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 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.{{Citation |first=H. S. M. |last=Coxeter |authorlink=Harold Scott MacDonald Coxeter |date=December 1992 |title=Affine regularity |journal=Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg |volume=62 |issue=1 |pages=249–253 |doi=10.1007/BF02941630|doi-access=|s2cid=186234003 }}. See in particular p. 249.
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 equal-area triangles if and only if is even, by affine invariance of equidissection and Monsky's theorem on equidissections of squares.{{Citation
| last1 = Monsky | first1 = P.
| title = On Dividing a Square into Triangles
| journal = The American Mathematical Monthly
| volume = 77
| issue = 2
| pages = 161–164
| doi = 10.2307/2317329
| year = 1970
| jstor = 2317329
| mr = 0252233
}}. More generally an -gon with may be equidissected into equal-area triangles if and only if is a multiple of .{{Citation |last=Kasimatis |first=Elaine A.|author-link= Elaine Kasimatis |date=December 1989 |title=Dissections of regular polygons into triangles of equal areas |journal=Discrete & Computational Geometry |volume=4 |issue=1 |pages=375–381 |doi=10.1007/BF02187738 |zbl=0675.52005 |url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN000364096|doi-access=free }}.