Affine-regular polygon properties of regular polygons are invariant under affine transformations, and affine-regular polygons share the same properties. For instance, an affine-regular quadrilateral can be equidissected into formula_1 equal-area triangles if and only if formula_1 is even, by affine invariance of equidissection and Monsky's theorem on equidissections of squares. More generally an formula_3-gon with formula_4 may be equidissected into formula_1 equal-area triangles if and only if formula_1 is a multiple of formula_3. Affine-regular polygon In geometry, an affine-regular polygon or affinely regular polygon is a polygon that is related to a regular polygon by an affine transformation. Affine transformations include translations, uniform
Geometry theory, lending credibility or possibly disproving string theory. The theme of symmetry in geometry is nearly as old as the science of geometry itself. Symmetric shapes such as the circle, regular polygons and platonic solids held deep significance for many ancient philosophers and were investigated in detail before the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in a multitude of forms, including the graphics of M.C. Escher. Nonetheless, it was not until the second half of 19th century that the unifying role of symmetry in foundations of geometry was recognized. Felix Klein's Erlangen program proclaimed