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The most familiar example is the leftright or mirror image symmetry exhibited for instance by the letter T: when this letter is reflected along a vertical axis, it remains 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 but no reflectional symmetry is the swastika.
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 groups attached to geometries, and the slogan transformation geometry[?] (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.
An example of a mathematical expression exhibiting symmetry is a^{2}c + 3ab + b^{2}c. If a and b are exchanged, the expression remains unchanged due to the commutativity of addition and multiplication.
In mathematics, one studies the symmetry of a given object by collecting all the operations that leave the object unchanged. These operations form a group. For a geometrical object, this is known as its symmetry group; for an algebraic object, one uses the term automorphism group. The whole subject of Galois theory deals with wellhidden symmetries of fields.
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.
See also: chirality, Bilateral symmetry
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