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 ...
,
polygons are associated into pairs called duals, where the
vertices of one correspond to the
edges of the other.
Properties
Regular polygons are
self-dual
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of is , then the ...
.
The dual of an
isogonal (vertex-transitive) polygon is an
isotoxal
In geometry, a polytope (for example, a polygon or a polyhedron) or a tiling is isotoxal () or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given ...
(edge-transitive) polygon. For example, the (isogonal)
rectangle and (isotoxal)
rhombus are duals.
In a
cyclic polygon, longer sides correspond to larger
exterior angle
In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) i ...
s in the dual (a
tangential polygon), and shorter sides to smaller angles. Further, congruent sides in the original polygon yields congruent angles in the dual, and conversely. For example, the dual of a highly acute
isosceles triangle is an obtuse isosceles triangle.
In the
Dorman Luke construction, each face of a
dual polyhedron
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the oth ...
is the dual polygon of the corresponding
vertex figure.
Duality in quadrilaterals
As an example of the side-angle duality of polygons we compare properties of the
cyclic
Cycle, cycles, or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in so ...
and
tangential quadrilateral
In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called ...
s.
[Michael de Villiers, ''Some Adventures in Euclidean Geometry'', , 2009, p. 55.]
This duality is perhaps even more clear when comparing an
isosceles trapezoid to a
kite.
Kinds of duality
Rectification
The simplest qualitative construction of a dual polygon is a
rectification
Rectification has the following technical meanings:
Mathematics
* Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points
* Rectifiable curve, in mathematics
* Recti ...
operation, where the edges of a polygon are
truncated down to vertices at the center of each original edge. New edges are formed between these new vertices.
This construction is not reversible. That is, the polygon generated by applying it twice is in general not similar to the original polygon.
Polar reciprocation
As with dual polyhedra, one can take a circle (be it the
inscribed circle
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is a triangle center called the triangle's incen ...
,
circumscribed circle, or if both exist, their
midcircle
In inversive geometry, the circle of antisimilitude (also known as mid-circle) of two circles, ''α'' and ''β'', is a reference circle for which ''α'' and ''β'' are inverses of each other. If ''α'' and ''β'' are non-intersecting or tangen ...
) and perform
polar reciprocation in it.
Projective duality
Under
projective duality
In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and (plane) duality is the formalization of this concept. There are two approaches to the subject of ...
, the dual of a point is a line, and of a line is a point – thus the dual of a polygon is a polygon, with edges of the original corresponding to vertices of the dual and conversely.
From the point of view of the
dual curve, where to each point on a curve one associates the point dual to its tangent line at that point, the projective dual can be interpreted thus:
* every point on a side of a polygon has the same tangent line, which agrees with the side itself – they thus all map to the same vertex in the dual polygon
* at a vertex, the "tangent lines" to that vertex are all lines through that point with angle between the two edges – the dual points to these lines are then the edge in the dual polygon.
Combinatorially
Combinatorially, one can define a polygon as a set of vertices, a set of edges, and an incidence relation (which vertices and edges touch): two adjacent vertices determine an edge, and dually, two adjacent edges determine a vertex. Then the dual polygon is obtained by simply switching the vertices and edges.
Thus for the triangle with vertices and edges , the dual triangle has vertices , and edges , where B connects AB & BC, and so forth.
This is not a particularly fruitful avenue, as combinatorially, there is a single family of polygons (given by number of sides); geometric duality of polygons is more varied, as are combinatorial
dual polyhedra.
See also
*
Dual curve
*
Dual polyhedron
In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the oth ...
*
Self-dual polygon
In Euclidean geometry, a regular polygon is a polygon that is direct equiangular (all angles are equal in measure) and equilateral (all sides have the same length). Regular polygons may be either convex, star or skew. In the limit, a sequence ...
References
External links
Dual Polygon Appletby
Don Hatch
{{DEFAULTSORT:Dual Polygon
Polygons