Transformations (mathematics).

In Transformations Representations mathematics, transformations have a rich history that connects various disciplines, including geometry, algebra, linear algebra, and analysis with applications in statistics, physics, computer science, architecture, art, astronomy, and optics. In general, a transformation changes some aspect while at the same time preserving some type of structure. For example, a dilation of an object will shrink or enlarge it but will preserve the basic shape; while a reflection of the plane will produce a mirror image, which flips figures while preserving distances between points. Mathematicians and geometers often transform an object, equation, or data to something that is easier to investigate, such as transforming coordinates to simplify algebraic expressions. The theory of transformations has important implications as well. There are many types of transformations including geometric transformations, conformal transformations, z-score transformations, linear transformations, and Möbius transformations, named for August Ferdinand Möbius. Geometric transformations have long been implicitly used in aesthetically pleasing design patterns in pottery, quilts, architecture, and art, such as tessellations in the MoorishAlhambra Fortress. Historians and anthropologists compare and contrast these patterns to track the spread of groups of people. Mathematical transformations can be represented in a variety of ways, such as matrix representations of linear transformations, which are useful in algorithms and computer graphics. In school, young children study geometric transformations and this study continues through high school, where students represent various geometric and algebraic transformations using coordinates, vectors, function notation, and matrices. Students also investigate transformations using computers and calculators.