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| {{Technical|date=June 2011}}
| | Hi, everybody! My name is Maude. <br>It is a little about myself: I live in Brazil, my city of Itajuba. <br>It's called often Northern or cultural capital of MG. I've married 1 years ago.<br>I have two children - a son (Lela) and the daughter (Larhonda). We all like Home Movies.<br><br>Here is my page: [http://fungonline.com/profile/124586/jumaestas Here is your mountain bike sizing.] |
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| In [[mathematics]], the '''pentagram map''' is a discrete [[dynamical system]] on the [[moduli space]] of [[polygons]] in the [[projective plane]]. The [[pentagram]] map takes a given polygon, finds the intersections of the shortest diagonals of the polygon, and constructs a new polygon from these intersections.
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| [[Richard Schwartz]] introduced the pentagram map for a general polygon in a 1992 paper
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| <ref name=SCH1>{{cite journal|title =The Pentagram Map|url =http://www.expmath.org/expmath/volumes/1/1.html|author =Schwartz, Richard Evan |journal=[[Experimental Mathematics (journal)|Journal of Experimental Math]] |year=1992 |volume=1|pages=90–95}}</ref>
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| though it seems that the special case, in which
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| the map is defined for [[pentagons]] only, goes back to an 1871 paper of [[Alfred Clebsch]]<ref name = CLE>{{cite journal|title=Ueber das ebene Funfeck|journal = Mathematische Annalen|volume=4|year = 1871|pages= 476–489|author=A. Clebsch}}</ref> and a 1945 paper of [[Theodore Motzkin]].<ref name=MOT>{{cite journal|doi=10.1090/S0002-9904-1945-08488-2|title=The pentagon in the projective plane, with a comment on Napier's rule|journal=[[Bulletin of the American Mathematical Society]]|volume=51|issue=12|year=1945|pages=985–989|author=Th. Motzkin|authorlink=Theodore Motzkin}}</ref>
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| 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
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| the diagonals of a polygon.
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| <ref name=ZAK>{{cite journal|title = On the products of cross-ratios on diagonals of polygons|author= Zaks, Joseph|url=http://www.springerlink.com/content/p592345k82444x61/|journal=[[Geometriae Dedicata]] |volume=60 |number=2 |pages=145–151 |doi=10.1007/BF00160619 |accessdate= 2010-02-12}}</ref>
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| ==Definition of the map==
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| ===Basic construction===
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| Suppose that the [[vertex (geometry)|vertices]] of the [[polygon]] P are given by <math> P_1,P_3,P_5,\ldots </math> The image of P under the pentagram map is the
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| polygon Q with vertices <math> Q_2,Q_4,Q_6,\ldots</math> as shown in the figure. Here <math> Q_4 </math> is the intersection of the diagonals
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| <math> (P_1P_5)</math> and <math>(P_3P_7) </math>, and so on.
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| [[File:penga3.svg|border|right|300px|test]]
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| On a basic level, one can think of the pentagram map as an operation defined on [[convex set|convex]] polygons in the [[Plane (geometry)|plane]]. From a more
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| sophisticated point of view,
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| the pentagram map is defined for a polygon contained in the [[projective plane]] over a [[Field (mathematics)|field]] provided that
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| the [[vertex (geometry)|vertices]] are in sufficiently [[general position]].
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| The pentagram map [[Commutative property|commutes]] with [[projective transformations]] and thereby induces a [[Map (mathematics)|mapping]] on the
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| [[moduli space]] of projective [[equivalence classes]] of polygons.
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| ===Labeling conventions===
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| The map <math> P \to Q </math> is slightly problematic, in the sense that the
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| indices of the P-vertices are naturally odd integers whereas the indices of
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| Q-vertices are naturally even
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| integers. A more conventional approach to the labeling would be to label the
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| vertices of P and Q by integers of the same parity. One can arrange this
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| either by adding or subtracting 1 from each of the indices of the Q-vertices.
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| Either choice is equally canonical. An even more conventional choice
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| would be to label the vertices of P and Q by consecutive integers, but
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| again there are 2 natural choices for how to align these labellings:
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| Either <math> Q_k </math> is just clockwise from <math> P_k </math>
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| or just counterclockwise. In most papers on the subject, some choice
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| is made once and for all at the beginning of the paper and then the
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| formulas are tuned to that choice.
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| There is a perfectly natural way to label the vertices of the
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| second iterate of the pentagram map by consecutive integers. For
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| this reason, the second iterate of the pentagram map is more
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| naturally considered as an iteration defined on labeled polygons.
