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2. Dihedral group of order 8 - Wikipedia

   en.wikipedia.org/wiki/Dihedral_group_of_order_8

   Applying two symmetry transformations in succession yields a symmetry transformation. For instance a ∘ a, also written as a 2, is a 180° degree turn. a 3 is a 270° clockwise rotation (or a 90° counter-clockwise rotation). We also see that b 2 = e and also a 4 = e. A horizontal flip followed by a rotation, a ∘ b is the same as b ∘ a 3.

3. Symmetry (physics) - Wikipedia

   en.wikipedia.org/wiki/Symmetry_(physics)

   The symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation. A family of particular transformations may be continuous (such as rotation of a circle) or discrete (e.g., reflection of a bilaterally symmetric figure, or rotation of a ...

4. List of character tables for chemically important 3D point groups

   en.wikipedia.org/wiki/List_of_character_tables...

   The two rightmost columns indicate which irreducible representations describe the symmetry transformations of the three Cartesian coordinates (x, y and z), rotations about those three coordinates (R x, R y and R z), and functions of the quadratic terms of the coordinates(x 2, y 2, z 2, xy, xz, and yz).

5. Symmetry operation - Wikipedia

   en.wikipedia.org/wiki/Symmetry_operation

   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.

6. Symmetry - Wikipedia

   en.wikipedia.org/wiki/Symmetry

   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]

7. Symmetry group - Wikipedia

   en.wikipedia.org/wiki/Symmetry_group

   In wider contexts, a symmetry group may be any kind of transformation group, or automorphism group. Each type of mathematical structure has invertible mappings which preserve the structure. Conversely, specifying the symmetry group can define the structure, or at least clarify the meaning of geometric congruence or invariance; this is one way ...