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.

References

External links

Dual Polygon Applet
by Don Hatch {{DEFAULTSORT:Dual Polygon Polygons