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| See the figure.
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| [[File:Penta8.svg|border|right|300px]]
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| ===Twisted polygons===
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| The pentagram map is also defined on the larger space of
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| twisted polygons.<ref name=SCH2/>
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| A twisted N-gon is a bi-infinite sequence of
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| points in the projective plane that is N-periodic modulo a [[projective transformation]]
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| That is, some projective transformation M carries
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| <math> P_k </math> to <math> P_{N+k} </math> for all k.
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| The map M is called the [[monodromy]] of the twisted N-gon.
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| When M is the identity, a twisted N-gon can be interpreted
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| as an ordinary N-gon whose vertices have been listed
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| out repeatedly. Thus, a twisted N-gon is a generalization
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| of an ordinary N-gon.
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| Two twisted N-gons are equivalent if a projective transformation
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| carries one to the other. The moduli space of
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| twisted N-gons is the set of equivalence classes of
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| twisted N-gons. The space of twisted N-gons contains
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| the space of ordinary N-gons as a sub-variety of
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| co-dimension 8.<ref name=SCH2/><ref name=OST1/>
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| ==Elementary properties==
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| ===Action on pentagons and hexagons===
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| The pentagram map is the identity on the moduli space of [[pentagon]]s.
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| <ref name=SCH1/>
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| <ref name = CLE/>
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| <ref name=MOT/>
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| This is to say that there is always a [[projective transformation]] carrying a
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| pentagon to its image under the pentagram map. It
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| is very likely that this (easy) result was known to the 19th century projective geometers.
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| Indeed, one can deduce this result from a theorem
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| of [[Jean Gaston Darboux|Darboux]] concerning [[Jean-Victor Poncelet|Poncelet]] polygons {{Citation needed|date=June 2011}}
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| The map <math>T^2</math> is the identity on the space of labeled
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| [[hexagon]]s.<ref name=SCH1/>
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| Here T is the second iterate of the pentagram map, which
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| acts naturally on labeled hexagons, as described above. This
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| is to say that the hexagons <math> H </math> and
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| <math> T^2(H) </math> are equivalent by a label-preserving
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| [[projective transformation]]. More precisely, the
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| hexagons <math> H' </math> and <math> T(H) </math> are
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| projectively equivalent, where <math> H' </math> is the labeled
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| hexagon obtained from <math> H </math> by shifting the labels by 3.
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| <ref name=SCH1/>
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| See the figure.
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| It seems entirely possible that this fact was also known | |
| in the 19th century.
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| [[File:penta hexagon.svg|border|right|300px]]
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| The action of the pentagram map on pentagons and hexagons is similar in spirit to
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| classical configuration theorems in projective geometry such as [[Pascal's theorem]],
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| [[Desargues's theorem]] and others.
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| <ref>{{cite arXiv|title = Elementary Surprises in Projective Geometry|first1= Richard Evan |last1=Schwartz |first2=Serge |last2=Tabachnikov |date=October 2009|eprint=0910.1952}}</ref>
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| ===Exponential shrinking===
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| The iterates of the pentagram map shrink any [[convex polygon]] exponentially fast to a point.
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| <ref name=SCH1/>
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| This is to say that the diameter of
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| the nth iterate of a convex polygon is less than
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| <math> K a^n </math>
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| for constants <math> K>0 </math> and
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| <math> 0<a<1 </math> which depend
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| on the initial polygon.
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| Here we are taking about the geometric
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| action on the polygons themselves, not on the moduli
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| space of projective equivalence classes of polygons.
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| ==Motivating discussion==
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| This section is meant to give a non-technical overview for much of the remainder of the article.
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| The context for the pentagram map is [[projective geometry]].
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| Projective geometry is the geometry of our vision. When one looks at the top of a glass,
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| which is a [[circle]], one typically sees an [[ellipse]]. When one looks at a [[rectangular]]
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| door, one sees a typically non-rectangular [[quadrilateral]]. [[Projective transformations]] convert between the
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| various shapes one can see when looking at same object from different points of view. This is why it
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| plays such an important role in old topics like [[perspective drawing]] and new ones like [[computer vision]].
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| Projective geometry is built around the fact that a straight [[line (geometry)|line]] looks
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| like a straight line from any perspective. The straight lines are the building blocks for the subject.
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| The pentagram map is defined entirely in terms of points and straight lines.
