# Dictionary Definition

tessellation

### Noun

1 the careful juxtaposition of shapes in a
pattern; "a tessellation of hexagons"

2 the act of adorning with mosaic

# User Contributed Dictionary

## English

- The property or fact of tessellating.
- Squares can be used for tessellation.

- A tiling pattern with no gaps; the result
of tessellating an area or plane.
- This is a tessellation of the plane with squares and regular octagons.

- A less common name for polygon tessellation.

### Translations

- French: pavage, dallage
- Italian: tessellazione, tassellazione

### Derived terms

### Related terms

# Extensive Definition

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.

## 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

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.
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

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,5), (3 \tfrac,4 \tfrac), (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 ) ( 1 - \frac )

where F is the number of faces and V the number
of vertices, and \chi 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

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 qs whenever \tfrac+\tfrac ; 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

## See also

- Convex uniform honeycomb - The 28 uniform 3-dimensional tessellations, a parallel construction to this plane set
- Honeycomb (geometry)
- Jig-saw puzzle
- List of uniform tilings
- Mathematics and fiber arts
- Mosaic
- Penrose tilings
- Polyiamond - tilings with equilateral triangles
- Polyomino
- Quilting
- Self-replication
- Tile
- Tiling puzzle
- Tiling, Aperiodic
- Tilings of regular polygons
- Trianglepoint - needlepoint with polyiamonds (equilateral triangles)
- Triangulation
- Uniform tessellation
- Voronoi tessellation
- Wallpaper group - seventeen types of two-dimensional repetitive patterns
- Wang tiles

## References

- Grunbaum, Branko and G. C. Shephard. Tilings and Patterns. New York: W. H. Freeman & Co., 1987. ISBN 0-7167-1193-1.
- Coxeter, H.S.M.. Regular Polytopes, Section IV : Tessellations and Honeycombs. Dover, 1973. ISBN 0-486-61480-8.

## External links

- Tilings Encyclopedia - Reference for Substitution Tilings
- 2D Puzzles generated using a tessellation of the plane
- Tessellation Video
- Math Forum Tessellation Tutorials - make your own
- Mathematical Art of M. C. Escher - tessellations in art
- The 14 Different Types of Convex Pentagons that Tile the Plane
- Tiling Plane & Fancy at Southern Polytechnic State University
- Grotesque Geometry, Andrew Crompton
- Tessellations.org - many examples and do it yourself tutorials from the artistic, not mathematical, point of view
- Tessellation.info A database with over 500 tessellations categorized by artist and depicted subjects
- Minesweeper Variants - A Minesweeper game using different tessellation patterns
- Semiregular pattern - This pattern can describe a collapsing cylinder
- Hyperbolic Tessellations, David E. Joyce, Clark University
- Some Special Radial and Spiral Tilings

tessellation in Arabic: فسيفساء (رياضيات)

tessellation in Danish: Tessellation

tessellation in German: Parkettierung

tessellation in Spanish: Teselación

tessellation in French: Pavage

tessellation in Korean: 쪽 맞추기

tessellation in Italian: Tassellatura

tessellation in Hebrew: ריצוף של המישור

tessellation in Japanese: 平面充填

tessellation in Norwegian: Tesselering

tessellation in Polish: Tesselacja

tessellation in Russian: Тесселяция

tessellation in Serbian: Теселација

tessellation in Swedish: Tessellation

tessellation in Tamil: தரைபாவுமை

tessellation in Thai: เทสเซลเลชัน

tessellation in Ukrainian: Теселяція

tessellation in Contenese:
密鋪平面