User:Tomruen/aa

Euclidean_group

The number of degrees of freedom for E(n) is n(n + 1)/2, which gives 3 in case n = 2, 6 for n = 3, and 10 for n = 4. Of these, n can be attributed to available translational symmetry, and the remaining n(n − 1)/2 to rotational symmetry.

Overview of isometries in up to three dimensions
E(1), E(2), and E(3) can be categorized as follows, with degrees of freedom:

Chasles' theorem asserts that any element of E+(3) is a screw displacement.

See also 3D isometries that leave the origin fixed, space group, involution.