Tessellation
Introduction
Tessellation, also known as tiling, is a process in geometry that involves the covering of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In other words, a tessellation is a way of tiling a plane using shapes such that there are no gaps or overlaps. The shapes used in tessellation are usually polygons, but they can also be more complex shapes.
History
The concept of tessellation has been present in various cultures throughout history. The earliest known tessellations were found in the Sumerian and Egyptian cultures, where they were used in wall decorations and floor mosaics. The Greeks further developed the concept, using tessellations in their architecture and pottery designs. In the Islamic world, tessellations were used extensively in architectural decoration, particularly in the form of complex geometric patterns.
Mathematical Foundations
Tessellation is a fundamental concept in the field of geometry. It is closely related to the principles of symmetry, regularity, and repetition. The mathematical study of tessellations involves the use of several branches of mathematics, including geometry, topology, and group theory.
Regular Tessellations
A regular tessellation is a tessellation that uses only one type of regular polygon, and the same number of polygons meet at each vertex. There are only three regular tessellations: those made up of equilateral triangles, squares, or regular hexagons.
Semi-Regular Tessellations
A semi-regular tessellation, also known as an Archimedean tessellation, is a tessellation made up of more than one type of regular polygon, but the arrangement of polygons around each vertex is the same. There are eight semi-regular tessellations.
Non-Regular Tessellations
Non-regular tessellations are tessellations that do not fit into the categories of regular or semi-regular tessellations. They can be made up of irregular polygons, or they can be made up of regular polygons but with different arrangements of polygons at each vertex.
Tessellations in Art and Architecture
Tessellations have been used in art and architecture throughout history. They are a common feature in Islamic art and architecture, where complex geometric patterns are used to decorate buildings. In the modern era, artists such as M.C. Escher have used tessellations in their work to create complex and visually striking images.
Tessellations in Nature
Tessellations also occur in nature. For example, honeycombs are a natural example of a regular hexagonal tessellation. Tessellations can also be seen in the patterns of certain animal skins, such as the scales of reptiles or the feathers of birds.
Tessellations in Computer Graphics
In computer graphics, tessellation is used to create complex surfaces from simpler shapes. This process is used in 3D modeling and animation to create detailed and realistic images.