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Symmetry groups of Euclidean objects may be completely classified as the subgroups of the Euclidean group E(n) (the isometry group of R n). Two geometric figures have the same symmetry type when their symmetry groups are conjugate subgroups of the Euclidean group: that is, when the subgroups H 1, H 2 are related by H 1 = g −1 H 2 g for some g ...
V is the symmetry group of this cross: flipping it horizontally (a) or vertically (b) or both (ab) leaves it unchanged.A quarter-turn changes it. In two dimensions, the Klein four-group is the symmetry group of a rhombus and of rectangles that are not squares, the four elements being the identity, the vertical reflection, the horizontal reflection, and a 180° rotation.
The symmetry group of a snowflake is D 6, a dihedral symmetry, the same as for a regular hexagon.. In mathematics, a dihedral group is the group of symmetries of a regular polygon, [1] [2] which includes rotations and reflections.
The RVB state on triangle lattice also realizes the Z2 spin liquid, [16] where different bond configurations only have real amplitudes. The toric code model is yet another realization of Z2 spin liquid (and Z2 topological order) that explicitly breaks the spin rotation symmetry and is exactly solvable. [17]
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 ...
In mathematics and theoretical physics, a superalgebra is a Z 2-graded algebra. [1] That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading.
If there are n different ways of moving the polygon to itself by a rotation (including the null rotation) then this symmetry group is isomorphic to Z/nZ. In three or higher dimensions there exist other finite symmetry groups that are cyclic, but which are not all rotations around an axis, but instead rotoreflections.
Z2 may refer to: Z2 (computer), a computer created by Konrad Zuse; Z2 (company), video game developer; Z2 Comics, a publisher of graphic novels