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Discrete symmetries sometimes involve some type of 'swapping', these swaps usually being called reflections or interchanges. In mathematics and theoretical physics , a discrete symmetry is a symmetry under the transformations of a discrete group —e.g. a topological group with a discrete topology whose elements form a finite or a countable set .
The proper symmetry group is then a subgroup of the special orthogonal group SO(n), and is called the rotation group of the figure. In a discrete symmetry group, the points symmetric to a given point do not accumulate toward a limit point. That is, every orbit of the group (the images of a given point under all group elements) forms a discrete ...
The symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation. A family of particular transformations may be continuous (such as rotation of a circle) or discrete (e.g., reflection of a bilaterally symmetric figure, or rotation of a ...
This article summarizes the classes of discrete symmetry groups of the Euclidean plane. The symmetry groups are named here by three naming schemes: International notation, orbifold notation, and Coxeter notation. There are three kinds of symmetry groups of the plane: 2 families of rosette groups – 2D point groups. 7 frieze groups – 2D line ...
Rotational symmetry of order n, also called n-fold rotational symmetry, or discrete rotational symmetry of the nth order, with respect to a particular point (in 2D) or axis (in 3D) means that rotation by an angle of (180°, 120°, 90°, 72°, 60°, 51 3⁄7 °, etc.) does not change the object. A "1-fold" symmetry is no symmetry (all ...
Discrete group. The integers with their usual topology are a discrete subgroup of the real numbers. In mathematics, a topological group G is called a discrete group if there is no limit point in it (i.e., for each element in G, there is a neighborhood which only contains that element). Equivalently, the group G is discrete if and only if its ...
There are ten discrete symmetry classes of topological insulators and superconductors, corresponding to the ten Altland–Zirnbauer classes of random matrices.They are defined by three symmetries of the Hamiltonian ^ =, †, (where , and †, are the annihilation and creation operators of mode , in some arbitrary spatial basis) : time-reversal symmetry, particle-hole (or charge conjugation ...
O h, (*432) [4,3] =. Icosahedral symmetry. I h, (*532) [5,3] =. In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O (3), the group of all isometries that leave the origin fixed, or ...