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Point groups are used to describe the symmetries of geometric figures and physical objects such as molecules. Each point group can be represented as sets of orthogonal matrices M that transform point x into point y according to y = Mx. Each element of a point group is either a rotation (determinant of M = 1), or it is a reflection or improper ...
When comparing the symmetry type of two objects, the origin is chosen for each separately, i.e., they need not have the same center. Moreover, two objects are considered to be of the same symmetry type if their symmetry groups are conjugate subgroups of O(3) (two subgroups H 1, H 2 of a group G are conjugate, if there exists g ∈ G such that H 1 = g −1 H 2 g).
The 54 hemisymmorphic space groups contain only axial combination of symmetry elements from the corresponding point groups. Example for point group 4/mmm (): hemisymmorphic space groups contain the axial combination 422, but at least one mirror plane m will be substituted with glide plane, for example P4/mcc (, 35h), P4/nbm (, 36h), P4/nnc ...
The Schoenflies (or Schönflies) notation, named after the German mathematician Arthur Moritz Schoenflies, is a notation primarily used to specify point groups in three dimensions. Because a point group alone is completely adequate to describe the symmetry of a molecule, the notation is often sufficient and commonly used for spectroscopy.
For example, as stated above, a rectangle swept along a straight axis would define a "brick" and the cross section would be straight. Axis: straight vs. curved. Size of cross-section as it is swept along an axis: constant vs. expanding (or contracting) vs. expanding then contracting vs. contracting then expanding.
In geometry, a polar point group is a point group in which there is more than one point that every symmetry operation leaves unmoved. [1] The unmoved points will constitute a line, a plane, or all of space. While the simplest point group, C 1, leaves all points invariant, most polar point groups will move some, but not all points. To describe ...
[2] [page needed] Unlike other subdivisions within biological psychology, the main focus of psychological research is the development of theories that describe brain-behavior relationships. Physiological psychology studies many topics relating to the body's response to a behavior or activity in an organism. [3]
The independent variable of a study often has many levels or different groups. In a true experiment, researchers can have an experimental group, which is where their intervention testing the hypothesis is implemented, and a control group, which has all the same element as the experimental group, without the interventional element.
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