In
Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
, a kite is a
quadrilateral
In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...
with
reflection symmetry
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a Reflection (mathematics), reflection. That is, a figure which does not change upon undergoing a reflection has reflecti ...
across a
diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek � ...
. 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.
[See H. S. M. Coxeter's review of in : "It is unfortunate that the author uses, instead of 'kite', the name 'deltoid', which belongs more properly to a curve, the three-cusped hypocycloid."] A kite may also be called a dart, particularly if it is not convex.
Every kite is an
orthodiagonal quadrilateral (its diagonals are at right angles) and, when convex, a
tangential quadrilateral
In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex polygon, convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This cir ...
(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
In plane Euclidean geometry, a rhombus (: 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. The rhom ...
, with two diagonal axes of symmetry; and the
square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
s, which are also special cases of both right kites and rhombi.
The quadrilateral with the greatest ratio of
perimeter
A perimeter is the length of a closed boundary that encompasses, surrounds, or outlines either a two-dimensional shape or a one-dimensional line. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimet ...
to
diameter
In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions a ...
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
A Penrose tiling is an example of an aperiodic tiling. Here, a ''tiling'' is a covering of two-dimensional space, the plane by non-overlapping polygons or other shapes, and a tiling is ''aperiodic'' if it does not contain arbitrarily large Perio ...
. Kites also form the faces of several
face-symmetric polyhedra and
tessellation
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ...
s, and have been studied in connection with
outer billiards, a problem in the advanced mathematics of
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
s.
Definition and classification

A kite is a
quadrilateral
In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...
with
reflection symmetry
In mathematics, reflection symmetry, line symmetry, mirror symmetry, or mirror-image symmetry is symmetry with respect to a Reflection (mathematics), reflection. That is, a figure which does not change upon undergoing a reflection has reflecti ...
across one of its diagonals. Equivalently, it is a quadrilateral whose four sides can be grouped into two pairs of adjacent equal-length sides. A kite can be constructed from the centers and crossing points of any two intersecting
circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
s. Kites as described here may be either
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
or
concave, although some sources restrict ''kite'' to mean only convex kites. A quadrilateral is a kite
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
any one of the following conditions is true:
* The four sides can be split into two pairs of adjacent equal-length sides.
* One diagonal crosses the midpoint of the other diagonal at a right angle, forming its
perpendicular bisector. (In the concave case, the line through one of the diagonals bisects the other.)
* One diagonal is a line of symmetry. It divides the quadrilateral into two congruent triangles that are mirror images of each other.
* One diagonal
bisects both of the angles at its two ends.
Kite quadrilaterals are named for the wind-blown, flying
kite
A kite is a tethered heavier than air flight, heavier-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. Kites often have ...
s, which often have this shape and which are in turn named for
a hovering bird and the sound it makes. According to
Olaus Henrici, the name "kite" was given to these shapes by
James Joseph Sylvester
James Joseph Sylvester (3 September 1814 – 15 March 1897) was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory, and combinatorics. He played a leadership ...
.
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
In plane Euclidean geometry, a rhombus (: 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. The rhom ...
(quadrilaterals with four equal sides),
squares
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
, and
Apollonius quadrilaterals (in which the products of opposite sides are equal). All
equilateral
An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
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.
Like kites, a
parallelogram
In Euclidean geometry, a parallelogram is a simple polygon, simple (non-list of self-intersecting polygons, self-intersecting) quadrilateral with two pairs of Parallel (geometry), parallel sides. The opposite or facing sides of a parallelogram a ...
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
In Euclidean geometry, an isosceles trapezoid 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 defined as a trapezoid in which both legs and bo ...
, with an axis of symmetry through the midpoints of two sides. These include as special cases the
rhombus
In plane Euclidean geometry, a rhombus (: 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. The rhom ...
and the
rectangle
In Euclidean geometry, Euclidean plane geometry, a rectangle is a Rectilinear polygon, rectilinear convex polygon or a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that a ...
respectively, and the square, which is a special case of both. The self-crossing quadrilaterals include another class of symmetric quadrilaterals, the
antiparallelogram
In geometry, an antiparallelogram is a type of list of self-intersecting polygons, 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 ...
s.
