In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ... , the pentagram map is a discrete
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. Examples include the mathematical models that describe the swinging of a ... on the
moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spac ... of
polygons
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 toge ... in the
projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ... . The
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 aroun ... map takes a given polygon, finds the intersections of the shortest
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 δ ... s of the polygon, and constructs a new polygon from these intersections.
Richard Schwartz introduced the pentagram map for a general polygon in a 1992 paper
though it seems that the special case, in which the map is defined for
pentagons
In geometry, a pentagon (from the Greek language, Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is ... only, goes back to an 1871 paper of
Alfred Clebsch
Rudolf Friedrich Alfred Clebsch (19 January 1833 – 7 November 1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory. He attended the University of Königsberg and was habilitated at Berlin. ... and a 1945 paper of
Theodore Motzkin
Theodore Samuel Motzkin (26 March 1908 – 15 December 1970) was an Israeli-American mathematician.
Biography
Motzkin's father Leo Motzkin, a Ukrainian Jew, went to Berlin at the age of thirteen to study mathematics. He pursued university studi ... .
The pentagram map is similar in spirit to the constructions underlying
Desargues' theorem
In projective geometry, Desargues's theorem, named after Girard Desargues, states:
:Two triangles are in perspective ''axially'' if and only if they are in perspective ''centrally''.
Denote the three vertices of one triangle by and , and tho ... and
Poncelet's porism
In geometry, Poncelet's closure theorem, also known as Poncelet's porism, states that whenever a polygon is inscribed figure, inscribed in one conic section and circumscribes another one, the polygon must be part of an infinite family of polygo ... . It echoes the rationale and construction underlying a conjecture of
Branko Grünbaum
Branko Grünbaum ( he, ברנקו גרונבאום; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descent[vertices of the ](_blank)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 toge ... P_1,P_3,P_5,\ldots The image of ''P'' under the pentagram map is the polygon ''Q'' with vertices Q_2,Q_4,Q_6,\ldots as shown in the figure. Here Q_4 is the intersection of the diagonals (P_1P_5) and (P_3P_7) , and so on.
On a basic level, one can think of the pentagram map as an operation defined on 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 polytope ... plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ... projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ... field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ... vertices are in sufficiently general position
In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the ''general case'' situation, as opposed to some more special or coincidental cases that ar ... commutes with projective transformations
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, s ... mapping on the moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spac ... equivalence classes
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ... Labeling conventions
The map P \to Q is slightly problematic, in the sense that the indices of the ''P''-vertices are naturally odd integers whereas the indices of ''Q''-vertices are naturally even integers. A more conventional approach to the labeling would be to label the vertices of P and Q by integers of the same parity. One can arrange this either by adding or subtracting 1 from each of the indices of the ''Q''-vertices. Either choice is equally canonical. An even more conventional choice would be to label the vertices of ''P'' and ''Q'' by consecutive integers, but again there are two natural choices for how to align these labellings: Either Q_k is just clockwise from P_k or just counterclockwise. In most papers on the subject, some choice is made once and for all at the beginning of the paper and then the formulas are tuned to that choice.
There is a perfectly natural way to label the vertices of the second iterate of the pentagram map by consecutive integers. For this reason, the second iterate of the pentagram map is more naturally considered as an iteration defined on labeled polygons. See the figure.
Twisted polygons
The pentagram map is also defined on the larger space of twisted polygons.[
A twisted ''N''-gon is a bi-infinite sequence of points in the projective plane that is ''N''-periodic modulo a ]projective transformation
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, s ... P_k to P_ for all ''k''. The map ''M'' is called the monodromy
In mathematics, monodromy is the study of how objects from mathematical analysis, algebraic topology, algebraic geometry and differential geometry behave as they "run round" a singularity. As the name implies, the fundamental meaning of ''mono ... [
] Elementary properties
Action on pentagons and hexagons
The pentagram map is the identity on the moduli space of pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simpl ... [ This is to say that there is always a ]projective transformation
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, s ... T^2 is the identity on the space of labeled hexagon
In geometry, a hexagon (from Ancient Greek, Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple polygon, simple (non-self-intersecting) hexagon is 720°.
