Kite (geometry)

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Kite
GeometricKite.svg
A kite, showing its pairs of equal-length sides and its inscribed circle.
Type Quadrilateral
Edges and vertices 4
Symmetry group D1 (*)

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, [1] but the word deltoid may also refer to a deltoid curve, an unrelated geometric object sometimes studied in connection with quadrilaterals. [2] [3] A kite may also be called a dart, [4] particularly if it is not convex. [5] [6]

Contents

Every kite is an orthodiagonal quadrilateral (its diagonals are at right angles) and, when convex, a tangential quadrilateral (its sides are tangent to an inscribed circle). The convex kites are exactly the quadrilaterals that are both orthodiagonal and tangential. They include as special cases the right kites, with two opposite right angles; the rhombi, with two diagonal axes of symmetry; and the squares, which are also special cases of both right kites and rhombi.

The quadrilateral with the greatest ratio of perimeter to diameter is a kite, with 60°, 75°, and 150° angles. Kites of two shapes (one convex and one non-convex) form the prototiles of one of the forms of the Penrose tiling. Kites also form the faces of several face-symmetric polyhedra and tessellations, and have been studied in connection with outer billiards, a problem in the advanced mathematics of dynamical systems.

Definition and classification

Convex and concave kites Deltoid.svg
Convex and concave kites

A kite is a quadrilateral with reflection symmetry across one of its diagonals. Equivalently, it is a quadrilateral whose four sides can be grouped into two pairs of adjacent equal-length sides. [1] [7] A kite can be constructed from the centers and crossing points of any two intersecting circles. [8] Kites as described here may be either convex or concave, although some sources restrict kite to mean only convex kites. A quadrilateral is a kite if and only if any one of the following conditions is true:

Kite quadrilaterals are named for the wind-blown, flying kites, which often have this shape [10] [11] and which are in turn named for a hovering bird and the sound it makes. [12] [13] According to Olaus Henrici, the name "kite" was given to these shapes by James Joseph Sylvester. [14]

Quadrilaterals can be classified hierarchically, meaning that some classes of quadrilaterals include other classes, or partitionally, meaning that each quadrilateral is in only one class. Classified hierarchically, kites include the rhombi (quadrilaterals with four equal sides) and squares. All equilateral kites are rhombi, and all equiangular kites are squares. When classified partitionally, rhombi and squares would not be kites, because they belong to a different class of quadrilaterals; similarly, the right kites discussed below would not be kites. The remainder of this article follows a hierarchical classification; rhombi, squares, and right kites are all considered kites. By avoiding the need to consider special cases, this classification can simplify some facts about kites. [15]

Like kites, a parallelogram also has two pairs of equal-length sides, but they are opposite to each other rather than adjacent. Any non-self-crossing quadrilateral that has an axis of symmetry must be either a kite, with a diagonal axis of symmetry; or an isosceles trapezoid, with an axis of symmetry through the midpoints of two sides. These include as special cases the rhombus and the rectangle respectively, and the square, which is a special case of both. [1] The self-crossing quadrilaterals include another class of symmetric quadrilaterals, the antiparallelograms. [16]

Special cases

Bicentric kite 001.svg
Right kite
Reuleaux kite.svg
Equidiagonal kite in a Reuleaux triangle

The right kites have two opposite right angles. [15] [16] The right kites are exactly the kites that are cyclic quadrilaterals, meaning that there is a circle that passes through all their vertices. [17] The cyclic quadrilaterals may equivalently defined as the quadrilaterals in which two opposite angles are supplementary (they add to 180°); if one pair is supplementary the other is as well. [9] Therefore, the right kites are the kites with two opposite supplementary angles, for either of the two opposite pairs of angles. Because right kites circumscribe one circle and are inscribed in another circle, they are bicentric quadrilaterals (actually tricentric, as they also have a third circle externally tangent to the extensions of their sides). [16] If the sizes of an inscribed and a circumscribed circle are fixed, the right kite has the largest area of any quadrilateral trapped between them. [18]

