The Limit Point of the Pentagram Map

Glick, Max

doi:10.1093/imrn/rny110

Cited by 11 publications

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“…In 1992 I wrote a paper [13] defining the pentagram map for general n-gons and proving in the convex case that the forward orbit shrinks to a point. Very recently, M. Glick [2] found a kind of formula for this collapse point.…”

Section: Contextmentioning

confidence: 95%

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A Textbook Case of Pentagram Rigidity

Schwartz¹

2021

Preprint

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Section: Contextmentioning

confidence: 95%

“…Each of these two level sets is the disjoint union of a diagonal line and a circle. The map T 2 3 preserves each component and acts there with order 3. For example…”

Section: Special Casesmentioning

confidence: 99%

A Textbook Case of Pentagram Rigidity

Schwartz¹

2021

Preprint

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“…The pentagram map, T 2 , is one of the best known discrete completely integrable system. See for instance, [1], [2], [3], [8], [9], [10], [11], [12], [14], [15], [21], [22]. The Pentagram Rigidity Conjecture would be a statement about the global topology of an abelian foliation associated to a completely integrable system if the map T 3 were known to be completely integrable.…”

Section: Contextmentioning

confidence: 99%

Pentagram Rigidity for Centrally Symmetric Octagons

Schwartz¹

2021

Preprint

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In this paper I will establish a special case of a conjecture that intertwines the deep diagonal pentagram maps and Poncelet polygons. The special case is that of the 3-diagonal map acting on affine equivalence classes of centrally symmetric octagons. This is the simplest case that goes beyond an analysis of elliptic curves. The proof involves establishing that the map is Arnold-Liouville integrable in this case, and then exploring the Lagrangian surface foliation in detail.

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“…The deeper diagonal maps D k , k > 2, are also completely integrable. Incidentally, the Poncelet grid theorem implies that the pentagram map sends a Poncelet polygon to a projectively equivalent one.2) As a by-product of the study of the pentagram map, eight new projective configuration theorems were found in [17]; see also [20].3) To quote from Kasner [9], Among the corollaries ... the most interesting is this (valid for at least convex pentagons): The limit point of the sequence of successive inscribed pentagons coincides with the limit point of the sequence of successive diagonal pentagons.In this respect, see [7] for a recent striking result on the limiting point of the pentagram map acting on convex n-gons (before factorization by the projective group). 4) Poncelet theorem and its ramifications continue to be an active research area.…”

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confidence: 99%

“…In this respect, see [7] for a recent striking result on the limiting point of the pentagram map acting on convex n-gons (before factorization by the projective group). 4) Poncelet theorem and its ramifications continue to be an active research area.…”

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Kasner Meets Poncelet

Tabachnikov

2019

Math Intelligencer

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Given a planar pentagon P , construct two new pentagons, D(P ) and I(P ): the vertices of D(P ) are the intersection points of the diagonals of P , and the vertices of I(P ) are the tangency points of the conic inscribed in P (the vertices and the tangency points are taken in their cyclic order). The following result is due to E. Kasner [9] (published in 1928, but discovered much earlier, in 1896). Theorem 1 The two operations on pentagons, D and I, commute: ID(P ) = DI(P ), see Figure 1.Figure 1: Kasner's theorem.A polygon inscribed into a conic and circumscribed about a conic is called a Poncelet polygon. The celebrated Poncelet porism states that the existence of a Poncelet n-gon on a pair of nested ellipses implies that every point of the outer ellipse is a vertex of such a Poncelet n-gon, see, e.g., [2,4].Since a (generic) quintuple of points lies on a conic, a (generic) quintuple of lines is tangent to a conic, a (generic) pentagon is a Poncelet polygon. We generalize Kasner's theorem from pentagons to Poncelet polygons.Let P be a Poncelet n-gon, n ≥ 5. As before, the n-gon I(P ) is formed by the consecutive tangency points of the sides of P with the inscribed conic. Fix 2 ≤ k < n/2, and draw the k-diagonals of P , that is, connect ith vertex with (i + k)th vertex for i = 1, . . . , n. The consecutive intersection points of these diagonals form a new n-gon, denoted by D k (P ). Theorem 2 The two operations on Poncelet polygons commute: ID k (P ) = D k I(P ), see Figure 2.Figure 2: Theorem 2: n = 7, k = 3.We shall deduce Theorem 2 from the Poncelet grid theorem [15] which we now describe. The statements slightly differ for odd and even n, and we assume that n is odd.2 Let 1 , . . . , n be the lines containing the sides of a Poncelet n-gon, enumerated in such a way that their tangency points with the inscribed ellipse are in the cyclic order. The Poncelet grid is the collection of n(n + 1)/2 points i ∩ j , where i ∩ i is, by definition, the tangency point of the line i with the inscribed ellipse.Partition the Poncelet grid in two ways. Define the setswhere the indices are understood mod n. One has (n + 1)/2 sets Q k , each containing n points, and n sets R k , each containing (n + 1)/2 points. These sets are called concentric and radial, respectively, see Figure 3.Figure 3: Poncelet grid, n = 9: the concentric sets Q 0 , Q 2 , Q 3 , and Q 4 are shown.Theorem 3 (i) The concentric sets lie on nested ellipses, and the radial sets lie on disjoint hyperbolas.(ii) The complexifications of these conics have four common tangent lines (invisible in Figure 3). (iii) All the concentric sets are projectively equivalent to each other, and so are all the radial sets.The Poncelet grid theorem is a result in projective geometry. One can apply a projective transformation so that the initial two ellipses that support the Poncelet polygon become confocal (this is the case in Figure 2). Under 3 this assumption, an Euclidean version of the Poncelet grid theorem, proved in [11], asserts that all the concentric and the radial sets ...

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The Limit Point of the Pentagram Map

Cited by 11 publications

References 13 publications

A Textbook Case of Pentagram Rigidity

A Textbook Case of Pentagram Rigidity

Pentagram Rigidity for Centrally Symmetric Octagons

Kasner Meets Poncelet

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