Regular hexa ... [ Here ''T'' is the second iterate of the pentagram map, which acts naturally on labeled hexagons, as described above. This is to say that the hexagons ] H and T^2(H) are equivalent by a label-preserving projective transformation
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, s ... H' and T(H) are projectively equivalent, where H' is the labeled hexagon obtained from H by shifting the labels by 3. [ See the figure. It seems entirely possible that this fact was also known in the 19th century.
The action of the pentagram map on pentagons and hexagons is similar in spirit to classical configuration theorems in projective geometry such as ]Pascal's theorem
In projective geometry, Pascal's theorem (also known as the ''hexagrammum mysticum theorem'') states that if six arbitrary points are chosen on a conic (which may be an ellipse, parabola or hyperbola in an appropriate affine plane) and joined by ... Desargues's theorem
In projective geometry, Desargues's theorem, named after Girard Desargues, states:
:Two triangles are in perspective ''axially'' if and only if they are in perspective ''centrally''.
Denote the three vertices of one triangle by and , and tho ... Exponential shrinking
The iterates of the pentagram map shrink any convex polygon
In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a ... [ This is to say that the diameter of the nth iterate of a convex polygon is less than ] K a^n for constants K>0 and 0 which depend on the initial polygon. Here we are taking about the geometric action on the polygons themselves, not on the moduli space of projective equivalence classes of polygons.
Motivating discussion
This section is meant to give a non-technical overview for much of the remainder of the article. The context for the pentagram map is projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pro ... circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ... ellipse
In mathematics, an ellipse is a plane curve surrounding two focus (geometry), focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It generalizes a circle, which is the special ty ... rectangular
In Euclidean geometry, 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 (360°/4 = 90°); or a par ... quadrilateral
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, ... Projective transformations
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, s ... perspective drawing
Linear or point-projection perspective (from la, perspicere 'to see through') is one of two types of 3D projection, graphical projection perspective in the graphic arts; the other is parallel projection. Linear perspective is an approximate r ... computer vision
Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the hum ... line
Line most often refers to:
* Line (geometry), object with zero thickness and curvature that stretches to infinity
* Telephone line, a single-user circuit on a telephone communication system
Line, lines, The Line, or LINE may also refer to:
Arts ... mapping on the moduli space of polygons
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 toge ... moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spac ... Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry: the ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small ... triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ... triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, an ... x,y,z such that x+y+z =180. So, each point (x,y,z) , satisfying the constraints just mentioned, indexes a triangle (up to scale). One might say that (x,y,z) are coordinates for the moduli space of scale equivalence classes of triangles. If you want to index all possible quadrilaterals, either up to scale or not, you would need some additional parameters
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ... higher-dimensional moduli space. The moduli space relevant to the pentagram map is the moduli space of projective equivalence classes of polygons. Each point in this space corresponds to a polygon, except that two polygons which are different views of each other are considered the same. Since the pentagram map is adapted to projective geometry, as mentioned above, it induces a mapping on this particular moduli space. That is, given any point in the moduli space, you can apply the pentagram map to the corresponding polygon and see what new point you get.
The reason for considering what the pentagram map does to the moduli space is that it gives more salient features of the map. If you just watch, geometrically, what happens to an individual polygon, say a convex polygon
In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a ... [ To see things more clearly, you might dilate the shrinking family of polygons so that they all have, say, the same ]area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape
A shape or figure is a graphics, graphical representation of an obje ... [ Now you can change the ]aspect ratio so as to try to get yet a better view of these polygons. If you do this process as systematically as possible, you find that you are simply looking at what happens to points in the moduli space. The attempts to zoom in to the picture in the most perceptive possible way lead to the introduction of the moduli space.
