Pentagram map

In mathematics, the pentagram map is a discrete dynamical system on the moduli space of polygons in the projective plane.

[3] The pentagram map is similar in spirit to the constructions underlying Desargues' theorem and Poncelet's porism.

It echoes the rationale and construction underlying a conjecture of Branko Grünbaum concerning the diagonals of a polygon.

On a basic level, one can think of the pentagram map as an operation defined on convex polygons in the plane.

From a more sophisticated point of view, the pentagram map is defined for a polygon contained in the projective plane over a field provided that the vertices are in sufficiently general position.

There is a perfectly natural way to label the vertices of the second iterate of the pentagram map by consecutive integers.

This is why it took people so long to figure out that the Earth was not flat; on small scales one cannot easily distinguish a sphere from a plane.

The Poisson bracket, discussed below, is a more sophisticated math gadget that sort of encodes the local geometry of the tori.

The inverse cross ratio is used in order to define a coordinate system on the moduli space of polygons, both ordinary and twisted.

When one defines the corner invariants of a twisted polygon, one obtains a 2N-periodic bi-infinite sequence of numbers.

There is a second set of coordinates for the moduli space of twisted polygons, developed by Sergei Tabachnikov and Valentin Ovsienko.

The (ab) coordinates bring out the close analogy between twisted polygons and solutions of 3rd order linear ordinary differential equations, normalized to have unit Wronskian.

[5] The equations work more gracefully when one considers the second iterate of the pentagram map, thanks to the canonical labelling scheme discussed above.

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.

The labels on the horizontal edges are simply auxiliary variables introduced to make the formulas simpler.

The pentagram map, when acting on the moduli space X of convex polygons, has an invariant volume form.

These two properties combine with the 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.[9] 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.

In a 2010 paper,[6] Valentin Ovsienko, Richard Schwartz and Sergei Tabachnikov produced a Poisson bracket on the space of twisted polygons which is invariant under the pentagram map.

on the moduli space, we have the so-called Hamiltonian vector field where a summation over the repeated indices is understood.

In practical terms, the fact that the monodromy invariants Poisson-commute means that the corresponding Hamiltonian vector fields define commuting flows.

The Hamiltonian vector fields associated to the remaining monodromy invariants generically span the tangent distribution to the iso-monodromy foliation.

The fact that the monodromy invariants Poisson-commute means that these vector fields define commuting flows.

These flows in turn define local coordinate charts on each iso-monodromy level such that the transition maps are Euclidean translations.

In a 2011 preprint,[10] Fedor Soloviev showed that the pentagram map has a Lax representation with a spectral parameter, and proved its algebraic-geometric integrability.

This means that the space of polygons (either twisted or ordinary) is parametrized in terms of a spectral curve with marked points and a divisor.

The algebraic-geometric methods guarantee that the pentagram map exhibits 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 (i.e., the variables that determine the polygon as explicit functions of time).

Max Glick used the cluster algebra formalism to find formulas for the iterates of the pentagram map in terms of alternating sign matrices.

When the time parameter is suitably chosen, the continuous limit of the pentagram map is the classical Boussinesq equation.

This geometric description makes it fairly obvious that the B-equation is the continuous limit of the pentagram map.

[14][15] In a 2010 paper [12] Max Glick identified the pentagram map as a special case of a cluster algebra.