Among all quadrilaterals, the shape that has the greatest ratio of its perimeter to its diameter is an equidiagonal kite with angles 60°, 75°, 150°, 75°. Its four vertices lie at the three corners and one of the side midpoints of the Reuleaux triangle. [19] [20] When an equidiagonal kite has side lengths less than or equal to its diagonals, like this one or the square, it is one of the quadrilaterals with the greatest ratio of area to diameter. [21]

A kite with three 108° angles and one 36° angle forms the convex hull of the lute of Pythagoras, a fractal made of nested pentagrams. [22] The four sides of this kite lie on four of the sides of a regular pentagon, with a golden triangle glued onto the fifth side. [16]

There are only eight polygons that can tile the plane such that reflecting any tile across any one of its edges produces another tile; this arrangement is called an edge tessellation. One of them is a tiling by a right kite, with 60°, 90°, and 120° angles. It produces the deltoidal trihexagonal tiling (see § Tilings and polyhedra). [23]

In non-Euclidean geometry, a kite can have three right angles and one non-right angle, forming a special case of a Lambert quadrilateral. The fourth angle is acute in hyperbolic geometry and obtuse in spherical geometry. [24]

Properties

Diagonals, angles, and area

Every kite is an orthodiagonal quadrilateral, meaning that its two diagonals are at right angles to each other. Moreover, one of the two diagonals (the symmetry axis) is the perpendicular bisector of the other, and is also the angle bisector of the two angles it meets. [1] Because of its symmetry, the other two angles of the kite must be equal. [10] [11] The diagonal symmetry axis of a convex kite divides it into two congruent triangles; the other diagonal divides it into two isosceles triangles. [1]

As is true more generally for any orthodiagonal quadrilateral, the area of a kite may be calculated as half the product of the lengths of the diagonals and : [10]

Alternatively, the area can be calculated by dividing the kite into two congruent triangles and applying the SAS formula for their area. If and are the lengths of two sides of the kite, and is the angle between, then the area is [25]

Inscribed circle

Kite inexcircles.svg
Dart inexcircles.svg
Antipar inexcircles.svg
Two circles tangent to the sides and extended sides of a convex kite (top), non-convex kite (middle), and antiparallelogram (bottom). The four lines through the sides of each quadrilateral are bitangents of the circles.

Every convex kite is also a tangential quadrilateral, a quadrilateral that has an inscribed circle. That is, there exists a circle that is tangent to all four sides. Additionally, if a convex kite is not a rhombus, there is a circle outside the kite that is tangent to the extensions of the four sides; therefore, every convex kite that is not a rhombus is an ex-tangential quadrilateral. The convex kites that are not rhombi are exactly the quadrilaterals that are both tangential and ex-tangential. [16] For every concave kite there exist two circles tangent to two of the sides and the extensions of the other two: one is interior to the kite and touches the two sides opposite from the concave angle, while the other circle is exterior to the kite and touches the kite on the two edges incident to the concave angle. [26]

For a convex kite with diagonal lengths and and side lengths and , the radius of the inscribed circle is

and the radius of the ex-tangential circle is [16]

A tangential quadrilateral is also a kite if and only if any one of the following conditions is true: [27]

If the diagonals in a tangential quadrilateral intersect at , and the incircles of triangles , , , have radii , , , and respectively, then the quadrilateral is a kite if and only if [27]

If the excircles to the same four triangles opposite the vertex have radii , , , and respectively, then the quadrilateral is a kite if and only if [27]

Duality

A kite and its dual isosceles trapezoid Kite isotrap duality.svg
A kite and its dual isosceles trapezoid

Kites and isosceles trapezoids are dual to each other, meaning that there is a correspondence between them that reverses the dimension of their parts, taking vertices to sides and sides to vertices. From any kite, the inscribed circle is tangent to its four sides at the four vertices of an isosceles trapezoid. For any isosceles trapezoid, tangent lines to the circumscribing circle at its four vertices form the four sides of a kite. This correspondence can also be seen as an example of polar reciprocation, a general method for corresponding points with lines and vice versa given a fixed circle. Although they do not touch the circle, the four vertices of the kite are reciprocal in this sense to the four sides of the isosceles trapezoid. [28] The features of kites and isosceles trapezoids that correspond to each other under this duality are compared in the table below. [7]