To explain how the pentagram map acts on the moduli space, one must say a few words about the torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ... donut
A doughnut or donut () is a type of food made from leavened fried dough. It is popular in many countries and is prepared in various forms as a sweet snack that can be homemade or purchased in bakeries, supermarkets, food stalls, and franc ... Asteroids
An asteroid is a minor planet of the inner Solar System. Sizes and shapes of asteroids vary significantly, ranging from 1-meter rocks to a dwarf planet almost 1000 km in diameter; they are rocky, metallic or icy bodies with no atmosphere.
... torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ... manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ... Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ... sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ... Earth
Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's surfa ... plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ... [ These invisible tori fill up the moduli space somewhat like the way the layers of an onion fill up the onion itself, or how the individual cards in a deck fill up the deck. The technical statement is that the tori make a ]foliation
In mathematics (differential geometry), a foliation is an equivalence relation on an ''n''-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the decomposition of ... 7 -gons is 6 dimensional and the tori in this case are 3 dimensional.
The tori are invisible subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ... dynamical systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a p ... integrable system
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ... Coordinates for the moduli space
Cross-ratio
When the field underlying all the constructions is ''F'', the affine line
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ... projective line
In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ... projective transformation
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, s ... t_1,t_2,t_3,t_4 in the affine line one defines the (inverse) cross ratio
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, the ... X=\frac.
Most authors consider 1/''X'' to be the cross-ratio
In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, the ... projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ... projective transformation
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, s ... The corner coordinates
The corner invariants are basic coordinates on the space of twisted polygons.[ Suppose that P is a ]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 toge ... flag
A flag is a piece of fabric (most often rectangular or quadrilateral) with a distinctive design and colours. It is used as a symbol, a signalling device, or for decoration. The term ''flag'' is also used to refer to the graphic design empl ... F_1,\ldots,F_ To each flag F, we associate the inverse cross ratio of the points t_1,t_2,t_3,t_4 shown in the figure at left. In this way, one associates numbers x_1,\ldots,x_ to an n-gon. If two n-gons are related by a projective transformation, they get the same coordinates. Sometimes the variables x_1,y_1,x_2,y_2,\ldots are used in place of x_1,x_2,x_3,x_4,\ldots\,.
The corner invariants make sense on the moduli space of twisted polygons. When one defines the corner invariants of a twisted polygon, one obtains a 2''N''-periodic bi-infinite sequence of numbers. Taking one period of this sequence identifies a twisted ''N''-gon with a point in F^ where ''F'' is the underlying field. Conversely, given almost any (in the sense of measure theory
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ... F^ one can construct a twisted ''N''-gon having this list of corner invariants. Such a list will not always give rise to an ordinary polygon; there are an additional 8 equations which the list must satisfy for it to give rise to an ordinary ''N''-gon.
(ab) coordinates
There is a second set of coordinates for the moduli space of twisted polygons, developed by Sergei Tabachnikov
Sergei Tabachnikov, also spelled Serge, (in Russian: Сергей Львович Табачников; born in 1956) is a Russian mathematician who works in geometry and dynamical systems. He is currently a Professor of Mathematics at Pennsylvan ... projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do ... \ldots V_1,V_2,V_3,\ldots in R^3 so that each consecutive triple of vectors spans a parallelepiped
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidea ... V_ = a_i V_ + b_i V_ + V_i
The coordinates a_1,b_1,a_2,b_2,\ldots serve as coordinates for the moduli space of twisted ''N''-gons as long as ''N'' is not divisible by 3.
The (ab) coordinates bring out the close analogy between twisted polygons and solutions of 3rd order linear ordinary differential equations
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast w ... Wronskian
In mathematics, the Wronskian (or Wrońskian) is a determinant introduced by and named by . It is used in the study of differential equations, where it can sometimes show linear independence in a set of solutions.
