Affine transformations include translations, uniform and non-uniform scaling, reflections, rotations, shears, and other similarities and some, but not all linear maps.
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] 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