Special cases
The
right kites have two opposite
right angle
In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
s. The right kites are exactly the kites that are
cyclic quadrilateral
In geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertex (geometry), vertices all lie on a single circle, making the sides Chord (geometry), chords of the circle. This circle is called ...
s, meaning that there is a circle that passes through all their vertices. 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. 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). 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.
Among all quadrilaterals, the shape that has the greatest ratio of its
perimeter
A perimeter is the length of a closed boundary that encompasses, surrounds, or outlines either a two-dimensional shape or a one-dimensional line. The perimeter of a circle or an ellipse is called its circumference.
Calculating the perimet ...
to its
diameter
In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions a ...
(maximum distance between any two points) 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. An equidiagonal kite is a special case of a
midsquare quadrilateral. 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.
A kite with three 108° angles and one 36° angle forms the
convex hull
In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, ...
of the
lute of Pythagoras, a
fractal
In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...
made of nested
pentagram
A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle around ...
s. The four sides of this kite lie on four of the sides of a
regular pentagon
In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°.
A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...
, with a
golden triangle glued onto the fifth side.

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 ). A
prototile made by eight of these kites tiles the plane only
aperiodically, key to a claimed solution of the
einstein problem.
In
non-Euclidean geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean ge ...
, a kite can have three right angles and one non-right angle, forming a special case of a
Lambert quadrilateral
In geometry, a Lambert quadrilateral (also known as Ibn al-Haytham–Lambert quadrilateral), is a quadrilateral in which three of its angles are right angles. Historically, the fourth angle of a Lambert quadrilateral was of considerable interest s ...
. The fourth angle is acute in
hyperbolic geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For a ...
and obtuse in
spherical geometry
300px, A sphere with a spherical triangle on it.
Spherical geometry or spherics () is the geometry of the two-dimensional surface of a sphere or the -dimensional surface of higher dimensional spheres.
Long studied for its practical applicati ...
.
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. Because of its symmetry, the other two angles of the kite must be equal. The diagonal symmetry axis of a convex kite divides it into two
congruent triangles
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.
More formally, two sets of points are called congruent if, and only if, one can be ...
; the other diagonal divides it into two
isosceles triangle
In geometry, an isosceles triangle () is a triangle that has two Edge (geometry), sides of equal length and two angles of equal measure. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at le ...
s.
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 :
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
In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
between, then the area is
Inscribed circle
Every ''convex'' kite is also a
tangential quadrilateral
In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex polygon, convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This cir ...
, a quadrilateral that has an
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 incente ...
. That is, there exists a circle that is
tangent
In geometry, the tangent line (or simply tangent) to a plane curve at a given point is, intuitively, the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points o ...
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. 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.
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
A tangential quadrilateral is also a kite
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
any one of the following conditions is true:
* The area is one half the product of the
diagonal
In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek � ...
s.
* The diagonals are
perpendicular
In geometry, two geometric objects are perpendicular if they intersect at right angles, i.e. at an angle of 90 degrees or π/2 radians. The condition of perpendicularity may be represented graphically using the '' perpendicular symbol'', � ...
. (Thus the kites are exactly the quadrilaterals that are both tangential and
orthodiagonal.)
* The two line segments connecting opposite points of tangency have equal length.
* The
tangent lengths, distances from a point of tangency to an adjacent vertex of the quadrilateral, are equal at two opposite vertices of the quadrilateral. (At each vertex, there are two adjacent points of tangency, but they are the same distance as each other from the vertex, so each vertex has a single tangent length.)
* The two
bimedians, line segments connecting midpoints of opposite edges, have equal length.
* The products of opposite side lengths are equal.
* The center of the incircle lies on a line of symmetry that is also a diagonal.
