Symmetry combinations
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This article discusses various symmetry combinations.
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 2n. For n > 2 an example is the n-sided regular polygon and various n-sided star polygons, 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:
- n=2: rectangle, rhombus
- n=5: regular Penrose tiling - infinite tiling, but no translational symmetry
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 with respect to two perpendicular lines of reflection implies rotational symmetry at the point of intersection for an angle of 180°. The symmetry group is the Klein four-group. This applies e.g. for a rectangle, a rhombus, and the letter H.
- Mirror-image symmetry of a square (with a pattern) with respect to the horizontal axis and to one diagonal, implies mirror-image symmetry with respect to the vertical axis and the other diagonal, and 4-fold rotational symmetry.
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 groups.
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 2n-fold rotation-reflection axis, this is Dnd; with only this 2n-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).
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[edit] 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:
- the translation vectors can be chosen to be perpendicular, and the rectangle spanned by these can be positioned with the axes of reflection along two opposite sides and halfway
- the second case concerns wallpaper group cm (and in the case of additional symmetry: cmm); one can choose different representations:
- the translation vectors can be chosen symmetrically with respect to an axis of reflection; then they have equal magnitude and the rhombus spanned by these has axes of reflection along a diagonal and through the other two vertices
- we can also choose one translation vector perpendicular to the axes of reflection, while they cross it at the ends and midway; then the other translation vector can be chosen such that it ends at the axis of reflection crossing the first translation vector midway; in that case the two span a parallelogram with one diagonal having equal length as each of one pair of sides (hence it is composed of two isosceles triangles) with the axes of reflection through all vertices
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one can consider a rectangle with one pair of sides perpendicular to the axes of reflection (while, again, they cross it at the ends and midway) and the other pair of sides parallel to it; in that case the latter are not translation vectors; from a vertex to halfway a side of the first pair is the other translation vector; such a rectangle (in the figure the left and right half of the full rectangle), reproduced by translation, fills the plane and forms a common tiling (see the picture of the brick wall; in relation to the upper image and the description, the image is rotated 90°, and the bricks in the image are horizontally symmetric); due to the symmetry, one half of it is a fundamental domain; this can e.g. be rectangular (one quadrant of the full image at the top, one half of a brick). If the bricks are vertically symmetric the brick's image without rotation represents another correspondence with the upper image, with the brick in the two strips in the center.
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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.
[edit] 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.
[edit] 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.
Only in group p4g (4*2) the properties really change when considering these larger, tilted squares: the lines of reflection, which were in diagonal direction, are horizontal and vertical relative to the larger squares, positioned at 1/4 and 3/4 of the square. The rotational centers midway on the sides of the larger squares are the mirror image of those in the corners and in the center. (The last image shows a version which is shifted 1/4 of the large square.)
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.