Definition
The Wronskian of ... Formula for the pentagram map
As a birational mapping
Here is a formula for the pentagram map, expressed in corner coordinates.[ The equations work more gracefully when one considers the second iterate of the pentagram map, thanks to the canonical labelling scheme discussed above. The second iterate of the pentagram map is the ]composition
Composition or Compositions may refer to:
Arts and literature
*Composition (dance), practice and teaching of choreography
*Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ... B \circ A . The maps A and B are birational mapping s of order 2, and have the following action.
: A(x_1,\ldots,x_)=(a_1,\ldots,a_)
: B(x_1,\ldots,x_)=(b_1,\ldots,b_)
where
:
\begin
a_ & = \frac x_ \\ pt a_ & = \frac x_ \\ pt b_ & =\frac x_ \\ pt b_ & = \frac x_
\end
(Note: the index 2''k'' + 0 is just 2''k''. The 0 is added to align the formulas.) In these coordinates, the pentagram map is a birational mapping of F^
As grid compatibility relations
The formula for the pentagram map has a convenient interpretation as a certain compatibility rule for labelings on the edges of triangular grid, as shown in the figure.[ In this interpretation, the corner invariants of a polygon P label the non-horizontal edges of a single row, and then the non-horizontal edges of subsequent rows are labeled by the corner invariants of ] A(P) , B(A(P)) , A(B(A(P))) , and so forth. the compatibility rules are
: c=1-ab
: wx=yz
These rules are meant to hold for all configurations which are congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ... Invariant structures
Corner coordinate products
It follows directly from the formula for the pentagram map, in terms of corner coordinates, that the two quantities
: O_N= x_1x_3\cdots x_
: E_N = x_2x_4\cdots x_
are invariant under the pentagram map. This observation is closely related to the 1991 paper of Joseph Zaks [ concerning the diagonals of a polygon.
When ''N'' = 2''k'' is even, the functions
:] O_k = x_1x_5x_9 \cdots x_+ x_3x_7x_ \cdots x_
: E_k = x_2x_6x_ \cdots x_+ x_4x_8x_ \cdots x_
are likewise seen, directly from the formula, to be invariant functions. All these products turn out to be Casimir invariant
In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operator ... O_k and E_k are the simplest examples of the monodromy invariants defined below.
The level set
In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is:
: L_c(f) = \left\~,
When the number of independent variables is two, a level set is calle ... f=O_NE_N are compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ... convex polygon
In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a ... [ Hence, each orbit of the pentagram map acting on this space has a ]compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ... closure .
Volume form
The pentagram map, when acting on the moduli space ''X'' of convex polygons, has an invariant volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of the ... f=O_NE_N has compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ... level sets
In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is:
: L_c(f) = \left\~,
When the number of independent variables is two, a level set is calle ... Poincaré recurrence theorem to imply that the action of the pentagram map on ''X'' is recurrent: The orbit of almost any equivalence class of convex polygon ''P'' returns infinitely often to every neighborhood of ''P''.[ This is to say that, modulo projective transformations, one typically sees nearly the same shape, over and over again, as one iterates the pentagram map. (It is important to remember that one is considering the projective equivalence classes of convex polygons. The fact that the pentagram map visibly shrinks a convex polygon is irrelevant.)
It is worth mentioning that the recurrence result is subsumed by the complete integrability results discussed below.][
] Monodromy invariants
The so-called monodromy invariants are a collection of functions on the moduli space
In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spac ... monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of variables with nonnegative integer exponent ... - x_1x_5x_6x_7
*123789 gives rise to + x_1x_2x_3x_7x_8x_9
The sign is determined by the parity of the number of single-digit blocks in the sequence. The monodromy invariant O_k is defined as the sum of all monomials coming from odd admissible sequences composed of k blocks. The monodromy invariant E_k is defined the same way, with even replacing odd in the definition.
When ''N'' is odd, the allowable values of ''k'' are 1, 2, ..., (''n'' − 1)/2. When ''N'' is even, the allowable values of k are 1, 2, ..., ''n''/2. When ''k'' = ''n''/2, one recovers the product invariants discussed above. In both cases, the invariants O_N and E_N are counted as monodromy invariants, even though they are not produced by the above construction.
