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All even-sided polygons have two simple reflective forms, one with lines of reflections through vertices, and one through edges. For an arbitrary shape, the axiality of the shape measures how close it is to being bilaterally symmetric. It equals 1 for shapes with reflection symmetry, and between two-thirds and 1 for any convex shape.
Let l, m, n be integers greater than or equal to 2. A triangle group Δ(l,m,n) is a group of motions of the Euclidean plane, the two-dimensional sphere, the real projective plane, or the hyperbolic plane generated by the reflections in the sides of a triangle with angles π/l, π/m and π/n (measured in radians).
C nh, [n +,2], (n*) of order 2n - prismatic symmetry or ortho-n-gonal group (abstract group Z n × Dih 1); for n=1 this is denoted by C s (1*) and called reflection symmetry, also bilateral symmetry. 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 ...
There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry. This article lists the groups by Schoenflies notation , Coxeter notation , [ 1 ] orbifold notation , [ 2 ] and order.
Its symmetry group has four elements, the identity, two reflections and the 180° rotation. It is the dihedral group of order 2, also known as the Klein four-group. X can be oriented in only one way. It has four axes of reflection symmetry, aligned with the gridlines and the diagonals, and rotational symmetry of order 4.
There are symmetry groups on the Euclidean plane constructed from fundamental triangles: (4 4 2), (6 3 2), and (3 3 3). Each is represented by a set of lines of reflection that divide the plane into fundamental triangles. These symmetry groups create 3 regular tilings, and 7 semiregular ones. A number of the semiregular tilings are repeated ...
An exploration of transformation geometry often begins with a study of reflection symmetry as found in daily life. The first real transformation is reflection in a line or reflection against an axis. The composition of two reflections results in a rotation when the lines intersect, or a translation when they are parallel.
For example, the face of a human being has a plane of symmetry down its centre, or a pine cone displays a clear symmetrical spiral pattern. Internal features can also show symmetry, for example the tubes in the human body (responsible for transporting gases, nutrients, and waste products) which are cylindrical and have several planes of symmetry.
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