Pentagram map

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 ${\displaystyle P}$ with ${\displaystyle n}$ 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 ${\displaystyle n}$-gon ${\displaystyle T(P)}$; 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 ${\displaystyle T(P)}$ can fail to be a new ${\displaystyle n}$-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.

Historical elements

The pentagram map for general polygons was introduced in (Schwartz 1992), but the simplest case is the one of pentagons, hence the name "pentagram". Their study goes back to (Clebsch 1871), (Kasner 1928) and (Motzkin 1945).

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.

Definition of the map

Let ${\displaystyle n\geq 5}$ be an integer. A polygon ${\displaystyle P}$ with ${\displaystyle n}$ sides, or ${\displaystyle n}$-gon, is a tuple of vertices ${\displaystyle (v_{1},\dots ,v_{n})}$ lying in some projective plane ${\displaystyle \mathbb {P} ^{2}}$, where the indices are understood modulo ${\displaystyle n}$. The dimension of the space of ${\displaystyle n}$-gons is ${\displaystyle 2n}$.

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 point$${\displaystyle w_{k}:={\overline {v_{k-1}v_{k+1}}}\cap {\overline {v_{k}v_{k+2}}}.}$$This procedure defines a new ${\displaystyle n}$-gon ${\displaystyle T(P)=(w_{1},\dots ,w_{n})}$.

The labeling of the indices of ${\displaystyle T(P)}$ 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 ${\displaystyle T:(\mathbb {P} ^{2})^{n}}$⇢${\displaystyle (\mathbb {P} ^{2})^{n}}$. Indeed, each coordinate of ${\displaystyle w_{k}}$ is given as a rational function of the coordinates of ${\displaystyle v_{k-1},\dots ,v_{k+2}}$, since it is defined as the intersection of lines passing by them. Moreover, the inverse map is given by taking the intersections ${\displaystyle {\overline {w_{k-2}w_{k-1}}}\cap {\overline {w_{k}w_{k+1}}}}$, 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 ${\displaystyle \mathbb {P} \mathrm {GL} _{3}}$. Such a transformation ${\displaystyle M}$ acts on a polygon ${\displaystyle P}$ by sending it to ${\displaystyle M\cdot P:=(Mv_{1},\dots ,Mv_{n})}$. 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 ${\displaystyle 2n-8}$.

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 ${\displaystyle n\geq 5}$, a twisted ${\displaystyle n}$-gon ${\displaystyle P}$ is the data of:

• a bi-infinite sequence of points ${\displaystyle (v_{k})_{k\in \mathbb {Z} }}$ in the projective plane (called the vertices),
• a projective transformation ${\displaystyle M\in \mathbb {P} \mathrm {GL} _{3}}$ (called the monodromy),

such that for any ${\displaystyle k\in \mathbb {Z} }$, the property ${\displaystyle v_{k+n}=Mv_{k}}$ is satisfied. The dimension of the space of twisted ${\displaystyle n}$-gons is ${\displaystyle 2n+8}$.

When ${\displaystyle M}$ is the identity, this gives back the initial definition of polygons (which are said to be closed). The space of closed ${\displaystyle n}$-gons is of codimension ${\displaystyle 8}$ 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 ${\displaystyle 2n}$.

Exponential shrinking

Let ${\displaystyle P}$ 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.
2. There exists a constant ${\displaystyle 0<\eta _{P}<1}$, depending on ${\displaystyle P}$, such that for any ${\displaystyle N\in \mathbb {N} }$, the diameters of the iterates verify the inequality ${\textstyle \operatorname {diam} (T^{N}(P))\leq \eta _{P}^{N}\operatorname {diam} (P).}$

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 (Glick 2020). 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 ${\displaystyle \mathbb {R} ^{3}}$.

This operator was reinterpreted in (Aboud & Izosimov 2022) as the infinitesimal monodromy of the polygon. The scaling symmetry is used to deform a closed polygon ${\displaystyle P}$ into a family of twisted ones ${\displaystyle (P_{z})_{z\in \mathbb {C} ^{*}}}$ with monodromy ${\displaystyle M_{z}}$. The infinitesimal monodromy is defined to be: $${\displaystyle \left.{\frac {dM_{z}}{dz}}\right|_{z=1}.}$$

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.

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 ${\displaystyle T}$ is the identity on the moduli space of pentagons. The second iterate ${\displaystyle T^{2}}$ 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.

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 ${\displaystyle n}$-gon ${\displaystyle P}$ lying on the real projective plane, the polygon ${\displaystyle T^{2}(P)}$ is projectively equivalent to ${\displaystyle P}$.

In fact, when ${\displaystyle n}$ 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.