The monodromy invariants are defined on the space of twisted polygons, and restrict to give invariants on the space of closed polygons. They have the following geometric interpretation. The monodromy M of a twisted polygon is a certain rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ... trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''Trace'' ... determinants
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ... matrices
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ... [
Whenever ''P'' has all its vertices on a ]conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ... O_k(P)=E_k(P) for all ''k''. [ ]
Poisson bracket
A Poisson bracket
In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. Th ... linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ... \ on the space of functions which satisfies the Leibniz Identity and the Jacobi identity
In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the asso ... [ Valentin Ovsienko, Richard Schwartz and Sergei Tabachnikov produced a ]Poisson bracket
In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. Th ... \=\=\=0
for all indices.
Here is a description of the invariant Poisson bracket in terms of the variables.
: x_1,y_1,x_2,y_2,\ldots\,.
: \ = -x_i\, x_
: \ = x_i\, x_
: \ = y_i\, y_
: \ = -y_i\, y_
: \ = \ = \ = 0 for all other i,j.
There is also a description in terms of the (ab) coordinates, but it is more complicated.[
Here is an alternate description of the invariant bracket. Given any function ] f on the moduli space, we have the so-called Hamiltonian vector field In mathematics and physics, a Hamiltonian vector field on a symplectic manifold is a vector field defined for any energy function or Hamiltonian. Named after the physicist and mathematician Sir William Rowan Hamilton, a Hamiltonian vector field is ... H(f) = \left( x_ \frac - x_ \frac \right) x_i \frac + \left( y_ \frac - y_ \frac \right) y_i \frac
where a summation over the repeated indices is understood. Then
: H(f) g = \
The first expression is the directional derivative
In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity s ... g in the direction of the vector field H(f) . In practical terms, the fact that the monodromy invariants Poisson-commute means that the corresponding Hamiltonian vector field s define commuting flows.
Complete integrability
Arnold–Liouville integrability
The monodromy invariants and the invariant bracket combine to establish Arnold–Liouville integrability of the pentagram map on the space of twisted ''N''-gons. [ The situation is easier to describe for N odd. In this case, the two products
: ] O_n =x_1\cdots x_n
: E_n = y_1\cdots y_n
are Casimir invariant
In mathematics, a Casimir element (also known as a Casimir invariant or Casimir operator) is a distinguished element of the center of the universal enveloping algebra of a Lie algebra. A prototypical example is the squared angular momentum operator ... \=\ =0
for all functions f. A Casimir level set
In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is:
: L_c(f) = \left\~,
When the number of independent variables is two, a level set is calle ... O_n and E_n .
Each Casimir level set has an iso-monodromy foliation
In mathematics (differential geometry), a foliation is an equivalence relation on an ''n''-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the decomposition of ... coordinate charts on each iso-monodromy level such that the transition maps are Euclidean translations. That is, the Hamiltonian vector fields impart a flat Euclidean structure on the iso-monodromy levels, forcing them to be flat tori when they are smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ... compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ... manifolds
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a Ne ... quasi-periodic motion . This explains the Arnold-Liouville integrability.
From the point of view of symplectic geometry
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed differential form, closed, nondegenerate form, nondegenerate different ... symplectic form In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form.
A symplectic bilinear form is a mapping that is
; Bilinear: Linear in each argument s ... Algebro-geometric integrability
In a 2011 preprint, Lax representation In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that satisfy a corresponding differential equation, called the ''Lax equation''. Lax pairs were introduced by Peter Lax to discuss sol ... divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ... quasi-periodic motion on a torus (both in the twisted and the ordinary case), and they allow one to construct explicit solutions formulas using Riemann theta functions
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum fiel ... Connections to other topics
The Octahedral recurrence
The octahedral recurrence is a dynamical system defined on the vertices of the octahedral tiling of space. Each octahedron has 6 vertices, and these vertices are labelled in such a way that
: a_1b_1 + a_2b_2 = a_3b_3
Here a_i and b_i are the labels of antipodal vertices. A common convention is that a_2,b_2,a_3,b_3 always lie in a central horizontal plane and a_1,b_1 are the top and bottom vertices. The octahedral recurrence is closely related to C. L. Dodgson's method of condensation for computing determinants
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and ... [ Typically one labels two horizontal layers of the tiling and then uses the basic rule to let the labels propagate dynamically.
