Pentagram map

Template:Short description 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.Template:Sfn

The pentagram map on the moduli space is famous for its complete integrability and its link with cluster algebras.Template:Sfn

It admits many generalizations in projective spaces and other settings.

Introduction

Informal definition

On polygons

The pentagram map applied on a convex pentagon.

Initially, the pentagram map was defined for convex polygons (with at least five sides) on the Euclidean plane. Given such a polygon $P$ with $n$ 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 $n$-gon $T(P)$; this new polygon is the output of the pentagram map.Template:Sfn

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 $T(P)$ can fail to be a new $n$-gon because some consecutive vertices could coincide. However, this generically doesn't happen.Template:Sfn 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.Template:Sfn

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.Template:Sfn

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,Template:Sfn whose behavior differs quite a lot from the initial one.Template:Efn The dynamic is trivial for the classes of pentagons and hexagons, but this is no longer the case for polygons with more vertices.Template:Efn

Historical elements

The pentagram map for general polygons was introduced in Template:Harvard citation, but the simplest case is the one of pentagons, hence the name "pentagram".Template:Sfn Their study goes back to Template:Harvard citation,Template:Sfn Template:Harvard citationTemplate:Sfn and Template:Harvard citation.Template:Sfn

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.Template:Sfn Some specific configurations make Desargues's theorem and Poncelet's porism appear.Template:SfnTemplate:Efn

Definitions and first properties

Definition of the map

The pentagram map on a convex pentagon, with vertices labeled.

The pentagram map applied on a self-intersecting (in particular, non-convex) pentagon. The vertex $w_2$ is on the line at infinity, because it is the intersection of two parallel lines.

Let $n\ge 5$ be an integer. A polygon $P$ with $n$ sides, or $n$-gon, is a tuple of vertices $(v_1,\dots ,v_n)$ lying in some projective plane ${\mathbb{P}}^2$,Template:Efn where the indices are understood modulo $n$. The dimension of the space of $n$-gons is $2n$.Template:Sfn

Suppose that the vertices are in sufficiently general position, meaning that no consecutive triple of points are collinear.Template:Sfn Taking the intersection of two consecutive "shortest" diagonalsTemplate:Efn defines a new point$$w_k:=\overline{v_{k-1}{v}_{k+1}}\cap \overline{v_k{v}_{k+2}}.$$This procedure defines a new $n$-gon $T(P)=(w_1,\dots ,w_n)$.Template:Sfn

The labeling of the indices of $T(P)$ is not canonical. In most papers, a choice is made at the beginning of the paper and the formulas are tuned accordingly.Template:Sfn

The pentagram map on polygons is a birational map $T:({\mathbb{P}}^2{)}^n$Template:Nowrap$({\mathbb{P}}^2{)}^n$. Indeed, each coordinate of $w_k$ is given as a rational function of the coordinates of $v_{k-1},\dots ,v_{k+2}$, since it is defined as the intersection of lines passing by them. Moreover, the inverse map is given by taking the intersections $\overline{w_{k-2}{w}_{k-1}}\cap \overline{w_k{w}_{k+1}}$, which is rational for the same reason.Template:Sfnp

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 $\mathbb{P}{\mathrm{G}\mathrm{L}}_3$. Such a transformation $M$ acts on a polygon $P$ by sending it to $M\cdot P:=(M{v}_1,\dots ,M{v}_n)$. 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 $2n-8$.Template:Sfn

Twisted polygons

An example of twisted heptagon on the real plane.

The pentagram map naturally generalizes to the larger space of twisted polygons (see the example of heptagon in the figure). For any integer $n\ge 5$, a twisted $n$-gon $P$ is the data of:

• a bi-infinite sequence of points $(v_k{)}_{k\in \mathbb{Z}}$ in the projective plane (called the vertices),
• a projective transformation $M\in \mathbb{P}{\mathrm{G}\mathrm{L}}_3$ (called the monodromy),

such that for any $k\in \mathbb{Z}$, the property $v_{k+n}=M{v}_k$ is satisfied. The dimension of the space of twisted $n$-gons is $2n+8$.Template:Sfn

When $M$ is the identity, this gives back the initial definition of polygons (which are said to be closed). The space of closed $n$-gons is of codimension $8$ in the space of twisted ones.Template:Sfn

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 $2n$.Template:Sfn

Collapsing of convex polygons

Exponential shrinking

The pentagram map iterated on a convex heptagon, exhibiting the convergence.