Isosceles trapezoidKite
Two pairs of equal adjacent anglesTwo pairs of equal adjacent sides
Two equal opposite sidesTwo equal opposite angles
Two opposite sides with a shared perpendicular bisectorTwo opposite angles with a shared angle bisector
An axis of symmetry through two opposite sidesAn axis of symmetry through two opposite angles
Circumscribed circle through all verticesInscribed circle tangent to all sides

Dissection

The equidissection problem concerns the subdivision of polygons into triangles that all have equal areas. In this context, the spectrum of a polygon is the set of numbers such that the polygon has an equidissection into equal-area triangles. Because of its symmetry, the spectrum of a kite contains all even integers. Certain special kites also contain some odd numbers in their spectra. [29] [30]

Every triangle can be subdivided into three right kites meeting at the center of its inscribed circle. More generally, a method based on circle packing can be used to subdivide any polygon with sides into kites, meeting edge-to-edge. [31]

Tilings and polyhedra

AnimSun2k.gif
Recursive construction of the kite and dart Penrose tiling
Fractal Penrose kite rosette.svg
Fractal rosette of Penrose kites

All kites tile the plane by repeated point reflection around the midpoints of their edges, as do more generally all quadrilaterals. [32] Kites and darts with angles 72°, 72°, 72°, 144° and 36°, 72°, 36°, 216°, respectively, form the prototiles of one version of the Penrose tiling, an aperiodic tiling of the plane discovered by mathematical physicist Roger Penrose. [5] When a kite has angles that, at its apex and one side, sum to for some positive integer , then scaled copies of that kite can be used to tile the plane in a fractal rosette in which successively larger rings of kites surround a central point. [33] These rosettes can be used to study the phenomenon of inelastic collapse, in which a system of moving particles meeting in inelastic collisions all coalesce at a common point. [34]

A kite with angles 60°, 90°, 120°, 90° can also tile the plane by repeated reflection across its edges; the resulting tessellation, the deltoidal trihexagonal tiling, superposes a tessellation of the plane by regular hexagons and isosceles triangles. [16] The deltoidal icositetrahedron, deltoidal hexecontahedron, and trapezohedron are polyhedra with congruent kite-shaped faces, [35] which can alternatively be thought of as tilings of the sphere by congruent spherical kites. [36] There are infinitely many face-symmetric tilings of the hyperbolic plane by kites. [37] These polyhedra (equivalently, spherical tilings), the square and deltoidal trihexagonal tilings of the Euclidean plane, and some tilings of the hyperbolic plane are shown in the table below, labeled by face configuration (the numbers of neighbors of each of the four vertices of each tile). Some polyhedra and tilings appear twice, under two different face configurations.

PolyhedraEuclidean
Rhombicdodecahedron.jpg
V4.3.4.3
Deltoidalicositetrahedron.jpg
V4.3.4.4
Deltoidalhexecontahedron.jpg
V4.3.4.5
Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg
V4.3.4.6
PolyhedraEuclideanHyperbolic tilings
Deltoidalicositetrahedron.jpg
V4.4.4.3
Square tiling uniform coloring 1.png
V4.4.4.4
H2-5-4-deltoidal.svg
V4.4.4.5
H2chess 246d.png
V4.4.4.6
PolyhedraHyperbolic tilings
Deltoidalhexecontahedron.jpg
V4.3.4.5
H2-5-4-deltoidal.svg
V4.4.4.5
H2-5-4-rhombic.svg
V4.5.4.5
Deltoidal pentahexagonal tiling.png
V4.6.4.5
EuclideanHyperbolic tilings
Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg
V4.3.4.6
H2chess 246d.png
V4.4.4.6
Deltoidal pentahexagonal tiling.png
V4.5.4.6
H2chess 266d.png
V4.6.4.6
Ten-sided dice 2 10-sided dice.jpg
Ten-sided dice