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| This makes it adapted to projective geometry. If you look at the pentagram
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| map from another point of view (''i.e.'', you tilt the paper on which it is drawn) then
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| you are still looking at the pentagram map. This explains the statement that the
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| pentagram map commutes with projective transformations.
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| The pentagram map is fruitfully considered as a [[Map (mathematics)|mapping]] on the
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| moduli space of [[polygons]].
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| A [[moduli space]] is an auxiliary space whose points index other objects.
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| For example, in [[Euclidean geometry]], the sum of the angles of a [[triangle]] is
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| always 180 degrees. You can specify a [[triangle]] (up to scale) by giving
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| 3 positive numbers, <math> x,y,z </math> such that <math> x+y+z =180. </math>
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| So, each point <math> (x,y,z) </math>, satisfying the constraints just mentioned,
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| indexes a triangle (up to scale). One might say that <math> (x,y,z) </math> are
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| coordinates for the moduli space of scale equivalence classes of triangles.
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| If you want to index all possible quadrilaterals, either up to scale or not, you
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| would need some additional [[parameters]]. This would lead to a higher [[dimension]]al
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| moduli space. The moduli space relevant to the pentagram map
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| is the moduli space of projective equivalence classes of polygons. Each point
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| in this space corresponds to a polygon, except that two polygons which are
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| different views of each other are considered the same. Since the pentagram
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| map is adapted to projective geometry, as mentioned above, it induces a
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| [[Map (mathematics)|mapping]] on this particular moduli space. That is, given any point
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| in the moduli space, you can apply the pentagram map to the corresponding
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| polygon and see what new point you get.
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| The reason for considering what the pentagram map does to the moduli
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| space is that it gives more salient features of the map. If you just watch,
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| geometrically, what happens to an individual polygon, say a [[convex polygon]], then repeated application shrinks the polygon to a point.<ref name=SCH1/>
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| To see things more clearly, you might dilate the shrinking family of
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| polygons so that they all have, say, the same [[area]]. If you do this,
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| then typically you will see that the family of polygons gets long and
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| thin.<ref name=SCH1/> Now you can change the [[aspect ratio]]
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| so as to try to get yet a better view of these polygons. If you do this
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| process as systematically as possible, you find that you are simply
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| looking at what happens to points in the moduli space. The attempts
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| to zoom in to the picture in the most perceptive possible way lead
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| to the introduction of the moduli space.
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| To explain how the pentagram map acts on the moduli space, one must say a few words about the [[torus]].
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| One way to roughly define the torus is to say that it is the surface of an idealized [[donut]].
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| Another way is that it is the playing field for the [[Asteroids (video game)|Asteroids]] video game.
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| Yet another way to describe the torus is to say that it is a computer screen with wrap, both left-to-right
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| and up-to-down.
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| The [[torus]] is a classical example of what is known in mathematics as a [[manifold]].
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| This is a space that looks somewhat like ordinary [[Euclidean space]] at each point, but
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| somehow is hooked together differently. A [[sphere]] is another example of a manifold.
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| This is why it took people so long to figure out that the [[Earth]] was not flat; on
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| small scales one cannot easily distinguish a sphere from a [[Plane (geometry)|plane]]. So, too, with
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| manifolds like the torus. There are higher dimensional tori as well.
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| You could imagine playing Asteroids in your room, where you can freely go through
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| the walls and ceiling/floor, popping out on the opposite side.
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| One can do experiments with the pentagram map, where one looks at how
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| this mapping acts on the moduli space of polygons. One starts with a point
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| and just traces what happens to it as the map is applied over and over
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| again. One sees a surprising thing: These points seem to line up along
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| multi-dimensional tori.<ref name=SCH1/> These invisible tori fill
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| up the moduli space somewhat like the way
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| the layers of an onion fill up the onion itself, or how the
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| individual cards in a deck fill up the deck. The technical statement
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| is that the tori make a [[foliation]] of the moduli space. The
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| tori have half the dimension of the moduli space. For instance,
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| the moduli space of <math> 7 </math>-gons is <math> 6 </math> dimensional and the
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| tori in this case are <math> 3 </math> dimensional.
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| The tori are invisible [[subsets]] of the moduli space. They are
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| only revealed when one does the pentagram map and watches a point
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| move round and round, filling up one of the tori.
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| Roughly speaking, when [[dynamical systems]] have these invariant
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| tori, they are called [[integrable systems]].
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| Most of the results in this article have to
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| do with establishing that the pentagram map is an integrable system, that
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| these tori really exist.
