Transformations (mathematics)
Transformations in mathematics refer to operations that change geometric objects, functions, or data while preserving certain structures. They find applications across various fields, including geometry, algebra, computer science, and physics. Common types of transformations include geometric transformations, which involve actions like rotation, reflection, and dilation, and linear transformations, which maintain vector addition and scalar multiplication properties.
Historically, transformations have roots in ancient mathematical practices, with figures like Euclid and Archimedes contributing early concepts. Over time, mathematicians like Isaac Newton and Felix Klein expanded the theory, leading to significant developments in understanding geometrical properties through transformation groups. These theories have practical implications, such as in computer graphics and architectural design, where transformation techniques create visually appealing designs and accurate models.
In educational contexts, transformations are introduced to students at a young age, providing a foundational understanding that evolves through high school mathematics, where students learn to apply transformations using various tools like matrices and coordinates. Overall, the study of transformations encompasses both theoretical and practical aspects, making it a vital area within mathematics.
Transformations (mathematics)
Summary: Numerous mathematicians since antiquity have studied and worked on the concept of transformations.
In 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.
Early History
The early development of geometric transformations is tied to motions that were useful in modeling the Earth and the stars and in creating artistic works, architectural buildings, and geometric objects. The Pythagoreans thought that points traced lines and lines traced surfaces. Aristotle objected to the use of physical concepts like movement in these abstract mathematical objects. Euclidof Alexandria mostly avoided the concept of motion in his work. However, he used the notion of “superposition,” where one object is placed on top of another, in triangle congruence theorems, such as in his proof of side-angle-side congruence. In modern proofs, mathematicians would likely use transformations in order to place these triangles on top of each other. Euclid also defined a sphere as the rotation of a semicircle, and he defined a cylinder as the rotation of a rectangle. Archimedes of Syracuse investigated axial affinity motions in his work on ellipses, and Apollonius of Perga explored inversion. Marcus Vitruvius described the projections that were important in architecture, and he also investigated the concept of stereographic projection, which was useful in astronomy and map making.
Mathematicians around the world generalized these motions and applied them to a variety of fields. Most mathematicians in later times relied on transformations in geometry, although Omar Khayyám criticized Ibn al-Haytham’s extensive use of motion by questioning how a line could be defined by a moving point when “it precedes a point by its essence and by its existence.” Both Thabit ibn Qurra and his grandson Ibrahim ibn Sinan investigated “affine” transformations of the plane that preserved straight lines, like dilations. Alexis Clairaut and Leonhard Euler defined and explored general affine transformations. Sir Isaac Newton investigated various coordinate systems and the transformations between them, such as what are referred to as “rectangular and polar coordinates.” Girard Desargues systematically investigated projective transformations, although many earlier mathematicians had investigated perspective drawing and projection in mathematics, art, and optics. Edward Waring and Gaspard Monge also studied projective transformations. Mobius represented affine and projective transformations analytically in terms of homogeneous coordinates.
Carl Friedrich Gauss linked transformations with linear algebra when he represented linear transformations of quadratic forms as rectangular arrays of numbers. A linear transformation of the plane is a map that preserves addition and scalar multiplication of vectors. Linear transformations of the plane are combinations of rotations, reflections, dilations, shears, and projections, and they are important in modeling movement in computer graphics. In general, a linear transformation is a map between vector spaces that preserves addition and scalar multiplication. Linear transformations of coordinates were important in the development of analytic geometry and some multivariate statistical methods and linear transformations were also linked to projective geometry and Möbius transformations, which are also called “fractional linear transformations.” Henri Poincaré connected these transformations to hyperbolic geometry. Gotthold Eisenstein and Charles Hermite tried to extend Gauss’s work on forms and in this context they defined the addition and multiplication of linear transformations. Arthur Cayley defined a general notion of matrices and recognized that the composition of linear transformations could be represented using them. James Sylvester explored properties of matrices that were preserved under transformations and defined the nullity of a matrix. Matrices continued to be connected to linear transformations and the theory of linear transformations extended to infinitely many dimensions.
Modern Developments
At the beginning of the twentieth century, Felix Klein revolutionized mathematics and physics with the idea of a transformation group. In his Erlanger Program, the properties of a space were now understood by the transformations that preserved them. Thus the classification, algebraic structure, and invariants of these transformations provided information about the corresponding geometries. His ideas unified Euclidean and non-Euclidean geometry and became the basis for geometry in the twentieth century. Klein’s collaboration with Sophus Lie impacted the development of the Erlanger program. Lie also developed the notion of continuous transformation groups and associated these with a differential equation. Physicists and mathematicians continue to study the local structure of a so-called Lie group by the infinitesimal transformations in the Lie algebra. Earlier mathematicians and physicists had already used invariants in a several ways. For instance, Cremona transformations are named for Luigi Cremona, who studied birational transformations. These transformations were important in the study of algebraic functions and integrals. Max Noether investigated the invariant properties of algebraic varieties using birational transformations. In physics, Hermann Minkowski explored Maxwell’s equations for electromagnetism, named after James Maxwell. These equations were invariant under Lorentz transformations, named for Hendrik Lorentz, and led to a geometry of space-time and the beginning of relativity theory.
Bibliography
Kastrup, H. A. “On the Advancements of Conformal Transformations and Their Associated Symmetries in Geometry and Theoretical Physics.” Annalen der Physik 17, no. 9–10 (2008).
Kleiner, Israel. A History of Abstract Algebra. Boston: Birkhauser, 2007.
Rosenfeld, B. A. A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space. New York: Springer, 1988.