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The two groups are obtained from it by changing 2-fold rotational symmetry to 4-fold, and adding 5-fold symmetry, respectively. There are two crystallographic point groups with the property that no crystallographic point group has it as proper subgroup: O h and D 6h. Their maximal common subgroups, depending on orientation, are D 3d and D 2h.
A crystal may have zero, one, or multiple axes of symmetry but, by the crystallographic restriction theorem, the order of rotation may only be 2-fold, 3-fold, 4-fold, or 6-fold for each axis. An exception is made for quasicrystals which may have other orders of rotation, for example 5-fold. An axis of symmetry is also known as a proper rotation.
3×4-fold, 4×3-fold, and 6×2-fold axes: the rotation group O of order 24 of a cube and a regular octahedron. The group is isomorphic to symmetric group S 4. 6×5-fold, 10×3-fold, and 15×2-fold axes: the rotation group I of order 60 of a dodecahedron and an icosahedron. The group is isomorphic to alternating group A 5.
Each colored cube has two opposite vertices on a 3-fold symmetry axis, which are shared with the black cube. (In the picture both 3-fold vertices of the green cube are visible.) The remaining six vertices of each colored cube correspond to the faces of the black cube. This compound shares these properties with the compound of five cubes ...
These axes are arranged as 3-fold axes in a cube, directed along its four space diagonals (the cube has 4 / m 3 2 / m symmetry). These symbols are constructed the following way: First position – symmetrically equivalent directions of the coordinate axes x, y, and z. They are equivalent due to the presence of diagonal 3-fold ...
We now confine our attention to the plane in which the symmetry acts (Scherrer 1946), illustrated with lattice vectors in the figure. Lattices restrict polygons Compatible: 6-fold (3-fold), 4-fold (2-fold) Incompatible: 8-fold, 5-fold. Now consider an 8-fold rotation, and the displacement vectors between adjacent points of the polygon.
Views from 2-fold, 5-fold and 3-fold symmetry axis If the shape is considered as a union of five cubes yielding a simple nonconvex solid without self-intersecting surfaces, then it has 360 faces (all triangles ), 182 vertices (60 with degree 3, 30 with degree 4, 12 with degree 5, 60 with degree 8, and 20 with degree 12), and 540 edges, yielding ...
In the figure at right, the (001) plane has a 3-fold symmetry: it remains unchanged by a rotation of 1/3 (2 π /3 rad, 120°). The [100], [010] and the [ 1 1 0] directions are really similar. If S is the intercept of the plane with the [ 1 1 0] axis, then