In mathematics, the pentagram map is a discrete dynamical system acting on polygons in the projective plane. It defines a new polygon whose vertices are obtained as the intersection points of the shortest diagonals of the initial polygon. This is a projectively equivariant procedure, hence it descends to the moduli space of polygons and defines another dynamical system (which is also referred to as the pentagram map). It was first introduced by Richard Schwartz in 1992.
The pentagram map on the moduli space is famous for its complete integrability and its link with cluster algebras.
It admits many generalizations in projective spaces and other settings.
Introduction Informal definition On polygons
Initially, the pentagram map was defined for convex polygons (with at least five sides) on the Euclidean plane. Given such a polygon with sides, one can draw the "shortest diagonals", meaning the segments whose endpoints are a vertex and one of its second neighbors (as in the figure). The intersections of the shortest diagonals are then taken as the vertices of a new -gon ; this new polygon is the output of the pentagram map.
The same construction can be done on non-convex polygons, but there are several complications. First, some consecutive short diagonals may not intersect, so one must extend the segments to lines. Second, the image can fail to be a new -gon because some consecutive vertices could coincide. However, this generically doesn't happen. Finally, it is possible that two diagonals are parallel and don't intersect on the Euclidean plane. This is resolved by extending the Euclidean plane to the real projective plane by the addition of a line at infinity, where the intersection point lies (see the figure in the next section). Hence, the pentagram map is defined for generic polygons in the real projective plane.
More generally, the construction of the pentagram map is well defined whenever the concepts of lines and their intersections make sense. This is encompassed by the notion of a general projective plane, of which the real projective plane is one example; but the pentagram map can also be considered over other fields, for instance the complex numbers, which give the complex projective plane.
On the moduli space of polygons Since the pentagram map is constructed by drawing lines and marking their intersections, it commutes with any transformation that sends lines to lines. Such maps are called projective transformations. Hence, polygons can be identified up to projective transformations. This identification gives the quotient space (technically called a moduli space) of classes of polygons.
The pentagram map on polygons induces another dynamical system on the moduli space, whose behavior differs quite a lot from the initial one. The dynamic is trivial for the classes of pentagons and hexagons, but this is no longer the case for polygons with more vertices.
Historical elements The pentagram map for general polygons was introduced in , but the simplest case is the one of pentagons, hence the name "pentagram". Their study goes back to , and .
The pentagram map interacts with some classical configuration theorems of projective geometry. It provides results analogous to the ones of Pascal's theorem and Brianchon's theorem. Some specific configurations make Desargues's theorem and Poncelet's porism appear.
Definitions and first properties Definition of the map
Let be an integer. A polygon with sides, or -gon, is a tuple of vertices lying in some projective plane , where the indices are understood modulo . The dimension of the space of -gons is .
Suppose that the vertices are in sufficiently general position, meaning that no consecutive triple of points are collinear. Taking the intersection of two consecutive "shortest" diagonals defines a new pointThis procedure defines a new -gon .
The labeling of the indices of is not canonical. In most papers, a choice is made at the beginning of the paper and the formulas are tuned accordingly.
The pentagram map on polygons is a birational map . Indeed, each coordinate of is given as a rational function of the coordinates of , since it is defined as the intersection of lines passing by them. Moreover, the inverse map is given by taking the intersections , which is rational for the same reason.
Moduli space The pentagram map is defined by taking lines and intersections of them. The biggest group which maps lines to lines is the one of projective transformations, denoted by . Such a transformation acts on a polygon by sending it to . The pentagram map commutes with this action, and thereby induces another dynamical system on the moduli space of projective equivalence classes of polygons, whose dimension is .
Twisted polygons
The pentagram map naturally generalizes to the larger space of twisted polygons (see the example of heptagon in the figure). For any integer , a twisted -gon is the data of:
a bi-infinite sequence of points in the projective plane (called the vertices), a projective transformation (called the monodromy),
such that for any , the property is satisfied. The dimension of the space of twisted -gons is .
When is the identity, this gives back the initial definition of polygons (which are said to be closed). The space of closed -gons is of codimension in the space of twisted ones.
The action of projective transformations over the space of closed polygons generalizes to the space of twisted ones (the monodromy is changed by conjugation). This provides again a moduli space, of dimension .
Collapsing of convex polygons Exponential shrinking
Let be a closed strictly convex polygon lying on the real plane. One of the first results proved by Richard Schwartz it that its iterates under the pentagram map shrink exponentially fast to a point, as illustrated in the figure. This follows from two facts.