The trapezohedra are another family of polyhedra that have congruent kite-shaped faces. In these polyhedra, the edges of one of the two side lengths of the kite meet at two "pole" vertices, while the edges of the other length form an equatorial zigzag path around the polyhedron. They are the dual polyhedra of the uniform antiprisms. [35] A commonly seen example is the pentagonal trapezohedron, used for ten-sided dice. [16]

Family of n-gonal trapezohedra
NameDigonal trapezohedron
(Tetrahedron)
Trigonal Tetragonal Pentagonal Hexagonal Heptagonal Octagonal ... Apeirogonal
Polyhedron Digonal trapezohedron.png TrigonalTrapezohedron.svg Tetragonal trapezohedron.png Pentagonal trapezohedron.svg Hexagonal trapezohedron.png Heptagonal trapezohedron.png Octagonal trapezohedron.png ...
Tessellation Spherical digonal antiprism.png Spherical trigonal trapezohedron.png Spherical tetragonal trapezohedron.png Spherical pentagonal trapezohedron.png Spherical hexagonal trapezohedron.png Spherical heptagonal trapezohedron.png Spherical octagonal trapezohedron.png ... Apeirogonal trapezohedron.svg
Face configuration V2.3.3.3V3.3.3.3V4.3.3.3V5.3.3.3V6.3.3.3V7.3.3.3V8.3.3.3...V∞.3.3.3

Outer billiards

Mathematician Richard Schwartz has studied outer billiards on kites. Outer billiards is a dynamical system in which, from a point outside a given compact convex set in the plane, one draws a tangent line to the convex set, travels from the starting point along this line to another point equally far from the point of tangency, and then repeats the same process. It had been open since the 1950s whether any system defined in this way could produce paths that get arbitrarily far from their starting point, and in a 2007 paper Schwartz solved this problem by finding unbounded billiards paths for the kite with angles 72°, 72°, 72°, 144°, the same as the one used in the Penrose tiling. [38] He later wrote a monograph analyzing outer billiards for kite shapes more generally. For this problem, any affine transformation of a kite preserves the dynamical properties of outer billiards on it, and it is possible to transform any kite into a shape where three vertices are at the points and , with the fourth at with in the open unit interval . The behavior of outer billiards on any kite depends strongly on the parameter and in particular whether it is rational. For the case of the Penrose kite, , an irrational number, where is the golden ratio. [39]

Related Research Articles

In geometry, a dodecahedron or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three regular star dodecahedra, which are constructed as stellations of the convex form. All of these have icosahedral symmetry, order 120.

<span class="mw-page-title-main">Quadrilateral</span> Polygon with four sides and four corners

In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words quadri, a variant of four, and latus, meaning "side". It is also called a tetragon, derived from greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons. Since "gon" means "angle", it is analogously called a quadrangle, or 4-angle. A quadrilateral with vertices , , and is sometimes denoted as .

<span class="mw-page-title-main">Hexagon</span> Shape with six sides

In geometry, a hexagon is a six-sided polygon or 6-gon creating the outline of a cube. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.

<span class="mw-page-title-main">Rectangle</span> Quadrilateral with four right angles

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.

<span class="mw-page-title-main">Parallelogram</span> Quadrilateral with two pairs of parallel sides

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 equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.

<span class="mw-page-title-main">Rhombus</span> Quadrilateral in which all sides have the same length

In plane Euclidean geometry, a rhombus 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. The rhombus is often called a "diamond", after the diamonds suit in playing cards which resembles the projection of an octahedral diamond, or a lozenge, though the former sometimes refers specifically to a rhombus with a 60° angle, and the latter sometimes refers specifically to a rhombus with a 45° angle.

<span class="mw-page-title-main">Cyclic quadrilateral</span> Quadrilateral whose vertices can all fall on a single circle

In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. The center of the circle and its radius are called the circumcenter and the circumradius respectively. Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.