Let $P$ 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.

1. The image of a strictly convex polygon is contained in its interior, and is also strictly convex.Template:Sfn
2. There exists a constant $0<{\eta }_P<1$, depending on $P$, such that for any $N\in \mathbb{N}$, the diameters of the iterates verify the inequality $\mathrm{diam}(T^N(P))\le {\eta }_P^N\mathrm{diam}(P).$Template:Sfn

Hence, by Cantor's intersection theorem, the sequence of polygons collapses toward a point.Template:Sfn

The behavior on the moduli space is very different, since the dynamics is recurrent.Template:Sfn It is even a quasiperiodic motion,Template:Sfn as discussed in the section about integrability.

Coordinates of the limit point

The limit point coordinates were given in Template:Harvard citation. 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 ${ℝ}^3$.Template:Sfn

This operator was reinterpreted in Template:Harvard citation as the infinitesimal monodromy of the polygon. The scaling symmetry is used to deform a closed polygon $P$ into a family of twisted ones $(P_z{)}_{z\in {ℂ}^{\ast }}$ with monodromy $M_z$. The infinitesimal monodromy is defined to be:Template:Sfn $${\frac{d{M}_z}{dz}|}_{z=1}.$$

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.Template:Sfn

Periodic orbits on the moduli space

For some configurations of closed polygons, the iterate of the pentagram map will send $P$ 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 $P$ is periodic.

Pentagons and hexagons

The outward hexagon is projectively equivalent to the inward one, with respect to their labeling.

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).Template:Sfn

The pentagram map $T$ is the identity on the moduli space of pentagons.Template:SfnTemplate:SfnTemplate:Sfn The second iterate $T^2$ is the identity on the space of labeled hexagons, up to a shift of labeling (see the figure).Template:Sfn This phenomenon doesn't generalize to generic polygons with at least seven sides, for which the motion is quasiperiodic.Template:Sfn

Generalization

The result about pentagons and hexagons generalizes to some higher pentagram maps in ${\mathbb{P}}^k$, for polygons with $k+3$ or $2k+2$ sides. The proof uses a generalization of the Gale transform.Template:Sfn

Poncelet polygons

A polygon is said to be PonceletTemplate:Efn if it is inscribed in a conic and circumscribed about another one.Template:SfnTemplate:Efn For a convex Poncelet $n$-gon $P$ lying on the real projective plane, the polygon $T^2(P)$ is projectively equivalent to $P$.Template:Sfn

In fact, when $n$ is odd, the converse is also true.Template:Sfn However, this converse statement is no longer true when the polygons are considered over the complex projective plane since there are explicit counterexamples.Template:Sfn

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

The geometric construction of the points defining the corner invariants.

Define the cross-ratio of four collinear points to be

$[a,b,c,d]=\frac{(a-b)(c-d)}{(a-c)(b-d)}.$

The corner invariants are a system of coordinates on the space of twisted polygons, constructed by taking intersections as in the figure.Template:Sfn The left and right invariants are respectively definedTemplate:Efn as the following cross-ratios:

$x_k:=[v_{k-2},v_{k-1},\overline{v_{k-2}{v}_{k-1}}\cap \overline{v_k{v}_{k+1}},\overline{v_{k-2}{v}_{k-1}}\cap \overline{v_{k+1}{v}_{k+2}}],$

$y_k:=[\overline{v_{k+1}{v}_{k+2}}\cap \overline{v_{k-2}{v}_{k-1}},\overline{v_{k+1}{v}_{k+2}}\cap \overline{v_{k-1}{v}_k},v_{k+1},v_{k+2}].$

Since the cross-ratio is projective invariant, the sequences $(x_k{)}_{k\in \mathbb{Z}}$ and $(y_k{)}_{k\in \mathbb{Z}}$ associated to a twisted $n$-gon are $n$-periodic.Template:Sfn

The corner invariants are elements of ${\mathbb{P}}^1\setminus \{0,1,\mathrm{\infty }\}$, and they realize an isomorphism of varieties between the moduli space of twisted $n$-gons and $({\mathbb{P}}^1\setminus \{0,1,\mathrm{\infty }\}{)}^{2n}$.Template:Sfn

ab-coordinates

There is a second set of coordinates for the moduli space of twisted $n$-gons defined over any field $F$ satisfying ${\mathrm{S}\mathrm{L}}_3(F)\cong \mathbb{P}{\mathrm{G}\mathrm{L}}_3(F)$,Template:Sfn and such that $n$ is not divisible by $3$.Template:Sfn

