# Tessellation pattern

The patterns are usually repeating.

Complex tessellations are those in which one or both of the rotation and reflection operations is used with the translation operation. The familiar "brick wall" tiling is not edge-to-edge because the long side of each rectangular brick is shared with two bordering bricks.

## Tessellation project

We can divide this by one diagonal, and take one half a triangle as fundamental domain. 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. If the arrangement produces an irregular or random pattern the tessellation is termed aperiodic. Choose a vertex and count the sides of the polygons that touch it. The four colour theorem states that for every tessellation of a normal Euclidean plane , with a set of four available colours, each tile can be coloured in one colour such that no tiles of equal colour meet at a curve of positive length. The reader should realise that polyiamonds of odd order cannot provide simple tessellations. 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. The Delaunay triangulation is a tessellation that is the dual graph of a Voronoi tessellation. All three of these tilings are isogonal and monohedral. A regular tessellation is a highly symmetric , edge-to-edge tiling made up of regular polygons , all of the same shape.

The Delaunay triangulation is a tessellation that is the dual graph of a Voronoi tessellation. Every polyiamond of odd order is by definition unbalanced. Further information: Euclidean tilings of regular polygonsUniform tilingand List of convex uniform tilings Mathematicians use some technical terms when discussing tilings.

A regular tessellation is a highly symmetricedge-to-edge tiling made up of regular polygonsall of the same shape. The word tessellation is derived from the Greek "tesseres", which means "four" and refers to the four sides of a square, the first shape to be tiled.

I propose the following classification of polyiamond tessellations which is based on the operations performed on the polyiamond being tessellated.

## Tessellation in nature

Where the shapes join together, the corner point, we call that the vertex. Certain polyhedra can be stacked in a regular crystal pattern to fill or tile three-dimensional space, including the cube the only Platonic polyhedron to do so , the rhombic dodecahedron , the truncated octahedron , and triangular, quadrilateral, and hexagonal prisms , among others. The recursive process of substitution tiling is a method of generating aperiodic tilings. Any triangle or quadrilateral even non-convex can be used as a prototile to form a monohedral tessellation, often in more than one way. Examples are restricted , with some noteable exceptions, to tessellations of individual polyiamonds. There are only three regular tessellations: those made up of equilateral triangles , squares , or regular hexagons. Regular tessellations in the mathematical sense are possible, however, with the moniamond, the triangular tetriamond and the hexagonal hexiamond. These patterns are called tessellations. The translation operation can be applied to all polyiamonds. Gardner 6th book p. 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 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.

The pattern at each vertex should be the same. By looking at the vertex and counting the sides of all the shapes that meet at the vertex you are able to name a tessellation.

Where the shapes join together, the corner point, we call that the vertex.

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