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In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere.It is a subgroup of the orthogonal group O(3), the group of all isometries that leave the origin fixed, or correspondingly, the group of orthogonal matrices.
The pyritohedral group T h with fundamental domain The seams of a volleyball have pyritohedral symmetry. T h, 3*2, [4,3 +] or m 3, of order 24 – pyritohedral symmetry. [1] This group has the same rotation axes as T, with mirror planes through two of the orthogonal directions. The 3-fold axes are now S 6 (3) axes
Finite spherical symmetry groups are also called point groups in three dimensions. 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.
The reflection point groups, defined by 1 to 3 mirror planes, can also be given by their Coxeter group and related polyhedra. The [3,3] group can be doubled, written as [[3,3]], mapping the first and last mirrors onto each other, doubling the symmetry to 48, and isomorphic to the [4,3] group.
D 2, [2,2] +, (222) of order 4 is one of the three symmetry group types with the Klein four-group as abstract group. It has three perpendicular 2-fold rotation axes. It is the symmetry group of a cuboid with an S written on two opposite faces, in the same orientation. D 2h, [2,2], (*222) of order 8 is the symmetry group of a cuboid.
In three dimensional geometry, there are four infinite series of point groups in three dimensions (n≥1) with n-fold rotational or reflectional symmetry about one axis (by an angle of 360°/n) that does not change the object. They are the finite symmetry groups on a cone. For n = ∞ they correspond to four frieze groups. Schönflies notation ...
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There are 230 space groups in three dimensions, given by a number index, and a full name in Hermann–Mauguin notation, and a short name (international short symbol). The long names are given with spaces for readability. The groups each have a point group of the unit cell.