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In geometry, polygons are associated into pairs called duals, where the vertices of one correspond to the edges of the other.
| | This article needs additional citations for verification .(November 2019) |
In geometry, polygons are associated into pairs called duals, where the vertices of one correspond to the edges of the other.
Regular polygons are self-dual.
The dual of an isogonal (vertex-transitive) polygon is an isotoxal (edge-transitive) polygon. For example, the (isogonal) rectangle and (isotoxal) rhombus are duals.
In a cyclic polygon, longer sides correspond to larger exterior angles in the dual (a tangential polygon), and shorter sides to smaller angles.[ citation needed ] 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 is the dual polygon of the corresponding vertex figure.
As an example of the side-angle duality of polygons we compare properties of the cyclic and tangential quadrilaterals. [1]
| Cyclic quadrilateral | Tangential quadrilateral |
|---|---|
| Circumscribed circle | Inscribed circle |
| Perpendicular bisectors of the sides are concurrent at the circumcenter | Angle bisectors are concurrent at the incenter |
| The sums of the two pairs of opposite angles are equal | The sums of the two pairs of opposite sides are equal |
This duality is perhaps even more clear when comparing an isosceles trapezoid to a kite.
| Isosceles trapezoid | Kite |
|---|---|
| Two pairs of equal adjacent angles | Two pairs of equal adjacent sides |
| One pair of equal opposite sides | One pair of equal opposite angles |
| An axis of symmetry through one pair of opposite sides | An axis of symmetry through one pair of opposite angles |
| Circumscribed circle | Inscribed circle |
The simplest qualitative construction of a dual polygon is a rectification 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.
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As with dual polyhedra, one can take a circle (be it the inscribed circle, circumscribed circle, or if both exist, their midcircle) and perform polar reciprocation in it.
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:
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 {A, B, C} and edges {AB, BC, CA}, the dual triangle has vertices {AB, BC, CA}, and edges {B, C, A}, 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.
In geometry, an n-gonal antiprism or n-antiprism is a polyhedron composed of two parallel direct copies of an n-sided polygon, connected by an alternating band of 2n triangles. They are represented by the Conway notation An.
A (symmetric) n-gonal bipyramid or dipyramid is a polyhedron formed by joining an n-gonal pyramid and its mirror image base-to-base. An n-gonal bipyramid has 2n triangle faces, 3n edges, and 2 + n vertices.
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 other. Such dual figures remain combinatorial or abstract polyhedra, but not all can also be constructed as geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron.
In geometry, a polyhedron is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.
In geometry, a hexagon is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
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 ; or a parallelogram containing a right angle. A rectangle with four sides of equal length is a square. The term "oblong" is occasionally used to refer to a non-square rectangle. A rectangle with vertices ABCD would be denoted as ABCD.
In Euclidean geometry, a kite is a quadrilateral with reflection symmetry across a diagonal. Because of this symmetry, a kite has two equal angles and two pairs of adjacent equal-length sides. Kites are also known as deltoids, but the word deltoid may also refer to a deltoid curve, an unrelated geometric object sometimes studied in connection with quadrilaterals. A kite may also be called a dart, particularly if it is not convex.
A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.
In geometry, a polytope or a tiling is isogonal or vertex-transitive if all its vertices are equivalent under the symmetries of the figure. This implies that each vertex is surrounded by the same kinds of face in the same or reverse order, and with the same angles between corresponding faces.
In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.
In geometry, lines in a plane or higher-dimensional space are concurrent if they intersect at a single point. They are in contrast to parallel lines.
In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent.
In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or solid. To say that "figure F is inscribed in figure G" means precisely the same thing as "figure G is circumscribed about figure F". A circle or ellipse inscribed in a convex polygon is tangent to every side or face of the outer figure. A polygon inscribed in a circle, ellipse, or polygon has each vertex on the outer figure; if the outer figure is a polygon or polyhedron, there must be a vertex of the inscribed polygon or polyhedron on each side of the outer figure. An inscribed figure is not necessarily unique in orientation; this can easily be seen, for example, when the given outer figure is a circle, in which case a rotation of an inscribed figure gives another inscribed figure that is congruent to the original one.
In geometry, the midsphere or intersphere of a convex polyhedron is a sphere which is tangent to every edge of the polyhedron. Not every polyhedron has a midsphere, but the uniform polyhedra, including the regular, quasiregular and semiregular polyhedra and their duals all have midspheres. The radius of the midsphere is called the midradius. A polyhedron that has a midsphere is said to be midscribed about this sphere.
In geometry, an antiparallelogram is a type of self-crossing quadrilateral. Like a parallelogram, an antiparallelogram has two opposite pairs of equal-length sides, but these pairs of sides are not in general parallel. Instead, each pair of sides is antiparallel with respect to the other, with sides in the longer pair crossing each other as in a scissors mechanism. Whereas a parallelogram's opposite angles are equal and oriented the same way, an antiparallelogram's are equal but oppositely oriented. Antiparallelograms are also called contraparallelograms or crossed parallelograms.
In geometry, a tessellation of dimension 2 or higher, or a polytope of dimension 3 or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent but must be transitive, i.e. must lie within the same symmetry orbit. In other words, for any two faces A and B, there must be a symmetry of the entire figure by translations, rotations, and/or reflections that maps A onto B. For this reason, convex isohedral polyhedra are the shapes that will make fair dice.
In geometry, a polytope 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 edges, there is a translation, rotation, and/or reflection that will move one edge to the other while leaving the region occupied by the object unchanged.
Jessen's icosahedron, sometimes called Jessen's orthogonal icosahedron, is a non-convex polyhedron with the same numbers of vertices, edges, and faces as the regular icosahedron. It is named for Børge Jessen, who studied it in 1967. In 1971, a family of nonconvex polyhedra including this shape was independently discovered and studied by Adrien Douady under the name six-beakedshaddock; later authors have applied variants of this name more specifically to Jessen's icosahedron.