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In geometry, circular symmetry is a type of continuous symmetry for a planar object that can be rotated by any arbitrary angle and map onto itself. Rotational circular symmetry is isomorphic with the circle group in the complex plane , or the special orthogonal group SO(2), and unitary group U(1).
Thus, a symmetry can be thought of as an immunity to change. [2] For instance, a circle rotated about its center will have the same shape and size as the original circle, as all points before and after the transform would be indistinguishable. A circle is thus said to be symmetric under rotation or to have rotational symmetry.
The circle is a highly symmetric shape: every line through the centre forms a line of reflection symmetry, and it has rotational symmetry around the centre for every angle. Its symmetry group is the orthogonal group O(2, R ).
There is no geometric figure that has as full symmetry group the circle group, but for a vector field it may apply (see the three-dimensional case below). the orthogonal group O(2) consisting of all rotations about a fixed point and reflections in any axis through that fixed point. This is the symmetry group of a circle.
This article describes the mathematics of the Standard Model of particle physics, a gauge quantum field theory containing the internal symmetries of the unitary product group SU(3) × SU(2) × U(1). The theory is commonly viewed as describing the fundamental set of particles – the leptons, quarks, gauge bosons and the Higgs boson.
A blue horocycle in the Poincaré disk model and some red normals. The normals converge asymptotically to the upper central ideal point.. In hyperbolic geometry, a horocycle (from Greek roots meaning "boundary circle"), sometimes called an oricycle or limit circle, is a curve of constant curvature where all the perpendicular geodesics through a point on a horocycle are limiting parallel, and ...
[116] [h] In music theory, tones that differ by an octave are generally considered to be equivalent notes, and the space of possible notes forms a circle, the chromatic circle. Because the Möbius strip is the configuration space of two unordered points on a circle, the space of all two-note chords takes the shape of a
P ' is the inverse of P with respect to the circle. To invert a number in arithmetic usually means to take its reciprocal. A closely related idea in geometry is that of "inverting" a point. In the plane, the inverse of a point P with respect to a reference circle (Ø) with center O and radius r is a point P ', lying on the ray from O through P ...