Tessellations• Combinatorics

• Designs and configurations For applications of design theory

By: Valerie Toothman

What are Tessellations

• A tessellation is created when a shape is repeated over and over again covering a plane without any gaps or overlaps.

• Meaning they have to share vertex points (a "corner point“) and edges (the side of the shape)

• 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.

Periodic

• Periodic – use tiles that form a repeating pattern

Regular Tessellation

• A regular tessellation is a tessellation made up of congruent regular polygons. – 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.]

Regular Tessellation

• There are only 3 regular tessellations:– Triangle -Square

– Hexagons

Regular Tessellation

Consider a two-dimensional tessellation with q regular p -gons at each polygon vertex. In the plane, (1− 2𝑝 )𝜋=2𝜋

𝑞1𝑝

+1𝑞

=12

(𝑝−2 ) (𝑞−2 )=4So

4

and the only factorizations are

Also called 666 because at 1 vertex point there are 3 hexagons with 6 sides

Also called 4444 because at 1 vertex point there are 4 squares with 4 sides

Also called 333333 because at 1 vertex point there are 6 triangles with 3 sides

• When the tessellation is made of regular polygons, the most common notation is the vertex configuration, which is simply a list of the number of sides of the polygons around a vertex

Semiregular Tessellation

• A semi-regular tessellation is made of two or more regular polygons. The pattern at each vertex must be the same! Sometimes called Archimedean tessellations. In the plane, there are eight such tessellations.

Semiregular Tessellation

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

Demiregular tessellation

• A demiregular tessellation is a type of tessellation whose definition is somewhat problematical.

• Some authors define them as orderly compositions of the three regular and eight semiregular tessellations (which is not precise enough to draw any conclusions from), while others defined them as a tessellation having more than one transitivity class of vertices (which leads to an infinite number of possible tilings).

Demiregular tessellation

• There are at least 14 demiregular tessellations.

• How was this determined? The process is almost fully trial-and-error and just requires a lot of time and effort.

Demiregular tessellation

Other Periodic

• Escher

Aperiodic

• Aperiodic - use tiles that cannot form a repeating pattern

Penrose tiling

• Penrose was not the first to discover aperiodic tilings, but his is probably the most well-known. In its simplest form, it consists of 36- and 72-degree rhombi, with "matching rules" forcing the rhombi to line up against each other only in certain patterns. It can also be formed by tiles in the shape of "kites" and "darts"

Penrose tiling

Penrose tiling

• Mathematicians have found no general rule for determining if a given shape can tile the plane or not, which means there are many unsolved problems concerning tessellations.

Sources

• http://mathforum.org/sum95/suzanne/whattess.html• http://mathworld.wolfram.com/Tessellation.html• http://www.mathsisfun.com/geometry/tessellation.html