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| The monodromy invariants, discussed
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| below, turn out to be the equations for the tori. The Poisson bracket, discussed below,
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| is a more sophisticated math gadget that sort of encodes the local geometry
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| of the tori. What is nice is that the various objects fit together exactly, and
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| together add up to a proof that this torus motion really exists.
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| ==Coordinates for the moduli space==
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| ===Cross-ratio===
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| When the field underlying all the constructions is ''F'', the [[affine line]] is just a copy of ''F''. The affine line is a subset of the [[projective line]]. Any finite list of points in the projective line can be moved into the affine
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| line by a suitable [[projective transformation]].
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| Given the four points <math> t_1,t_2,t_3,t_4 </math> in the affine line one
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| defines the (inverse) [[cross ratio]]
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| : <math> X=\frac{(t_1 - t_2)(t_3 - t_4)}{(t_1 - t_3)(t_2 - t_4)}. </math>
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| Most authors consider 1/X to be the [[cross-ratio]], and that is why X is called the inverse cross ratio. The inverse cross ratio is invariant under projective transformations and thus makes sense for points in the projective line. However,
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| the formula above only makes sense for points in the affine line.
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| In the slightly more general set-up below, the cross ratio makes sense
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| for any four collinear points in [[projective space]] One just identifies the
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| line containing the points with the projective line by a suitable [[projective transformation]] and then uses the formula above.
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| The result is independent of any choices made in the identification.
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| The inverse cross ratio is used in order to define a coordinate system on the moduli space
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| of polygons, both ordinary and twisted.
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| ===The corner coordinates===
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| The corner invariants are basic coordinates on the space of twisted polygons.<ref name=SCH2/><ref name=OST1/><ref name = ST1/>
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| Suppose that P is a [[polygon]]. A [[Flag (geometry)|flag]] of P is a pair (p,L), where p is a vertex of P and L is an adjacent line of P.
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| Each vertex of P is involved in 2 flags, and likewise each edge of P is involved in 2 flags.
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| The flags of P are ordered according to the orientation of P, as shown in the figure.
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| In this figure, a flag is represented by a thick arrow. Thus, there are 2N flags associated
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| to an N-gon.
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| [[File:Penta flag2.svg|border|right|300px]]
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| [[File:Penta corner7.svg|border|right|300px]]
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| Let P be an ''N''-gon, with flags <math> F_1,\ldots,F_{2N} </math>
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| To each flag F, we associate the inverse cross ratio of the points <math> t_1,t_2,t_3,t_4</math> shown in the figure
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| at left.
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| In this way, one associates numbers <math> x_1,\ldots,x_{2n} </math> to an n-gon. If two n-gons are
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| related by a projective transformation, they get the same coordinates. Sometimes the
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| variables <math> x_1,y_1,x_2,y_2,\ldots </math> are used in place of
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| <math> x_1,x_2,x_3,x_4,\ldots\,. </math>
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| The corner invariants make sense on the moduli space of twisted polygons.
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| When one defines the corner invariants of a twisted polygon, one obtains
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| a 2N-periodic bi-infinite sequence of numbers. Taking one period
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| of this sequence identifies a twisted N-gon with a point
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| in <math> F^{2N} </math> where F is the underlying field.
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| Conversely, given almost any (in the sense of [[measure theory]]) point in
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| <math> F^{2N} </math> one can construct a twisted N-gon having
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| this list of corner invariants. Such a list will not always give rise
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| to an ordinary polygon; there are an additional 8 equations which
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| the list must satisfy for it to give rise to an ordinary N-gon.
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| ===(ab) coordinates===
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| There is a second set of coordinates for the moduli space of twisted polygons,
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| developed by Sergei Tabachnikov and Valentin Ovsienko.
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| <ref name=OST1>{{cite journal|title = The Pentagram Map, A Discrete Integrable System|first1=Valentin |last1=Ovsienko |first2=Richard Evan |last2=Schwartz |first3=Serge |last3=Tabachnikov |url=http://math.univ-lyon1.fr/~ovsienko/Publis/Penta.pdf |format=pdf |journal=Comm. Math. Phys. 299 |year=2010 |issue=2 |pages=409–446 |accessdate=June 26, 2011}}</ref>
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| One describes a polygon in the [[projective plane]] by a sequence of vectors <math> \ldots V_1,V_2,V_3,\ldots </math> in
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| <math> R^3 </math> so that each consecutive triple of vectors
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| spans a [[parallelopiped]] having unit volume. This leads to the
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| relation
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| * <math> V_{i+3} = a_i V_{i+2} + b_i V_{i+1} + V_i </math>
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| The coordinates <math> a_1,b_1,a_2,b_2,\ldots </math>
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| serve as coordinates for the moduli space of twisted
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| N-gons as long as N is not divisible by 3.