The vertices $v_k$ in the projective plane ${\mathbb{P}}^2(F)$ can be lifted to vectors $V_k$ in the affine space $F^3$ so that each consecutive triple of vectors spans a parallelepiped having determinant equal to $1$. This leads to the relation defining the $ab$-coordinates:Template:Sfn

$V_{k+3}=a_k{V}_{k+2}+b_k{V}_{k+1}+V_k.$

This bring out an analogy between twisted polygons and solutions of third order linear ordinary differential equations, normalized to have unit Wronskian.Template:Sfn

They are linked to the corner coordinates by:Template:Sfn

$x_k=\frac{a_{k-2}}{b_{k-2}{b}_{k-1}},$

$y_k=-\frac{b_{k-1}}{a_{k-2}{a}_{k-1}}.$

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.Template:Sfn The corner invariants change in the following way:Template:Sfn

${x_k}^{\prime }=x_k\frac{1-x_{k-1}{y}_{k-1}}{1-x_{k+1}{y}_{k+1}},$

${y_k}^{\prime }=y_{k+1}\frac{1-x_{k+2}{y}_{k+2}}{1-x_k{y}_k}.$

The scaling symmetry

The multiplicative group $F\setminus \{0\}$ acts on the moduli space in the following way:

$R_s\cdot (x_1,\dots ,x_n,y_1,\dots ,y_n)=(s{x}_1,\dots ,s{x}_n,s^{-1}{y}_1,\dots ,s^{-1}{y}_n),$

where $R$ is called the scaling action and $s$ 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.Template:Sfn

Invariant structures

Monodromy invariants

The monodromy invariants, introduced in Template:Harvard citation, are a collection of functions on the moduli space that are invariant under the pentagram map.Template:Sfn The simplest examples of them are

$O_n=x_1{x}_2\cdots x_n,E_n=y_1{y}_2\cdots y_n.$

The other monodromy invariants can be retrieved through different points of view: through the scaling symmetry, as combinatorial objects, or as some determinants.Template:Sfn The one involving scaling symmetry is presented here.

Let $M\in {\mathrm{G}\mathrm{L}}_3$ be a lift of the monodromy of a twisted $n$-gon. The quantities

${\mathrm{\Omega }}_1=\frac{{\mathrm{trace}}^3(M)}{det(M)},{\mathrm{\Omega }}_2=\frac{{\mathrm{trace}}^3(M^{-1})}{det(M^{-1})},$

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.Template:Sfn Now, the quantities

${\tilde{\mathrm{\Omega }}}_1=O_n^2{E}_n{\mathrm{\Omega }}_1,{\tilde{\mathrm{\Omega }}}_2=O_n{E}_n^2{\mathrm{\Omega }}_2,$

have the same properties, but turn out to be polynomials in the corner invariants.Template:Efn They can be written asTemplate:Sfn

${\tilde{\mathrm{\Omega }}}_1=(\sum _{k=0}^{\lfloor n/2\rfloor }{O}_k{)}^3,{\tilde{\mathrm{\Omega }}}_2=(\sum _{k=0}^{\lfloor n/2\rfloor }{E}_k{)}^3,$

where each $O_k$ and $E_k$ are homogeneous polynomials respectively of weight $k$ and $-k$,Template:Sfn meaning they change under the rescaling action on variables byTemplate:Sfn

$R_s(O_k)=s^k{O}_k,R_s(E_k)=s^{-k}{E}_k.$

The quantities $O_1,\dots ,O_{\lfloor n/2\rfloor },O_n,E_1,\dots ,E_{\lfloor n/2\rfloor },E_n,$ are unchanged by the dynamics, and are called the monodromy invariants. Moreover, they are algebraically independent.Template:Sfn

Polygons on conics

Whenever $P$ is inscribed in a conic section, one has $O_k(P)=E_k(P)$ for all $k$.Template:Sfn Moreover, if $P$ is circumscribed about another conic,Template:Efn then its monodromy invariants are characterized by the pair of conics.Template:Sfn For such odd-gons, the translation on the Jacobian varietyTemplate:Efn is restricted to the Prym variety (which is a half-dimensional torus in the Jacobian).Template:Sfn

