Affine-regular polygon

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

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]

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.[2] More generally an -gon with may be equidissected into equal-area triangles if and only if is a multiple of .[3]

References

  1. {{#invoke:citation/CS1|citation |CitationClass=citation }}. See in particular p. 249.
  2. {{#invoke:citation/CS1|citation |CitationClass=citation }}.
  3. {{#invoke:citation/CS1|citation |CitationClass=citation }}.


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