<span class="mw-page-title-main">Isosceles triangle</span> Triangle with at least two sides congruent

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 version thus including the equilateral triangle as a special case. Examples of isosceles triangles include the isosceles right triangle, the golden triangle, and the faces of bipyramids and certain Catalan solids.

<span class="mw-page-title-main">Rhombic dodecahedron</span> Catalan solid with 12 faces

In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron.

<span class="mw-page-title-main">Square</span> Regular quadrilateral

In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles. It can also be defined as a rectangle with two equal-length adjacent sides. It is the only regular polygon whose internal angle, central angle, and external angle are all equal (90°), and whose diagonals are all equal in length. A square with vertices ABCD would be denoted ABCD.

<span class="mw-page-title-main">Rhombic triacontahedron</span> Catalan solid with 30 faces

In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Catalan solid, and the dual polyhedron of the icosidodecahedron. It is a zonohedron.

<span class="mw-page-title-main">Trigonal trapezohedron</span> Polyhedron with 6 congruent rhombus faces

In geometry, a trigonal trapezohedron is a rhombohedron in which, additionally, all six faces are congruent. Alternative names for the same shape are the trigonal deltohedron or isohedral rhombohedron. Some sources just call them rhombohedra.

<span class="mw-page-title-main">Antiparallelogram</span> Polygon with four crossed edges of two lengths

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, sides in the longer pair cross each other as in a scissors mechanism. Antiparallelograms are also called contraparallelograms or crossed parallelograms.

<span class="mw-page-title-main">Tangential quadrilateral</span> Polygon whose four sides all touch a circle

In Euclidean geometry, a tangential 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 the incircle of the quadrilateral or its inscribed circle, its center is the incenter and its radius is called the inradius. Since these quadrilaterals can be drawn surrounding or circumscribing their incircles, they have also been called circumscribable quadrilaterals, circumscribing quadrilaterals, and circumscriptible quadrilaterals. Tangential quadrilaterals are a special case of tangential polygons.

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

<span class="mw-page-title-main">Orthodiagonal quadrilateral</span>

In Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicular) to each other.

<span class="mw-page-title-main">Ex-tangential quadrilateral</span>

In Euclidean geometry, an ex-tangential quadrilateral is a convex quadrilateral where the extensions of all four sides are tangent to a circle outside the quadrilateral. It has also been called an exscriptible quadrilateral. The circle is called its excircle, its radius the exradius and its center the excenter. The excenter lies at the intersection of six angle bisectors. These are the internal angle bisectors at two opposite vertex angles, the external angle bisectors at the other two vertex angles, and the external angle bisectors at the angles formed where the extensions of opposite sides intersect. The ex-tangential quadrilateral is closely related to the tangential quadrilateral.

<span class="mw-page-title-main">Tangential trapezoid</span>

In Euclidean geometry, a tangential trapezoid, also called a circumscribed trapezoid, is a trapezoid whose four sides are all tangent to a circle within the trapezoid: the incircle or inscribed circle. It is the special case of a tangential quadrilateral in which at least one pair of opposite sides are parallel. As for other trapezoids, the parallel sides are called the bases and the other two sides the legs. The legs can be equal, but they don't have to be.

<span class="mw-page-title-main">Tangential polygon</span> Convex polygon that contains an inscribed circle

In Euclidean geometry, a tangential polygon, also known as a circumscribed polygon, is a convex polygon that contains an inscribed circle. This is a circle that is tangent to each of the polygon's sides. The dual polygon of a tangential polygon is a cyclic polygon, which has a circumscribed circle passing through each of its vertices.

<span class="mw-page-title-main">Right kite</span> Symmetrical quadrilateral

In Euclidean geometry, a right kite is a kite that can be inscribed in a circle. That is, it is a kite with a circumcircle. Thus the right kite is a convex quadrilateral and has two opposite right angles. If there are exactly two right angles, each must be between sides of different lengths. All right kites are bicentric quadrilaterals, since all kites have an incircle. One of the diagonals divides the right kite into two right triangles and is also a diameter of the circumcircle.

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