Affine-regular polygon explained

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 and non-uniform scaling, reflections, rotations, shears, and other similarities and some, but not all linear maps.

Examples

All triangles are affine-regular. In other words, all triangles can be generated by applying affine transformations to an equilateral triangle. A quadrilateral is affine-regular if and only if it is a parallelogram, which includes rectangles and rhombuses as well as squares. In fact, affine-regular polygons may be considered a natural generalization of parallelograms.^[1]

Properties

Many 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

m equal-area triangles if and only if m is even, by affine invariance of equidissection and Monsky's theorem on equidissections of squares.^[2] More generally an n -gon with n > 4 may be equidissected into m equal-area triangles if and only if m is a multiple of n .^[3]

Notes and References

1. . See in particular p. 249.
2. .
3. .