Poisson bracket

An invariant Poisson bracket on the space of twisted polygons was found in Template:Harvard citation. The monodromy invariants commute with respect to it: $$\{O_i,O_j\}=\{O_i,E_j\}=\{E_i,E_j\}=0$$for all $i,j$.Template:Sfn

The Poisson bracket is defined in terms of the corner coordinates by: $$\begin{array}{cc}\{x_i,x_{i\pm 1}\}& =\mp x_i{x}_{i+1},\\ \{y_i,y_{i\pm 1}\}& =\mp y_i{y}_{i+1},\end{array}$$and $$\{x_i,x_j\}=\{y_i,y_j\}=\{x_i,y_j\}=0$$for all other $i,j.$Template:Sfn

The spectral curve

Let $\zeta$ be an element of the multiplicative group and $P_{\zeta }$ be the polygon obtained by applying the rescaling action $R_{\zeta }$ on $P$. A Lax matrix $\hat{T}(\zeta )\in {\mathrm{G}\mathrm{L}}_3$ is a lift of the monodromy of $P_{\zeta }$ satisfying a zero-curvature equation.Template:Sfn Then, the spectral function is the bivariate characteristic polynomial$$Q(\lambda ,\zeta ):=det(\lambda \mathrm{Id}-\hat{T}(\zeta )),$$or some renormalization of it. The spectral curve is the projective completion of the affine curve defined by the equation $Q(\lambda ,\zeta )=0$.Template:Sfn It is invariant under the pentagram map, and the monodromy invariants appear as the coefficients of $Q$.Template:Sfn Its geometric genus is $n-1$ if $n$ is odd, and $n-2$ if $n$ is even.Template:Sfn

It was first introduced in Template:Harv for his proof of algebro-geometric integrability.Template:Sfn

Complete integrability

The pentagram map on the moduli space has been proved to be a completely integrable discrete dynamical system, both in the Arnold-LiouvilleTemplate:Efn and the algebro-geometricTemplate:Efn senses. In any case, this means that the moduli space is almost everywhere foliated by flat tori (or in the algebraic setting, Abelian varieties), where the motion is a translation. This generically induces a quasiperiodic motion on the corresponding torus.Template:Sfn

Arnold–Liouville integrability

The proof of the integrability of the pentagram map on a real twisted polygon was achieved in Template:Harvard citation. This is done by noticing that the monodromy invariants $O_n$ and $E_n$ are Casimir invariants for the Poisson bracket, meaning (in this context) that$$\{O_n,f\}=\{E_n,f\}=0$$for every smooth function $f$.Template:Sfn When $n$ is even, this is also true for the monodromy invariants $O_{\lfloor n/2\rfloor }$ and $E_{\lfloor n/2\rfloor }$.Template:Sfn

This allows to consider the Casimir level set, where each Casimir invariant has a specified value. Because of Sard's theorem, any generic level set is a smooth manifold.Template:Sfn This family of level sets forms a foliation in symplectic leaves, on which the Poisson bracket gives rise to a symplectic form.Template:Sfn

Each of these symplectic leaves has an iso-monodromy foliation, namely, a decomposition into the common level sets of the remaining monodromy functions. By using again Sard's theorem, they are generically Lagrangian manifolds.Template:Sfn Moreover, they are compact.Template:Sfn Since the monodromy invariants Poisson-commute and there are enough of them, the discrete Liouville–Arnold theorem can be applied to prove that the level sets are flat tori over which the dynamics is a translation.Template:Sfn

Algebro-geometric integrability

In Template:Harvard citation, it was shown that the pentagram map admits a Lax representation with a spectral parameter, which allows to prove its algebro-geometric integrability. This means that the space of polygons (either twisted or closed) is parametrized by its spectral data, consisting of its spectral curve, with marked points and a divisor given by a Floquet–Bloch equation. This gives an embedding to the Jacobian variety through the Abel–Jacobi map, where the motion is expressed in terms of translation.Template:Sfn The previously defined Poisson bracket is also retrieved.Template:Sfn

This integrability was generalized in Template:Harvard citation from the field of complex numbers to any algebraically closed field of characteristic different from 2. The translation on a torus is replaced by a translation on an Abelian variety (in fact, a Jacobian variety again).Template:Sfn