The image of a strictly convex polygon is contained in its interior, and is also strictly convex. There exists a constant , depending on , such that for any , the diameters of the iterates verify the inequality
Hence, by Cantor's intersection theorem, the sequence of polygons collapses toward a point.
The behavior on the moduli space is very different, since the dynamics is recurrent. It is even a quasiperiodic motion, as discussed in the section about integrability.
Coordinates of the limit point The limit point coordinates were given in . They satisfy some degree 3 polynomial equations, whose coefficients are rational functions in the coordinates of the vertices of the starting polygon. The proof relies on the fact that the limit point must be an eigenline of a certain linear operator of .
This operator was reinterpreted in as the infinitesimal monodromy of the polygon. The scaling symmetry is used to deform a closed polygon into a family of twisted ones with monodromy . The infinitesimal monodromy is defined to be:
Generalization The collapsing of polygons may also happen in some generalization of the pentagram map, when considering some specific configurations of polygons in the real plane. The coordinates of the collapse point are given by a formula analogous to the one for the original pentagram map.
Periodic orbits on the moduli space For some configurations of closed polygons, the iterate of the pentagram map will send to a projectively equivalent polygon (up to some shift of the indices). This means that, on the moduli space, the orbit of the class of is periodic.
Pentagons and hexagons
The following two facts are proved by checking cross-ratio equalities, so they are true for polygons in any projective plane (not just the real one).
The pentagram map is the identity on the moduli space of pentagons. The second iterate is the identity on the space of labeled hexagons, up to a shift of labeling (see the figure). This phenomenon doesn't generalize to generic polygons with at least seven sides, for which the motion is quasiperiodic.
Generalization The result about pentagons and hexagons generalizes to some higher pentagram maps in , for polygons with or sides. The proof uses a generalization of the Gale transform.
Poncelet polygons A polygon is said to be Poncelet if it is inscribed in a conic and circumscribed about another one. For a convex Poncelet -gon lying on the real projective plane, the polygon is projectively equivalent to .
In fact, when is odd, the converse is also true. However, this converse statement is no longer true when the polygons are considered over the complex projective plane since there are explicit counterexamples.
Coordinates for the moduli space The moduli space can be described by different coordinate systems. The following ones give simple expressions for the dynamics, as presented in the next section.
Corner coordinates
Define the cross-ratio of four collinear points to be
The corner invariants are a system of coordinates on the space of twisted polygons, constructed by taking intersections as in the figure. The left and right invariants are respectively defined as the following cross-ratios:
Since the cross-ratio is projective invariant, the sequences and associated to a twisted -gon are -periodic.
The corner invariants are elements of , and they realize an isomorphism of varieties between the moduli space of twisted -gons and .
ab-coordinates There is a second set of coordinates for the moduli space of twisted -gons defined over any field satisfying , and such that is not divisible by .
The vertices in the projective plane can be lifted to vectors in the affine space so that each consecutive triple of vectors spans a parallelepiped having determinant equal to . This leads to the relation defining the -coordinates:
This bring out an analogy between twisted polygons and solutions of third order linear ordinary differential equations, normalized to have unit Wronskian.
They are linked to the corner coordinates by:
Formulas on the moduli space As a birational map The pentagram map is a birational map on the moduli space, because it can be decomposed as the composition of two birational involutions. The corner invariants change in the following way:
The scaling symmetry The multiplicative group acts on the moduli space in the following way:
where is called the scaling action and is the scaling parameter. This action commutes with the pentagram map on the moduli space (as presented in the previous formulas). This property is called the scaling symmetry, and is instrumental in proving the complete integrability of the dynamics.
Invariant structures Monodromy invariants The monodromy invariants, introduced in , are a collection of functions on the moduli space that are invariant under the pentagram map. The simplest examples of them are
The other monodromy invariants can be retrieved through different points of view: through the scaling symmetry, as combinatorial objects, or as some determinants. The one involving scaling symmetry is presented here.
Let be a lift of the monodromy of a twisted -gon. The quantities
are independent of the choice of lift and are invariant under conjugation, so they are well defined for the projective class of the polygon. They are invariant under the pentagram map, since the monodromy matrix doesn't change. Now, the quantities
have the same properties, but turn out to be polynomials in the corner invariants. They can be written as
where each and are homogeneous polynomials respectively of weight and , meaning they change under the rescaling action on variables by
The quantities are unchanged by the dynamics, and are called the monodromy invariants. Moreover, they are algebraically independent.
Polygons on conics