often used in crystallography. In the infinite limit, these groups become the one-dimensional line groups. If a group is a symmetry of a two-dimensional...

In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry...

181440. Point groups in two dimensions Point groups in three dimensions Point groups in four dimensions Crystallography Crystallographic point group Molecular...

In geometry, a point group in four dimensions is an isometry group in four dimensions that leaves the origin fixed, or correspondingly, an isometry group...

reflection groups and analogues of reflection groups over a finite field. In two dimensions, the finite reflection groups are the dihedral groups, which are...

group of Euclidean plane isometries which keep the origin fixed. These groups form one of the two series of discrete point groups in two dimensions....

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...

isometry group is a Lie group. Point group Point groups in two dimensions Point groups in three dimensions Fixed points of isometry groups in Euclidean...

In geometry, dihedral symmetry in three dimensions is one of three infinite sequences of point groups in three dimensions which have a symmetry group...

studied – see point groups in three dimensions, polyhedral groups, and list of spherical symmetry groups. In 2 dimensions, the finite groups are either cyclic...

In mathematics, a rotation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x′y′-Cartesian coordinate system in which...

distinct. Space groups are discrete cocompact groups of isometries of an oriented Euclidean space in any number of dimensions. In dimensions other than 3...

line fixed. Point The point groups in two dimensions with respect to any point leave that point fixed. Space Only the trivial isometry group C1 leaves the...

matrix : In mathematics, a rotation of axes in two dimensions is a mapping from an xy-Cartesian coordinate system to an x′y′-Cartesian coordinate system in which...

Finite spherical symmetry groups are also called point groups in three dimensions. There are five fundamental symmetry classes which have triangular fundamental...

and in the case of the dihedral groups, one more for the positions of the mirrors. The remaining isometry groups in two dimensions with a fixed point are:...

system Frieze group Isometry Lattice Point group Point groups in two dimensions Point groups in three dimensions Space group Symmetry group Translational...

mirror planes remain unchanged. Molecular symmetry Point group Space group Point groups in three dimensions Crystal system Authier, André (2015). "12. The...

two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in...

reference results in the body being at the same coordinates. For example, in two dimensions rotating a body clockwise about a point keeping the axes fixed...

Schoenflies, is a notation primarily used to specify point groups in three dimensions. Because a point group alone is completely adequate to describe the symmetry...

finite, and in this case the two dimensions coincide. Classical physics theories describe three physical dimensions: from a particular point in space, the...

respect to a circle. In two dimensions, a point reflection is the same as a rotation of 180 degrees. In three dimensions, a point reflection can be described...

reflection, the symmetry group is isomorphic with Cs (see point groups in three dimensions for more), one of the three types of order two (involutions), hence...

indirect isometry. The resulting chirality definition works in spacetime. In two dimensions, every figure which possesses an axis of symmetry is achiral...

perpendicular unit vector. In two dimensions, angular velocity is a number with plus or minus sign indicating orientation, but not pointing in a direction. The...

anticonformal if at every point the Jacobian is a scalar times an orthogonal matrix with negative determinant: in two dimensions the Jacobian must be a scalar...

categorized based on what point group or translational symmetry applies to them. In two dimensions, the most basic point group corresponds to rotational...

of simple groups*; they provide a linear representation of the group of rotations in a space with any number n {\displaystyle n} of dimensions, each spinor...

tetragonal Bravais lattice in two dimensions: the square lattice. Bravais lattices Crystal system Crystal structure Point groups Cubic-to-Tetragonal Transition...