Dimension of the invariant manifold

For twisted $n$-gons, the dimension of the invariant tori (or Jacobian varieties) isTemplate:Sfn

$\{\begin{array}{cc}n-1& \text{when }n\text{ is odd,}\\ n-2& \text{when }n\text{ is even.}\end{array}$

Moreover, when $n$ is even, there are two isomorphic Jacobians on which the iterates of the pentagram map alternate. But on each of them, the second iterate is a translation.Template:Sfn

For closed polygons

There is no Poisson structure on the space of closed polygons.Template:Sfn Nevertheless, the one from twisted polygons can be used to prove integrability.Template:Sfn

Algebro-geometric integrability holds for closed polygons in a same manner as for the twisted ones.Template:Sfn However, Arnold-Liouville integrability is proved for real closed polygons only when they are convex. This is done by restricting the Hamiltonian vector fields of monodromy functions to smaller dimensional tori, and showing that enough of them are still independent.Template:Sfn

In both situation, the dimension of the invariant manifolds decreases by $3$ for closed $n$-gons (compared to the twisted case), and is equal toTemplate:SfnTemplate:Sfn

$\{\begin{array}{cc}n-4& \text{when }n\text{ is odd,}\\ n-5& \text{when }n\text{ is even.}\end{array}$

Connections to other topics

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 Boussinesq equation. This partial differential equation models water waves under some conditions, which is a classical example of integrable system.Template:Sfn

Here is a description of the geometric action of the Boussinesq equation. Given a locally convex curve $C:ℝ\to {ℝ}^2$ and real numbers $x$ and $t$, 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 construction is also similar to the pentagram map. Moreover, the pentagram invariant bracket is a discretization of an invariant Poisson bracket associated to the Boussinesq equation.Template:Sfn

Cluster algebras

The pentagram mapTemplate:Sfn and some of its generalizationsTemplate:SfnTemplate:Sfn are identified as special cases of discrete dynamical systems powered by cluster algebras. Using the results from Template:Harvard citation, this provides a link with the Poisson–Lie groups, dimer models and other so-called cluster-integrable systems.Template:SfnTemplate:Sfn These methods allow to retrieve the Poisson bracket and Hamiltonians used to prove complete integrabilityTemplate:Sfn and provide Lax representations.Template:Sfn

Octahedron recurrence

Using a method to compute determinants called Dodgson condensation, Template:Harvard citation proves that the pentagram map satisfies a property called the "octahedron recurrence".Template:Sfn This property turns out to be shared by other dynamical systems defined by geometric constructions similar to one of the pentagram map.Template:Sfn It is also shared by higher dimensional pentagram maps defined through cluster algebras mutations, referred as $T$-systems.Template:Sfn

Singularity theory

The pentagram map exhibits a property called singularity confinement, which is a typical phenomenon in integrable systems.Template:Sfn It states that if a polygon $P$ is singular for the pentagram map $T$, then there exists an integer $m$ such that $P$ not singular for the iterate map $T^m$.Template:Sfn

Moreover, the pentagram map (along with some of its generalizations and other discrete dynamical systems) exhibit the Devron property.Template:Efn This means that if a polygon $P$ is singular for some iterate of the pentagram map $T^m$, then it will also be singular for some iterate of the inverse map $T^{-m^{\prime }}$.Template:Sfn

Generalizations

The definition of polygons still makes sense in any projective space ${\mathbb{P}}^d$ under the action of the projective group $\mathbb{P}{\mathrm{G}\mathrm{L}}_{d+1}$, and even in other spaces with their associated groups.

The pentagram map can be generalized in many ways, and some of them are presented here. Not all of them are integrable.Template:Sfn Some are discretizations of PDEs from the KdV hierarchy, seen as higher dimensional version of Boussinesq or KP equations.Template:SfnTemplate:Sfn The description of all generalized pentagram maps in terms of cluster algebras is still an open question.Template:Sfn

Polygons in general positions

Let $d\ge 2$ and $P$ be a twisted polygon of ${\mathbb{P}}^d$ in general position.