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| The (ab) coordinates bring out the close analogy between twisted polygons
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| and solutions of 3rd order linear [[ordinary differential equations]], normalized
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| to have unit [[Wronskian]].
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| | |
| ==Formula for the pentagram map==
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| ===As a birational mapping ===
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| Here is a formula for the pentagram map, expressed in
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| corner coordinates.<ref name=SCH2/> The
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| equations work more gracefully when one considers the second
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| iterate of the pentagram map, thanks to the
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| canonical labelling scheme discussed above. The second iterate of the
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| pentagram map is the [[function composition|composition]] <math> B \circ A</math>.
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| The maps <math> A </math> and <math> B </math> are [[birational mapping]]s
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| of order 2, and have the following action.
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| *<math> A(x_1,\ldots,x_{2N})=(a_1,\ldots,a_{2N}) </math>
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| *<math> B(x_1,\ldots,x_{2N})=(b_1,\ldots,b_{2N}) </math>
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| where
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| *<math> a_{2k-1}=\frac{(1-x_{2k+1}x_{2k+2})}{(1-x_{2k-3}x_{2k-2})} x_{2k+0}</math>
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| | |
| *<math> a_{2k+0}=\frac{(1-x_{2k-3}x_{2k-2})}{(1-x_{2k+1}x_{2k+2})} x_{2k-1}</math>
| |
| | |
| *<math> b_{2k+1}=\frac{(1-x_{2k-2}x_{2k-1})}{(1-x_{2k+2}x_{2k+3})} x_{2k+0}</math>
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| | |
| *<math> b_{2k+0}=\frac{(1-x_{2k+2}x_{2k+3})}{(1-x_{2k-2}x_{2k-1})} x_{2k-1}</math>
| |
| | |
| (Note: the index 2k+0 is just 2k. The 0 is added to align the formulas.)
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| In these coordinates, the pentagram map is a birational mapping of <math> F^{2N} </math>
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| | |
| ===As grid compatibility relations===
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| [[File:penta relations2.svg|border|300px|right]]
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| The formula for the pentagram map has a convenient interpretation as
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| a certain compatibility rule for labelings on the [[edge (geometry)|edges]] of triangular grid,
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| as shown in the figure.<ref name=SCH2/> In this interpretation, the corner invariants of a polygon
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| P label the non-horizontal edges of a single row, and then the non-horizontal
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| edges of subsequent rows are labeled by the corner invariants of
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| <math> A(P) </math>,
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| <math>B(A(P))</math>,
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| <math> A(B(A(P))) </math>,
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| and so forth. the compatibility rules are
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| * c=1-ab
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| * wx=yz
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| These rules are meant to hold for all configurations which are
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| [[isometry|congruent]] to the ones shown in the figure.
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| In other words, the figures involved in the relations can be
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| in all possible positions and orientations.
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| The labels on the horizontal edges are simply
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| auxiliary variables introduced to make the formulas simpler.
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| Once a single row of non-horizontal edges is provided,
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| the remaining rows are uniquely determined by the
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| compatibility rules.
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| ==Invariant structures==
| |
| | |
| ===Corner coordinate products===
| |
| | |
| It follows directly from the formula for the pentagram map, in terms of corner coordinates,
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| that the two quantities
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| *<math> O_N= x_1x_3\cdots x_{2N-1} </math>
| |
| *<math> E_N = x_2x_4\cdots x_{2N} </math>
| |
| are invariant under the pentagram map.
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| This observation is closely related to the 1991 paper of Joseph Zaks
| |
| <ref name=ZAK/> concerning the diagonals of a polygon.
| |
| | |
| When ''N'' = 2''k'' is even, the functions
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| *<math> O_k = x_1x_5x_9 \cdots x_{2N-3}+ x_3x_7x_{11} \cdots x_{2N-1}</math>
| |
| *<math> E_k = x_2x_6x_{10} \cdots x_{2N-2}+ x_4x_8x_{12} \cdots x_{2N}</math>
| |
| are likewise seen, directly from the formula, to be invariant
| |
| functions. All these products turn out
| |
| to be [[Casimir invariant]]s with respect to the invariant
| |
| Poisson bracket discussed below. At the same time,
| |
| the functions <math> O_k </math> and <math> E_k </math> are
| |
| the simplest examples of the monodromy invariants defined below.
