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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. Such a transformation is an invertible mapping of the ambient space which takes the object to itself, and which preserves all the relevant structure of the object.
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.
The symmetry group of an n-sided regular polygon is the dihedral group D n (of order 2n): D 2, D 3, D 4, ... It consists of the rotations in C n, together with reflection symmetry in n axes that pass through the center. If n is even then half of these axes pass through two opposite vertices, and the other half through the midpoint of opposite ...
A drawing of a butterfly with bilateral symmetry, with left and right sides as mirror images of each other.. In geometry, an object has symmetry if there is an operation or transformation (such as translation, scaling, rotation or reflection) that maps the figure/object onto itself (i.e., the object has an invariance under the transform). [1]
The group order is defined as the subscript, unless the order is doubled for symbols with a plus or minus, "±", prefix, which implies a central inversion. [ 3 ] Hermann–Mauguin notation (International notation) is also given.
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry.In particular, all its elements or j-faces (for all 0 ≤ j ≤ n, where n is the dimension of the polytope) — cells, faces and so on — are also transitive on the symmetries of the polytope, and are themselves regular polytopes of dimension j≤ n.
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).
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags.A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive.