The Illustration Below is an Example of a Semi-regular Tessellation

The Illustration Below is an Example of a Semi-regular Tessellation.

Patterns 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 elementary 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, simply artistic, mosaics created past 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 whatever interstices (gaps) or overlapping. The patterns are usually repeating. At that place 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 3 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 upwardly with two or more than types of regular polygon which are fitted together in such a way that the same polygons in the same cyclic order surround every vertex. In that location are eight semi-regular tessellations which incorporate unlike 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 brake on the order of the polygons around vertices. At that place is an infinite number of such tessellations.

Taking account of the above mathematical definitions information technology will exist readily appreciated that most patterns made up with one or more than polyiamonds are not strictly tessellations considering the component polyiamonds are non regular polygons. The patterns might more than 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. Still I will apply the term tessellation (equally other authors have) to describe the patterns resulting from the system of one or more than 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 exist 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 airplane. The translation operation can be applied to all polyiamonds.

Rotation – rotating the polyiamond in the airplane. The rotation operation can be applied to all polyiamonds which practice not possess circular symmetry, for example the hexagonal hexiamond, which remains unchanged following rotation through lxo
or multiples thereof.

Reflection – reflecting the polyiamond in the plane, every bit if beingness viewed in a mirror. The reflection performance is express to polyiamonds which are enantiomorphic. An enantiomorphic polyiamond is one which cannot be superimposed on its reflection, its mirror image.

I advise the post-obit classification of polyiamond tessellations which is based on the operations performed on the polyiamond being tessellated..

Uncomplicated tessellations
are those in which only the translation performance is used.

Complex tessellations
are those in which i or both of the rotation and reflection operations is used with the translation operation.

A unmarried or multiple of a polyiamond may be combined to form a figure which is capable of tessellating the plane using only the translation performance. This effigy will be chosen the
unit cell.

A particular unit of measurement cell may be filled by multiples of different polyiamonds. Gardner described how five pairs of heptiamonds could be used to fill the aforementioned unit cell tessellation pattern. You volition be able to find many other examples in the illustrations subsequently.

Tessellations may be farther classified according to how the unit cells containing i or more polyiamonds are arranged. If the unit of measurement cells are arranged such that a regular repeating design is produced the tessellation is termed
periodic. If the arrangement produces an irregular or random design 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 volition incorporate ii iii or vi unit cells each containing an infinite number of poyiamonds.

All tesselations which are regular belong to a prepare of seventeen unlike symmetry groups which exhaust all the means in which patterns can be repeated endlessly in two dimensions.

The reader should realise that
polyiamonds of odd order cannot provide uncomplicated tessellations. Every polyiamond of odd order is past definition
unbalanced. The rotation and reflection operations must be used in club to provide counterbalanced 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 Five-shaped heptiamond. Gardner (6th volume 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.

The Illustration Below is an Example of a Semi-regular Tessellation

Source: https://www.mathpuzzle.com/Tessel.htm