Rational Symmetry

Rational symmetry is general solution of the system corresponding to its Janet base is rational. In the former case, there is only a two parameter group of rational generators, the remaining ones require a larger function field. As it has been explained, the largest rational symmetry group of an ode, in particular if the symmetry group is the projective one.

Brief Explanation of Rational Theory

Formally, rational symmetry is symmetry with respect to some or all rotations in m-dimensional Euclidean space. Rotations are direct isometrics that is isometrics preserving orientation. Therefore a symmetry group of rational symmetry is a subgroup of E+ (m) (see Euclidean group).
 Symmetry with respect to all this rotations of all points which mean translational symmetry with respect to all translations, so space is identical, and the symmetry group is the whole E(m). With the modified this notion of symmetry for vector fields of the symmetry group can also be E+(m).
 With respect to rotations of the point we can take that point as origin. These rotations form the special orthogonal group SO(m) and the group of m×m orthogonal matrices with determinant 1. For m=3 this is the regular changing group.

Rational Theory Example

Examples without extra reflection symmetry:
  * n = 2, 180°: the dyad, quadrilaterals by this symmetry are the parallelograms; extra examples: letters Z, N, S; apart from the colors: yin and yang

•  n = 3, 120°: triad, triskelion, Borromean rings; sometimes the term trilateral regularly is used;
•  Here n = 4, 90°: tetrad, swastika
•  n = 6, 60°: hexad, raelian symbol, new and original version
•  n = 8, 45°: octad, Octagonal muqarnas, computer-generated (CG), and ceiling

My forthcoming post is on Sinusoidal Function Examples and 6th grade math problems and answers will give you more understanding about Algebra.

Summary of Rational symmetry

Finally Rational symmetry at one point and 2-fold at another one (or ditto in 3D with respect to parallel axes) implies rotation group p6, that is double translational symmetry and 6-fold rational symmetry at some point (or, in 3D, parallel axis). The translation distance of this rational symmetry generated by one such pair of rotocenters is 2√3 times their distance.