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The axis of symmetry of a two-dimensional figure is a line such that, if a perpendicular is constructed, any two points lying on the perpendicular at equal distances from the axis of symmetry are identical. Another way to think about it is that if the shape were to be folded in half over the axis, the two halves would be identical as mirror ...
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a reflection. That is, a figure which does not change upon undergoing a reflection has reflectional symmetry. In 2-dimensional space, there is a line/axis of symmetry, in 3-dimensional space, there is a plane of symmetry
Axial symmetry is symmetry around an axis; an object is axially symmetric if its appearance is unchanged if rotated around an axis. [1] For example, a baseball bat without trademark or other design, or a plain white tea saucer , looks the same if it is rotated by any angle about the line passing lengthwise through its center, so it is axially ...
It has reflection symmetry with respect to a plane perpendicular to the n-fold rotation axis. C nv, [n], (*nn) of order 2n - pyramidal symmetry or full acro-n-gonal group (abstract group Dih n); in biology C 2v is called biradial symmetry. For n=1 we have again C s (1*). It has vertical mirror planes. This is the symmetry group for a regular n ...
In two dimensions, every figure which possesses an axis of symmetry is achiral, and it can be shown that every bounded achiral figure must have an axis of symmetry. (An axis of symmetry of a figure is a line , such that is invariant under the mapping (,) (,), when is chosen to be the -axis of the coordinate system.)
The set of all possible reflection symmetry lines in the plane is (by projective duality) a two-dimensional space, which they partition into cells within which the pattern of crossings of the polygon with its reflection is fixed, causing the axiality to vary smoothly within each cell. They thus reduce the problem to a numerical computation ...
Example subgroups from a hexagonal dihedral symmetry. D 1 is isomorphic to Z 2, the cyclic group of order 2. D 2 is isomorphic to K 4, the Klein four-group. D 1 and D 2 are exceptional in that: D 1 and D 2 are the only abelian dihedral groups. Otherwise, D n is non-abelian. D n is a subgroup of the symmetric group S n for n ≥ 3.
As an example, the six points (0,0,±1), (0,±1,0), and (±1,0,0) form the vertices of a regular octahedron, with each point opposite in the octahedron to its negation, but this is not flexible. Instead, these same six points can be paired up differently to form a Bricard octahedron, with a diagonal axis of symmetry.