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Symmetry
Symmetry
Symmetry is a characteristic of geometrical
shapes, equations and other objects; we say that such an object is symmetric
with respect to a given operation if this operation, when applied to the object,
does not appear to change it. The three main symmetrical operations are
reflection, rotation and translation. A
reflection "flips" an object over a line, inverting it as if in a mirror. A
rotation rotates an object using a point as its center. A
translation "slides" an object from one area to another by a
vector. Even more complex operations on a geometric object, like shrinking
or shape warping, can be reduced to the operation of translation of every point
within the object. Symmetry occurs in geometry, mathematics, physics, biology,
art, literature (palindromes),
etc.
Although two objects with great similarity appear the same,
they must logically be different. For example, if one rotates an
equilateral triangle around its center 120 degrees, it will appear the same
as it was before the rotation to an observer. In theoretical
euclidean geometry, such a rotation would be unrecognizable from its
previous form. In reality however, each corner of any equilateral triangle
composed of matter must be composed of separate molecules in separate locations.
Symmetry therefore, is a matter of similarity instead of sameness. The
difficulty for an
intelligence to differentiate such a seemingly exact similarity might be
responsible for the mild
altered state of consciousness one gets by observing intricate patterns
based on symmetry.
The generalisation of symmetry in
physics to mean
invariance under any kind of transformation has become one of the most
powerful tools of theoretical physics. See
Noether's theorem for more details. This has led to
group theory being one of the areas of mathematics most studied by
physicists.
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