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Symmetry occurs not only in geometry, but also in other branches of mathematics. Symmetry is a type of invariance: the property that a mathematical object remains unchanged under a set of operations or transformations. [1] Given a structured object X of any sort, a symmetry is a mapping of the object onto itself which preserves the structure.
This article describes symmetry from three perspectives: in mathematics, including geometry, the most familiar type of symmetry for many people; in science and nature; and in the arts, covering architecture, art, and music. The opposite of symmetry is asymmetry, which refers to the absence of symmetry.
A drawing of a butterfly with bilateral symmetry, with left and right sides as mirror images of each other.. In geometry, an object has symmetry if there is an operation or transformation (such as translation, scaling, rotation or reflection) that maps the figure/object onto itself (i.e., the object has an invariance under the transform). [1]
Following the life and work of famous mathematicians from antiquity to the present, Stewart traces mathematics' developing handling of the concept of symmetry.One of the first takeaways, established in the preface of this book, is that it dispels the idea of the origins of symmetry in geometry, as is often the first context in which the term is introduced.
In geometry, groups first became important in projective geometry and, later, non-Euclidean geometry. Felix Klein's Erlangen program proclaimed group theory to be the organizing principle of geometry. Galois, in the 1830s, was the first to employ groups to determine the solvability of polynomial equations.
The above ideas lead to the useful idea of invariance when discussing observed physical symmetry; this can be applied to symmetries in forces as well.. For example, an electric field due to an electrically charged wire of infinite length is said to exhibit cylindrical symmetry, because the electric field strength at a given distance r from the wire will have the same magnitude at each point on ...
Rotations define important classes of symmetry: rotational symmetry is an invariance with respect to a particular rotation. The circular symmetry is an invariance with respect to all rotation about the fixed axis. As was stated above, Euclidean rotations are applied to rigid body dynamics.
Thus inversive geometry, a larger study than grade school transformation geometry, is usually reserved for college students. Experiments with concrete symmetry groups make way for abstract group theory. Other concrete activities use computations with complex numbers, hypercomplex numbers, or matrices to express