Corner coordinates

Define the cross-ratio of four collinear points to be

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

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

${\displaystyle x_{k}:=[v_{k-2},v_{k-1},{\overline {v_{k-2}v_{k-1}}}\cap {\overline {v_{k}v_{k+1}}},{\overline {v_{k-2}v_{k-1}}}\cap {\overline {v_{k+1}v_{k+2}}}],}$

${\displaystyle y_{k}:=[{\overline {v_{k+1}v_{k+2}}}\cap {\overline {v_{k-2}v_{k-1}}},{\overline {v_{k+1}v_{k+2}}}\cap {\overline {v_{k-1}v_{k}}},v_{k+1},v_{k+2}].}$

Since the cross-ratio is projective invariant, the sequences ${\displaystyle (x_{k})_{k\in \mathbb {Z} }}$ and ${\displaystyle (y_{k})_{k\in \mathbb {Z} }}$ associated to a twisted ${\displaystyle n}$-gon are ${\displaystyle n}$-periodic.

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

ab-coordinates

There is a second set of coordinates for the moduli space of twisted ${\displaystyle n}$-gons defined over any field ${\displaystyle F}$ satisfying ${\displaystyle \mathrm {SL} _{3}(F)\cong \mathbb {P} \mathrm {GL} _{3}(F)}$, and such that ${\displaystyle n}$ is not divisible by ${\displaystyle 3}$.

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

${\displaystyle V_{k+3}=a_{k}V_{k+2}+b_{k}V_{k+1}+V_{k}.}$

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

They are linked to the corner coordinates by:

${\displaystyle x_{k}={\frac {a_{k-2}}{b_{k-2}b_{k-1}}},}$

${\displaystyle y_{k}=-{\frac {b_{k-1}}{a_{k-2}a_{k-1}}}.}$

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:

${\displaystyle x_{k}'=x_{k}{\frac {1-x_{k-1}y_{k-1}}{1-x_{k+1}y_{k+1}}},}$

${\displaystyle y_{k}'=y_{k+1}{\frac {1-x_{k+2}y_{k+2}}{1-x_{k}y_{k}}}.}$

The scaling symmetry

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

${\displaystyle R_{s}\cdot (x_{1},\dots ,x_{n},y_{1},\dots ,y_{n})=(sx_{1},\dots ,sx_{n},s^{-1}y_{1},\dots ,s^{-1}y_{n}),}$

where ${\displaystyle R}$ is called the scaling action and ${\displaystyle s}$ 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.

Monodromy invariants

The monodromy invariants, introduced in (Schwartz 2008), are a collection of functions on the moduli space that are invariant under the pentagram map. The simplest examples of them are

${\displaystyle O_{n}=x_{1}x_{2}\cdots x_{n},\quad E_{n}=y_{1}y_{2}\cdots y_{n}.}$

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

Let ${\displaystyle M\in \mathrm {GL} _{3}}$ be a lift of the monodromy of a twisted ${\displaystyle n}$-gon. The quantities

${\displaystyle \Omega _{1}={\frac {\operatorname {trace} ^{3}(M)}{\det(M)}},\quad \Omega _{2}={\frac {\operatorname {trace} ^{3}(M^{-1})}{\det(M^{-1})}},}$

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. Now, the quantities

${\displaystyle {\tilde {\Omega }}_{1}=O_{n}^{2}E_{n}\Omega _{1},\quad {\tilde {\Omega }}_{2}=O_{n}E_{n}^{2}\Omega _{2},}$

have the same properties, but turn out to be polynomials in the corner invariants. They can be written as

${\displaystyle {\tilde {\Omega }}_{1}={\biggl (}\sum _{k=0}^{\lfloor n/2\rfloor }O_{k}{\biggr )}^{3},\quad {\tilde {\Omega }}_{2}={\biggl (}\sum _{k=0}^{\lfloor n/2\rfloor }E_{k}{\biggr )}^{3},}$

where each ${\displaystyle O_{k}}$ and ${\displaystyle E_{k}}$ are homogeneous polynomials respectively of weight ${\displaystyle k}$ and ${\displaystyle -k}$, meaning they change under the rescaling action on variables by

${\displaystyle R_{s}(O_{k})=s^{k}O_{k},\quad R_{s}(E_{k})=s^{-k}E_{k}.}$

The quantities ${\displaystyle O_{1},\dots ,O_{\lfloor n/2\rfloor },O_{n},E_{1},\dots ,E_{\lfloor n/2\rfloor },E_{n},}$ are unchanged by the dynamics, and are called the monodromy invariants. Moreover, they are algebraically independent.

Poisson bracket

An invariant Poisson bracket on the space of twisted polygons was found in (Ovsienko, Schwartz & Tabachnikov 2010). The monodromy invariants commute with respect to it: $${\displaystyle \{O_{i},O_{j}\}=\{O_{i},E_{j}\}=\{E_{i},E_{j}\}=0}$$for all ${\displaystyle i,j}$.

