations

There are also "demiregular" tessellations, but mathematicians disagree on what they

actually are!

And some people allow curved shapes (not just polygons) so you can have tessellations like

these:

   

Curvy Shapes Circles Eagles?

Example Of Tessellation