In
geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
, an affine-regular polygon or affinely regular polygon is a
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed ''polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two toge ...
that is related to a
regular polygon by an
affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generally, ...
. Affine transformations include
translation
Translation is the communication of the Meaning (linguistic), meaning of a #Source and target languages, source-language text by means of an Dynamic and formal equivalence, equivalent #Source and target languages, target-language text. The ...
s, uniform and non-uniform
scaling
Scaling may refer to:
Science and technology
Mathematics and physics
* Scaling (geometry), a linear transformation that enlarges or diminishes objects
* Scale invariance, a feature of objects or laws that do not change if scales of length, energ ...
,
reflections,
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s,
shears
Shears may refer to:
Cutting devices
* Scissors, also called shears
* Hair-cutting shears
* Blade shears, typically used for shearing animals
* Grass shears, for lawn trimming
* Kitchen shears, scissors used in the kitchen for food preparation
* ...
, and other
similarities and some, but not all
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
s.
Examples
All
triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ...
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
In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equa ...
, which includes
rectangle
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°); or a parallelogram containi ...
s and
rhombuses as well as
squares. In fact, affine-regular polygons may be considered a natural generalization of parallelograms.
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
In geometry, Monsky's theorem states that it is not possible to dissect a square into an odd number of triangles of equal area. In other words, a square does not have an odd equidissection.
The problem was posed by Fred Richman in the '' American ...
on equidissections of squares. More generally an
-gon with
may be
equidissected into
equal-area triangles if and only if
is a multiple of
.
[.]
References
{{Reflist
Affine geometry
Types of polygons