The Poisson bracket is defined in terms of the corner coordinates by: $${\displaystyle {\begin{aligned}\{x_{i},x_{i\pm 1}\}&=\mp x_{i}x_{i+1},\\\{y_{i},y_{i\pm 1}\}&=\mp y_{i}y_{i+1},\\\end{aligned}}}$$and $${\displaystyle \{x_{i},x_{j}\}=\{y_{i},y_{j}\}=\{x_{i},y_{j}\}=0}$$for all other ${\displaystyle i,j.}$

The spectral curve

Let ${\displaystyle \zeta }$ be an element of the multiplicative group and ${\displaystyle P_{\zeta }}$ be the polygon obtained by applying the rescaling action ${\displaystyle R_{\zeta }}$ on ${\displaystyle P}$. A Lax matrix ${\displaystyle {\hat {T}}(\zeta )\in \mathrm {GL} _{3}}$ is a lift of the monodromy of ${\displaystyle P_{\zeta }}$ satisfying a zero-curvature equation. Then, the spectral function is the bivariate characteristic polynomial$${\displaystyle Q(\lambda ,\zeta ):=\det(\lambda \operatorname {Id} -{\hat {T}}(\zeta )),}$$or some renormalization of it. The spectral curve is the projective completion of the affine curve defined by the equation ${\displaystyle Q(\lambda ,\zeta )=0}$. It is invariant under the pentagram map, and the monodromy invariants appear as the coefficients of ${\displaystyle Q}$. Its geometric genus is ${\displaystyle n-1}$ if ${\displaystyle n}$ is odd, and ${\displaystyle n-2}$ if ${\displaystyle n}$ is even.

It was first introduced in (Soloviev 2013) for his proof of algebro-geometric integrability.

Arnold–Liouville integrability

The proof of the integrability of the pentagram map on a real twisted polygon was achieved in (Ovsienko, Schwartz & Tabachnikov 2010). This is done by noticing that the monodromy invariants ${\displaystyle O_{n}}$ and ${\displaystyle E_{n}}$ are Casimir invariants for the Poisson bracket, meaning (in this context) that$${\displaystyle \{O_{n},f\}=\{E_{n},f\}=0}$$for every smooth function ${\displaystyle f}$. When ${\displaystyle n}$ is even, this is also true for the monodromy invariants ${\displaystyle O_{\lfloor n/2\rfloor }}$ and ${\displaystyle E_{\lfloor n/2\rfloor }}$.

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. This family of level sets forms a foliation in symplectic leaves, on which the Poisson bracket gives rise to a symplectic form.

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. Moreover, they are compact. 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.

Algebro-geometric integrability

In (Soloviev 2013), 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. The previously defined Poisson bracket is also retrieved.

This integrability was generalized in (Weinreich 2022) 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).

Dimension of the invariant manifold

For twisted ${\displaystyle n}$-gons, the dimension of the invariant tori (or Jacobian varieties) is

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

Moreover, when ${\displaystyle n}$ 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.

For closed polygons

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

Algebro-geometric integrability holds for closed polygons in a same manner as for the twisted ones. 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.

In both situation, the dimension of the invariant manifolds decreases by ${\displaystyle 3}$ for closed ${\displaystyle n}$-gons (compared to the twisted case), and is equal to

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

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.

Here is a description of the geometric action of the Boussinesq equation. Given a locally convex curve ${\displaystyle C:\mathbb {R} \to \mathbb {R} ^{2}}$ and real numbers ${\displaystyle x}$ and ${\displaystyle t}$, consider the chord connecting ${\displaystyle C(x-t)}$ to ${\displaystyle C(x+t)}$. The envelope of all these chords is a new curve ${\displaystyle C_{t}(x)}$. When ${\displaystyle t}$ is extremely small, the curve ${\displaystyle C_{t}(x)}$ is a good model for the time ${\displaystyle t}$ evolution of the original curve ${\displaystyle C_{0}(x)}$ 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.

Cluster algebras

The pentagram map and some of its generalizations are identified as special cases of discrete dynamical systems powered by cluster algebras. Using the results from (Goncharov & Kenyon 2013), this provides a link with the Poisson–Lie groups, dimer models and other so-called cluster-integrable systems. These methods allow to retrieve the Poisson bracket and Hamiltonians used to prove complete integrability and provide Lax representations.

Octahedron recurrence

Using a method to compute determinants called Dodgson condensation, (Schwartz 2008) proves that the pentagram map satisfies a property called the "octahedron recurrence". This property turns out to be shared by other dynamical systems defined by geometric constructions similar to one of the pentagram map. It is also shared by higher dimensional pentagram maps defined through cluster algebras mutations, referred as ${\displaystyle T}$-systems.

Singularity theory

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

Moreover, the pentagram map (along with some of its generalizations and other discrete dynamical systems) exhibit the Devron property. This means that if a polygon ${\displaystyle P}$ is singular for some iterate of the pentagram map ${\displaystyle T^{m}}$, then it will also be singular for some iterate of the inverse map ${\displaystyle T^{-m'}}$.

Polygons in general positions

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

Corrugated polygons

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

${\displaystyle T_{\text{cor}}v_{k}:={\overline {v_{k}v_{k+d}}}\cap {\overline {v_{k+1}v_{k+d+1}}}.}$

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

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

Grassmannian polygons

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

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

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

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.

Notes