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 ...

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 t ...

s are associated into pairs called duals, where the vertices of one correspond to the edges of the other.

Properties

Regular 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 ...

s are self-dual. 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 two ...

(edge-transitive) polygon. For example, the (isogonal) rectangle and (isotoxal)

rhombus In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. Th ...

are duals. In a

cyclic polygon In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every polyg ...

, longer sides correspond to larger exterior angles 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 In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...

is an obtuse isosceles triangle. In the

Dorman Luke construction 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 ...

, 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 othe ...

is the dual polygon of the corresponding

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines ...

Duality in quadrilaterals

As an example of the side-angle duality of polygons we compare properties of the cyclic and tangential quadrilaterals.Michael de Villiers, ''Some Adventures in Euclidean Geometry'', , 2009, p. 55. This duality is perhaps even more clear when comparing an

isosceles trapezoid In Euclidean geometry, an isosceles trapezoid (isosceles trapezium in British English) is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides. It is a special case of a trapezoid. Alternatively, it can be defi ...

to a

kite A kite is a tethered heavier than air flight, heavier-than-air or lighter-than-air craft with wing surfaces that react against the air to create Lift (force), lift and Drag (physics), drag forces. A kite consists of wings, tethers and anchors. ...

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,

circumscribed circle In geometry, the circumscribed circle or circumcircle of a polygon is a circle that passes through all the vertices of the polygon. The center of this circle is called the circumcenter and its radius is called the circumradius. Not every poly ...

, or if both exist, their midcircle) and perform polar reciprocation in it.

Projective duality

Under projective duality, 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 Don, don or DON and variants may refer to: Places *County Donegal, Ireland, Chapman code DON *Don (river), a river in European Russia *Don River (disambiguation), several other rivers with the name *Don, Benin, a town in Benin *Don, Dang, a vill ...

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