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| 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 [[translation (geometry)|translation]]s, uniform and non-uniform [[scaling (geometry)|scaling]], [[reflection (mathematics)|reflection]]s, [[rotation]]s, [[shear mapping|shears]], and other [[similarity (geometry)|similarities]] and some, but not all [[linear map]]s.
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| All [[triangle]]s 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 [[rectangle]]s and [[rhombus]]es as well as [[square]]s. In fact, affine-regular polygons may be considered a natural generalization of parallelograms.<ref>{{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}}. See in particular p. 249.</ref>
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| Many properties of regular polygons are invariant under affine transformations, and affine-regular polygons share the same properties. For instance,
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| an affine-regular quadrilateral can be [[equidissection|equidissected]] into <math>m</math> equal-area triangles if and only if <math>m</math> is even, by affine invariance of equidissection and [[Monsky's theorem]] on equidissections of squares.<ref>{{Citation
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| | last1 = Monsky | first1 = P.
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| | title = On Dividing a Square into Triangles
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| | journal = The American Mathematical Monthly
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| | volume = 77
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| | issue = 2
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| | pages = 161–164
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| | doi = 10.2307/2317329
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| | year = 1970
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| | mr = 0252233
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| }}.</ref> More generally an <math>n</math>-gon with <math>n > 4</math> may be [[equidissection|equidissected]] into <math>m</math> equal-area triangles if and only if <math>m</math> is a multiple of <math>n</math>.<ref>{{Citation |last=Kasimatis |first=Elaine A. |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}}.</ref>
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| ==References==
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| {{Reflist}}
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| [[Category:Affine geometry]]
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| [[Category:Polygons]]
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| {{elementary-geometry-stub}}
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