Short diagonal pentagram maps

The $k$-th short diagonal hyperplane $H_k^{sh}$ is uniquely defined by passing through the vertices $v_k,v_{k+2},\dots ,v_{k+2d-2}$. Generically, the intersection of $d$ consecutive hyperplanes uniquely defines a new point

$T_{sh}{v}_k:=H_k^{sh}\cap H_{k+1}^{sh}\cap \dots \cap H_{k+d-1}^{sh}.$

Doing this for every vertex defines a new twisted polygon. This map, denoted by $T_{sh}$, is again projectively equivariant.Template:Sfn

Generalized pentagram maps

The previous procedure can be generalized. Let $I=(i_1,\dots ,i_{d-1}),J=(j_1,\dots ,j_{d-1})$ be two sets of integers, respectively called the jump tuple and the intersection tuple. Define $H_k^I$ to be the unique hyperplane passing through the vertices $v_k,v_{k+i_1},\dots ,v_{k+i_1+\dots +i_{d-1}}$. A new point is given by the intersection

$T_{I,J}{v}_k:=H_k^I\cap H_{k+j_1}^I\cap \dots \cap H_{k+j_1+\dots +j_{d-1}}^I.$

The map $T_{I,J}$ is called a generalized pentagram map.Template:Sfn The original pentagram map is recovered by considering$d=2,I=(2),J=(1)$.

Integrability can be numerically tested by picking a random polygon $P$ with rational coordinates and studying the growth rate of the height of its iterates. This is called the diophantine integrability test, and some generalized pentagram maps don't seem to pass it.Template:Sfn However, it is conjectured that the maps $T_{I,I}$ are integrable for any $I$.Template:Sfn

Some of these maps are discretizations of higher dimensional counterpart of the Boussinesq equation in the KdV hierarchy.Template:SfnTemplate:Sfn

Dented pentagram maps

Fix an integer $m\in \{1,\dots ,d-1\}$. Consider the jump tuple $I_m:=(1,\dots ,1,2,1,\dots ,1)$, where the $2$ is at the $m$-th place, and the intersection tuple $J:=(1,\dots ,1)$. The dented pentagram map is $T_m:=T_{I_m,J}$. They are proved to be integrable.Template:Sfn

For an integer $p\ge 2$, the deep dented pentagram map (of depth $p$) $T_m^p$ is the same map as before, but the number $2$ in the definition of $I_m$ is replaced by $p$. This kind of pentagram maps are integrable too.Template:Sfn

Corrugated polygons

A twisted polygon $P$ lying in ${\mathbb{P}}^d$ is said to be corrugated if for any $k\in \mathbb{Z}$, the vertices $v_k,v_{k+1},v_{k+d},v_{k+d+1}$ span a projective two-dimensional plane. Such polygons are not in general position. A new point is defined by

$T_{\text{cor}}{v}_k:=\overline{v_k{v}_{k+d}}\cap \overline{v_{k+1}{v}_{k+d+1}}.$

The map $T_{\text{cor}}$ yields a new corrugated polygon. This dynamics is completely Liouville-integrable.Template:Sfn

In fact, they can be retrieved as some dented pentagram map applied on corrugated polygons.Template:Sfn

Grassmannian polygons

Let $d\ge 3,m\ge 1$ be integers. The pentagram map can also be generalized to the Grassmannian space $\mathrm{G}\mathrm{r}(m,md)$, which consists of $m$-dimensional linear subspaces of an $md$-dimensional vector space. When $m=1$, the linear subspaces are lines, which retrieves the definition of projective spaces ${\mathbb{P}}^d$.Template:Sfn

A point $v\in \mathrm{Gr}(m,md)$ is represented by an $md\times m$ matrix $X_v$ such that its columns form a basis of $v$. Consider the action of the general linear group ${\mathrm{G}\mathrm{L}}_{md}$ by multiplication on the left of $X_v$. This defines an action on the Grassmannian, even though it is not faithful.Template:Efn Hence, the polygons of $\mathrm{G}\mathrm{r}(m,md)$ and their moduli spaces are defined as before, after the change of underlying group.Template:Sfn

Depending on the parity of $d$, one can define linear subspaces spanned by some $X_{v_k}$'s such that taking their intersection generically defines a new point $v\in \mathrm{G}\mathrm{r}(m,md)$.Template:Sfn This generalization of the pentagram map is integrable in a noncommutative sense.Template:Sfn

Over rings

The pentagram map admits a generalization by considering projective planes over stably finite rings, instead of fields. In particular, this retrieves the pentagram map over Grassmannians. Again, it admits a Lax representation.Template:Sfn

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