Max Glick used the ]cluster algebra Cluster algebras are a class of commutative rings introduced by . A cluster algebra of rank ''n'' is an integral domain ''A'', together with some subsets of size ''n'' called clusters whose union generates the algebra ''A'' and which satisfy variou ... alternating sign matrices .[ These formulas are similar in spirit to the formulas found by ]David P. Robbins
David Peter Robbins (12 August 1942 in Brooklyn – 4 September 2003 in Princeton) was an American mathematician. He is most famous for introducing alternating sign matrices.. He is also known for his work on generalizations of Heron's formula o ... [ Let ] T be the octahedral tiling. Let \pi: T \to R^2 be the linear projection
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ... T to the configuration of 6 points shown in the first figure. Say that an adapted labeling of T is a labeling so that all points in the (infinite) inverse image
In mathematics, the image of a function is the set of all output values it may produce.
More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ... G=\pi(T) get the same numerical label. The octahedral recurrence applied to an adapted labeling is the same as a recurrence on G in which the same rule as for the octahedral recurrence is applied to every configuration of points congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ... G which obeys the planar octahedral recurrence, one can create a labeling of the edges of G by applying the rule
: v=AD/BC
to every edge. This rule refers to the figure at right and is meant to apply to every configuration that is congruent
Congruence may refer to:
Mathematics
* Congruence (geometry), being the same size and shape
* Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure
* In mod ... The Boussinesq equation
The continuous limit of a convex polygon is a parametrized convex curve in the plane. When the time parameter is suitably chosen, the continuous limit of the pentagram map is the classical Boussinesq equation .[ This equation is a classical example of an ]integrable
In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ... partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ... locally convex
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological ve ... C:R\to R^2 , and real numbers x and t, we consider the chord connecting C(x-t) to C(x+t) . The envelope of all these chords is a new curve C_t(x) . When t is extremely small, the curve C_t(x) is a good model for the time t evolution of the original curve C_0(x) under the Boussinesq equation. This geometric description makes it fairly obvious that the B-equation is the continuous limit of the pentagram map. At the same time, the pentagram invariant bracket is a discretization of a well known invariant Poisson bracket associated to the Boussinesq equation. [
Recently, there has been some work on higher-dimensional generalizations of the pentagram map and its connections to Boussinesq-type partial differential equations ] Projectively natural evolution
The pentagram map and the Boussinesq equation are examples of projectively natural geometric evolution equations. Such equations arise in diverse fields of mathematics, such as projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pro ... computer vision
Computer vision is an interdisciplinary scientific field that deals with how computers can gain high-level understanding from digital images or videos. From the perspective of engineering, it seeks to understand and automate tasks that the hum ... Cluster algebras
In a 2010 paper [*] Max Glick identified the pentagram map as a special case of a cluster algebra Cluster algebras are a class of commutative rings introduced by . A cluster algebra of rank ''n'' is an integral domain ''A'', together with some subsets of size ''n'' called clusters whose union generates the algebra ''A'' and which satisfy variou ... See also
* Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ... Periodic table of shapes
Notes
References
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*{{cite journal, title = On the products of cross-ratios on diagonals of polygons, author= Zaks, Joseph, s2cid= 123626706, journal=Geometriae Dedicata
''Geometriae Dedicata'' is a mathematical journal, founded in 1972, concentrating on geometry and its relationship to topology, group theory and the theory of dynamical systems. It was created on the initiative of Hans Freudenthal in Utrecht, the N ... Projective geometry
Dynamical systems
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