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Benzene is a centrosymmetric molecule having a centre of symmetry at the centre. In crystallography, a centrosymmetric point group contains an inversion center as one of its symmetry elements. [1] In such a point group, for every point (x, y, z) in the unit cell there is an indistinguishable point (-x, -y, -z).
In a symmetry group, the group elements are the symmetry operations (not the symmetry elements), and the binary combination consists of applying first one symmetry operation and then the other. An example is the sequence of a C 4 rotation about the z-axis and a reflection in the xy-plane, denoted σ(xy)C 4. By convention the order of operations ...
For objects with several symmetries, the centre of symmetry is the point left unchanged by the symmetric actions. So the centre of a square, rectangle, rhombus or parallelogram is where the diagonals intersect, this is (among other properties) the fixed point of rotational symmetries.
The centroid and centre of mass, if defined, are this point. Another meaning of "centre of symmetry" is a point with respect to which inversion symmetry applies. Such a point needs not be unique; if it is not, there is translational symmetry, hence there are infinitely many of such points. On the other hand, in the cases of e.g. C 3h and D 2 ...
Cayley table for D 4 showing elements of the center, {e, a 2}, commute with all other elements (this can be seen by noticing that all occurrences of a given center element are arranged symmetrically about the center diagonal or by noticing that the row and column starting with a given center element are transposes of each other).
In a tetrahedral molecular geometry, a central atom is located at the center with four substituents that are located at the corners of a tetrahedron.The bond angles are arccos(− 1 / 3 ) = 109.4712206...° ≈ 109.5° when all four substituents are the same, as in methane (CH 4) [1] [2] as well as its heavier analogues.
The centroid of a ring or a bowl, for example, lies in the object's central void. If the centroid is defined, it is a fixed point of all isometries in its symmetry group. In particular, the geometric centroid of an object lies in the intersection of all its hyperplanes of symmetry.
Symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations. [17] This concept has become one of the most powerful tools of theoretical physics , as it has become evident that practically all laws of nature originate in symmetries.