If the diagonals in a tangential quadrilateral intersect at , and the
incircle
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 incenter ...
s of triangles }, , , have radii , , , and respectively, then the quadrilateral is a kite if and only if
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
Duality

Kites and
isosceles trapezoid
In Euclidean geometry, an isosceles trapezoid 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 defined as a trapezoid in which both legs and bo ...
s 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. The features of kites and isosceles trapezoids that correspond to each other under this duality are compared in the table below.
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.
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
In geometry, circle packing is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that no circle can be enlarged without creating an overlap. The associated ''packing den ...
can be used to subdivide any polygon with
sides into
kites, meeting edge-to-edge.
Tilings and polyhedra
All kites
tile the plane by repeated
point reflection
In geometry, a point reflection (also called a point inversion or central inversion) is a geometric transformation of affine space in which every point is reflected across a designated inversion center, which remains fixed. In Euclidean or ...
around the midpoints of their edges, as do more generally all quadrilaterals. 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
A Penrose tiling is an example of an aperiodic tiling. Here, a ''tiling'' is a covering of two-dimensional space, the plane by non-overlapping polygons or other shapes, and a tiling is ''aperiodic'' if it does not contain arbitrarily large Perio ...
, an
aperiodic tiling
An aperiodic tiling is a non-periodic Tessellation, tiling with the additional property that it does not contain arbitrarily large periodic regions or patches. A set of tile-types (or prototiles) is aperiodic set of prototiles, aperiodic if copie ...
of the plane discovered by mathematical physicist
Roger Penrose
Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, Philosophy of science, philosopher of science and Nobel Prize in Physics, Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics i ...
. 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
In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...
rosette in which successively larger rings of
kites surround a central point. 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.
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. The
deltoidal icositetrahedron,
deltoidal hexecontahedron, and
trapezohedron are
polyhedra
In geometry, a polyhedron (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. The term "polyhedron" may refer either to a solid figure or to its boundary su ...
with congruent kite-shaped
faces, which can alternatively be thought of as tilings of the sphere by congruent spherical kites. There are infinitely many
face-symmetric tilings of the
hyperbolic plane by kites. 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
In geometry, a vertex configuration is a shorthand notation for representing a polyhedron or Tessellation, tiling as the sequence of Face (geometry), faces around a Vertex (geometry), vertex. It has variously been called a vertex description, vert ...
(the numbers of neighbors of each of the four vertices of each tile). Some polyhedra and tilings appear twice, under two different face configurations.

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
antiprism
In geometry, an antiprism or is a polyhedron composed of two Parallel (geometry), parallel Euclidean group, direct copies (not mirror images) of an polygon, connected by an alternating band of triangles. They are represented by the Conway po ...
s. A commonly seen example is the
pentagonal trapezohedron, used for ten-sided
dice
A die (: dice, sometimes also used as ) is a small, throwable object with marked sides that can rest in multiple positions. Dice are used for generating random values, commonly as part of tabletop games, including dice games, board games, ro ...
.
Outer billiards
Mathematician
Richard Schwartz has studied
outer billiards on kites. Outer billiards is a
dynamical system
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
in which, from a point outside a given
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact, a type of agreement used by U.S. states
* Blood compact, an ancient ritual of the Philippines
* Compact government, a t ...
convex set
In geometry, a set of points is convex if it contains every line segment between two points in the set.
For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is n ...
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. He later wrote a
monograph
A monograph is generally a long-form work on one (usually scholarly) subject, or one aspect of a subject, typically created by a single author or artist (or, sometimes, by two or more authors). Traditionally it is in written form and published a ...
analyzing outer billiards for kite shapes more generally. For this problem, any
affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, '' affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More general ...
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
Rationality is the quality of being guided by or based on reason. In this regard, a person acts rationally if they have a good reason for what they do, or a belief is rational if it is based on strong evidence. This quality can apply to an ...
. For the case of the Penrose kite, , an irrational number, where is the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if
\fr ...
.
References
External links
*
area formulaewith interactive animation at Mathopenref.com
{{Polygons
Elementary shapes
Types of quadrilaterals
Kites