The reader should realise that polyiamonds of odd order cannot provide simple tessellations. 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 will comprise two three or six unit cells each containing an infinite number of poyiamonds.Īll 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. If the arrangement produces an irregular or random pattern the tessellation is termed aperiodic. If the unit cells are arranged such that a regular repeating pattern is produced the tessellation is termed periodic. Tessellations may be further classified according to how the unit cells containing one or more polyiamonds are arranged. You will be able to find many other examples in the illustrations later. Gardner described how five pairs of heptiamonds could be used to fill the same unit cell tessellation pattern. This figure will be called the unit cell.Ī particular unit cell may be filled by multiples of different polyiamonds. Simple tessellations are those in which only the translation operation is used.Ĭomplex tessellations are those in which one or both of the rotation and reflection operations is used with the translation operation.Ī 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. I propose the following classification of polyiamond tessellations which is based on the operations performed on the polyiamond being tessellated. An enantiomorphic polyiamond is one which cannot be superimposed on its reflection, its mirror image. The reflection operation is limited to polyiamonds which are enantiomorphic. R eflection - reflecting the polyiamond in the plane, as if being viewed in a mirror. 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 60 o or multiples thereof.
R otation - rotating the polyiamond in the plane. The translation operation can be applied to all polyiamonds. Tr anslation - sliding the polyiamond along the plane. Tessellations can be created by performing one or more of three basic operations, translation, rotation and reflection, on a polyiamond (see Figure). Examples are restricted, with some noteable exceptions, to tessellations of individual polyiamonds. The following definitions and descriptions refer to tessellations of polyiamonds. Nevertheless I will apply the term tessellation (as other authors have) to describe the patterns resulting from the arrangement of one or more polyiamonds to cover the plane without any interstices or overlapping.
Semi-regular tesselations are possible with combinations of the moniamond and the hexagonal hexiamond. Regular tessellations in the mathematical sense are possible, however, with the moniamond, the triangular tetriamond and the hexagonal hexiamond. The patterns might more accurately be called mosaics or tiling patterns. Taking account of the above mathematical definitions it will be readily appreciated that most patterns made up with one or more polyiamonds are not strictly tessellations because the component polyiamonds are not regular polygons. There is an infinite number of such tessellations. Non-regular tessellations are those in which there is no restriction on the order of the polygons around vertices. There are eight semi-regular tessellations which comprise different combinations of equilateral triangles, squares, hexagons, octagons and dodecagons. Semi-regular tessellations are made up with two or more types of regular polygon which are fitted together in such a way that the same polygons in the same cyclic order surround every vertex. There are only three regular tessellations which use a network of equilateral triangles, squares and hexagons. Regular tessellations are made up entirely of congruent regular polygons all meeting vertex to vertex. In geometrical terminology a tessellation is the pattern resulting from the arrangement of regular polygons to cover a plane without any interstices (gaps) or overlapping. Examples range from the simple 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, but artistic, mosaics created by Maurits Escher this century. 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.
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