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Noting that any identity matrix is a rotation matrix, and that matrix multiplication is associative, we may summarize all these properties by saying that the n × n rotation matrices form a group, which for n > 2 is non-abelian, called a special orthogonal group, and denoted by SO(n), SO(n,R), SO n, or SO n (R), the group of n × n rotation ...
4 × rotation by 120° clockwise (seen from a vertex): (234), (143), (412), (321) 4 × rotation by 120° counterclockwise (ditto) 3 × rotation by 180° The rotations by 180°, together with the identity, form a normal subgroup of type Dih 2, with quotient group of type Z 3. The three elements of the latter are the identity, "clockwise rotation ...
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.
The twelve rotations form the rotation (symmetry) group of the figure. In group theory , the symmetry group of a geometric object is the group of all transformations under which the object is invariant , endowed with the group operation of composition .
Rotation can have a sign (as in the sign of an angle): a clockwise rotation is a negative magnitude so a counterclockwise turn has a positive magnitude. A rotation is different from other types of motions: translations , which have no fixed points, and (hyperplane) reflections , each of them having an entire ( n − 1) -dimensional flat of ...
The angle θ and axis unit vector e define a rotation, concisely represented by the rotation vector θe.. In mathematics, the axis–angle representation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector e indicating the direction of an axis of rotation, and an angle of rotation θ describing the magnitude and sense (e.g., clockwise) of the ...
Rotational symmetry of order n, also called n-fold rotational symmetry, or discrete rotational symmetry of the n th order, with respect to a particular point (in 2D) or axis (in 3D) means that rotation by an angle of (180°, 120°, 90°, 72°, 60°, 51 3 ⁄ 7 °, etc.) does not change the object. A "1-fold" symmetry is no symmetry (all ...
The symmetry group of a snowflake is D 6, a dihedral symmetry, the same as for a regular hexagon.. In mathematics, a dihedral group is the group of symmetries of a regular polygon, [1] [2] which includes rotations and reflections.