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In 2D, mirror-image symmetry in combination with ''n''-fold rotational symmetry, with the center of rotational symmetry on the line of symmetry, implies mirror-image symmetry with respect to lines of reflection rotated by multiples of 180°/''n'', i.e. n reflection lines which are radially spaced evenly (for odd n this already follows from applying the rotational symmetry to a single reflection axis, but it also holds for even n). The symmetry group is the Dihedral Group of order 2''n''. For ''n'' > 2 an example is the ''n''-sided Regular Polygon and various ''n''-sided Star Polygon s, including complex ones, which are a combination of simple ones for a divisor of ''n''; also we have the simple "star" of ''n'' radial line segments (for even ''n'' this is a degenerate star polygon, for odd ''n'' it is not). Also multiple regular ''n''-sided polygons with common center, differing by arbitrary rotations, as long as these rotation angles have mirror-image symmetry, e.g. two squares differing by a rotation angle of 10°, or three squares differing by two successive rotation angles of 10°. Other examples:
Conversely, mirror-image symmetry with respect to two lines of reflection at an angle of 180°/''n'' implies n-fold rotational symmetry ( Kaleidoscope effect). In particular:
Mirror-image symmetry in combination with 2-fold rotational symmetry, with the point of symmetry ''not'' on the line of symmetry, implies an infinite sequence of alternating centers of symmetry and parallel lines of reflection, evenly spaced, with all these centers on a line perpendicular to the lines of reflection (the lines of reflection are the perpendicular bisectors of the line segments between adjacent copies of the points of symmetry). It also implies translational symmetry with as translation vector twice the difference in position between adjacent centers. This is Frieze Group nr. 6. Translational Symmetry can only be combined with 2-, 3-, 4-, and 6-fold rotational symmetry (angles of 180°, 120°, 90°, and 60°), see Crystallographic Restriction Theorem . In these cases the translational symmetry applies along lines in 1, 3, 2, and 3 directions, respectively. This applies for 13 of the 17 Wallpaper Group s. In the case of translational symmetry combined with 2-fold rotational symmetry, other centers of this symmetry can be found by translations by half the distances (the linear or 2D grid of rotocenters is twice as dense in each dimension as that of replicas of any given point by translation). ''n''-fold rotational symmetry with respect to two points of rotation implies translational symmetry. In 3D we can distinguish a plane of reflection through the axis of rotational symmetry, and hence ''n'' of them, similar to the 2D case (in Schoenflies Notation ''Cnv''), perpendicular to it ('''''Cnh'''''), and both ('''''Dnh'''''). In the latter case there are ''n'' perpendicular 2-fold rotation axes in the ''n'' planes of reflection. If, instead, the 2-fold rotation axes are in between the planes of reflection, hence we have a 2''n''-fold Rotation-reflection Axis , this is '''''Dnd'''''; with only this 2''n''-fold Rotation-reflection Axis we have '''''S2n'''''. Also there may be no plane of reflection, but just an additional, perpendicular 2-fold axis of rotation, and hence ''n'' of them (''Dn''). See also Chirality , Chirality (mathematics) , and Rigid Body . Mirror-image symmetry in combination with translational symmetry Mirror-image symmetry in combination with translational symmetry, with the translational vector not along the line or plane of reflection, implies that there are infinitely many parallel lines or planes of reflection, with a spacing such that one half of the translational vector, starting at one, ends at the next. In 2D, with translation in one direction, this is freeze group 4, or in the case of additional symmetry, 6 or 7. In 2D, with translation in two directions there are two cases:
. Group ''cm'' can also be described as a rectangular Checkerboard pattern, where the pattern of each of the two tiles is symmetric in, say, the horizontal direction, or looking at it differently (by shifting half a tile) a checkerboard pattern where the two tiles are each other's mirror image. With additional reflection axes perpendicular to the other ones, we have ''cmm''; in the case of the bricks this corresponds to homogeneous bricks, or, more generally, double symmetric ones. Group ''cmm'' can be described as a checkerboard pattern of 2-fold rotational tiles and their mirror image, or looking at it differently (by shifting half a tile in both directions) a checkerboard pattern of two horizontally and vertically symmetric tiles. Rotational symmetry of order 3 and/or 6 in combination with translational symmetry Of course rotational symmetry of order 3 or more in combination with translational symmetry implies translational symmetry in two directions. Rotational symmetry of order 3 at one center of rotation and order 2 at another one implies rotational symmetry of order 6 at some center of rotation. In the case of rotational symmetry of order 6, the centers of rotation of order 3 are arranged in a Honeycomb structure, and the centers of rotation of order 2 in little triangles around them, touching each other, and also forming hexagons, rotated 30° and a little smaller. See also Hexagonal Lattice . Rotational symmetry of order 4 in combination with translational symmetry ]] ]] ]] Of course rotational symmetry of order 4 in combination with translational symmetry implies translational symmetry in two directions. There are two different rotational centers of order 4, each in an upright square lattice, and together in a denser diagonal square lattice (orientations are expressed relative to the translational cells), each as many as there are translational cells. Also there is one kind of rotational center of order 2, there are as many of them as the other two together. In the figures the two kinds of rotational centers of order 4 are distinguished by color (red and green), except in p4g, where the two kinds are each other's mirror image, both shown in green. There is, of course, also translational symmetry with translations √2 times as large as the minimum, diagonally. Therefore the symmetries mentioned in the previous paragraph also apply in these larger translational squares. The two rotational centers of order 4 mentioned there are of the same kind in the larger squares, and the rotational centers midway on the sides are also of order 4.
In p4g there is a Checkerboard pattern of 4-fold rotational tiles and their mirror image, or looking at it differently (by shifting half a tile) a checkerboard pattern of horizontally and vertically symmetric tiles and their 90° rotated version. Note that neither applies for a plain checkerboard pattern of black and white tiles, this is group p4m (with diagonal translation cells). See also Square Lattice . SEE ALSO |
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