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A rotation rotates an object using a point as its center. An equilateral triangle has rotational symmetry with respect to an angle of 120°. Pentamerism is a body symmetry exhibited primarily by starfish; it is rotational symmetry with respect to an angle of 72°.
Rotational symmetry with respect to any angle is, in 2D, circular symmetry. In 3D we can distinguish cylindrical symmetry and spherical symmetry (no change when rotating about one axis, or for any rotation). I.e., no dependence on the angle using cylindrical coordinates and no dependence on either angle using spherical coordinates .
A translation "slides" an object from one area to another by a vector. 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. Therefore, symmetry in real physical objects 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 object with the most symmetry is empty space because any part of can be rotated, reflected or translated without apparent change.
The most familiar and conventionally taught type of symmetry is the left-right or mirror image symmetry exhibited for instance by the letter T: when this letter is reflected along a vertical axis, it appears the same.
An equilateral triangle exhibits such a reflection symmetry along three axes, and in addition it shows rotational symmetry: if rotated by 120 or 240 degrees, it remains unchanged. An instance of a shape which exhibits only rotational symmetry (w.r.t. an angle of 90 degrees) but no reflectional symmetry is the swastika.
An example of translational symmetry is:
wikipedia wikipedia wikipedia wikipedia(get the same by moving one line down and two positions to the right), and of two-fold translational symmetry:
* |* |* |* | |* |* |* |* |* |* |* |* * |* |* |* | |* |* |* |* |* |* |* |*(get the same by moving three positions to the right, or one line down and two positions to the right; consequently get also the same moving three lines down).
In both cases there is neither mirror-image symmetry nor rotational symmetry.
The German geometer Felix Klein enunciated a very influential Erlangen programme in 1872, suggesting symmetry as unifying and organising principle in geometry (at a time when that was read 'geometries'). This is a broad rather than deep principle. Initially it led to interest in the groupsIn mathematics, a group is a set, together with a binary operation satisfying certain axioms, detailed below. The branch of mathematics which studies groups is called group theory. The historical origin of group theory goes back to the works of Evariste G attached to geometries, and the slogan transformation geometryGeometry In mathematics, transformation geometry is a name for a pedagogic theory for teaching Euclidean geometry, based on the Erlangen programme. Felix Klein who put forward that point of view was himself much interested in mathematical education. It to (an aspect of the New Math, but hardly controversial in modern mathematical practice). By now it has been applied in numerous forms, as kind of standard attack on problems.
A fractalA fractal is a geometric object which is "broken up" in a radical way. The term fractal was coined in 1975 by Benoit Mandelbrot, from the Latin fractus or "broken", in order to call attention to such objects. They are in a number of major aspects differen, as conceived by MandelbrotSee: Benoit Mandelbrot a French mathematician largely responsible for the present interest in fractal geometry Mandelbrot set a fractal discovered by the above mathematician., has symmetry involving scaling. For example an equilateral triangle can be shrunk so that each of its sides are one third the length of the original's sides. These smaller triangles can be rotated and translated until they are adjacent and in the center of each of the larger triangle's lines. The smaller triangles can repeat the process, resulting in even smaller triangles on their sides. Fascinating intricate structures can be created by repeating such scaling symmetrical operations many times.