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In particle physics, the family symmetries or horizontal symmetries are various discrete, global, or local symmetries between quark-lepton families or generations.In contrast to the intrafamily or vertical symmetries (collected in the conventional Standard Model and Grand Unified Theories) which operate inside each family, these symmetries presumably underlie physics of the family flavors.
The type of symmetry is determined by the way the pieces are organized, or by the type of transformation: An object has reflectional symmetry (line or mirror symmetry) if there is a line (or in 3D a plane) going through it which divides it into two pieces that are mirror images of each other. [6]
The terms horizontal (h) and vertical (v), and the corresponding subscripts, refer to the additional mirror plane, that can be parallel to the rotation axis (vertical) or perpendicular to the rotation axis (horizontal). The simplest nontrivial axial groups are equivalent to the abstract group Z 2: C i (equivalent to S 2) – inversion symmetry
A symmetry plane parallel with the principal axis is dubbed vertical (σ v) and one perpendicular to it horizontal (σ h). A third type of symmetry plane exists: If a vertical symmetry plane additionally bisects the angle between two 2-fold rotation axes perpendicular to the principal axis, the plane is dubbed dihedral (σ d). A symmetry plane ...
A drawing of a butterfly with bilateral symmetry, with left and right sides as mirror images of each other.. In geometry, an object has symmetry if there is an operation or transformation (such as translation, scaling, rotation or reflection) that maps the figure/object onto itself (i.e., the object has an invariance under the transform). [1]
In mathematics, a symmetry operation is a geometric transformation of an object that leaves the object looking the same after it has been carried out. For example, a 1 ⁄ 3 turn rotation of a regular triangle about its center, a reflection of a square across its diagonal, a translation of the Euclidean plane, or a point reflection of a sphere through its center are all symmetry operations.
A pentagonal bipyramid and the Schoenflies notation that defines its symmetry: D 5h (a vertical quintuple axis of symmetry and a plane of horizontal symmetry equidistant from the two vertices) The Schoenflies (or Schönflies ) notation , named after the German mathematician Arthur Moritz Schoenflies , is a notation primarily used to specify ...
Any plane going through P, normal to the horizontal plane is a vertical plane at P. Through any point P, there is one and only one horizontal plane but a multiplicity of vertical planes. This is a new feature that emerges in three dimensions. The symmetry that exists in the two-dimensional case no longer holds.