| |
| | |
| The [[level sets]] of the function
| |
| <math> f=O_NE_N </math> are [[Compact space|compact]], when f is restricted to
| |
| the moduli space of real [[convex polygon]]s.
| |
| <ref name=SCH1/>
| |
| Hence, each orbit
| |
| of the pentagram map acting on this space has a [[Compact space|compact]] [[Closure (mathematics)|closure]].
| |
| | |
| ===Volume form===
| |
| | |
| The pentagram map, when acting on the moduli space X of
| |
| convex polygons, has an invariant [[volume form]].
| |
| <ref name=SCH3>{{cite journal|title =Recurrence of the Pentagram Map|url =http://www.expmath.org/expmath/volumes/10/10.4/Schwartz.pdf|author =Schwartz, Richard Evan |journal=Journal of Experimental Math |format=pdf |year=2001 |volume=10.4 |pages=519–528 |accessdate=June 30, 2011}}</ref>
| |
| At the same time, as was already mentioned, the function <math>f=O_NE_N </math> has
| |
| [[compact]]{{disambiguation needed|date=November 2012}} [[level sets]] on X. 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.<ref name=SCH3/>
| |
| 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.
| |
| (It is important to remember that one is considering the projective
| |
| equivalence classes of convex polygons. The fact that the pentagram map
| |
| visibly shrinks a convex polygon is irrelevant.)
| |
| | |
| It is worth mentioning that the recurrence result is
| |
| subsumed by the complete integrability results discussed below.<ref name=OST1/><ref name=SOL/>
| |
| | |
| ===Monodromy invariants===
| |
| | |
| The so-called monodromy invariants are a collection of [[Function (mathematics)|functions]] on the [[moduli space]] that are invariant under the pentagram map.
| |
| <ref name=SCH2>{{cite journal|title = Discrete monodromy, pentagrams, and the method of condensation|author= Schwartz, Richard Evan|journal=
| |
| journal of Fixed Point Theory and Applications (2008)|url= http://www.springerlink.com/content/627311749037p274/|accessdate= 2010-02-12}}</ref>
| |
| | |
| With a view towards defining the monodromy invariants,
| |
| say that a block is either a single integer
| |
| or a triple of consecutive integers, for instance 1 and 567. Say that a block is odd if it starts with | |
| an odd integer. Say that two blocks are well-separated if they have at least 3 integers between them.
| |
| For instance 123 and 567 are not well separated but 123 and 789 are well separated. Say that an
| |
| odd admissible sequence is a finite sequence of integers that decomposes into well separated odd blocks.
| |
| When we take these sequences from the set 1, ..., 2''N'', the notion of well separation is meant in the
| |
| cyclic sense. Thus, 1 and 2N-1 are not well separated.
| |
| | |
| Each odd admissible sequence gives rise to a [[monomial]] in the corner invariants. This is best illustrated by
| |
| example
| |
| * 1567 gives rise to <math> - x_1x_5x_6x_7 </math>
| |
| *123789 gives rise to <math> + x_1x_2x_3x_7x_8x_9 </math>
| |
| The sign is determined by the [[Parity (mathematics)|parity]] of the
| |
| number of single-digit blocks in the sequence.
| |
| The monodromy invariant <math> O_k </math> is defined as the sum of all
| |
| monomials coming from odd admissible sequences composed of k blocks.
| |
| The monodromy invariant <math> E_k </math> is defined the same way,
| |
| with even replacing odd in the definition.
| |
| | |
| When ''N'' is odd, the allowable values of ''k'' are 1, 2, ..., (''n'' − 1)/2. When ''N'' is even, the allowable values of k are 1, 2, ..., ''n''/2. When ''k'' = ''n''/2, one recovers the product invariants discussed above. In both cases, the invariants
| |
| <math> O_N </math> and <math> E_N </math> are counted
| |
| as monodromy invariants, even though they are not produced by the above construction.
| |
| | |
| The monodromy invariants are defined on the space of twisted polygons, and
| |
| restrict to give invariants on the space of closed polygons. They have
| |
| the following geometric interpretation. The monodromy M of a twisted
| |
| polygon is a certain [[rational function]] in the corner coordinates.
| |
| The monodromy invariants are essentially the homogeneous parts of the [[Trace (linear algebra)|trace]]
| |
| of ''M''.
| |
| There is also a description of the monodromy invariants in terms of the (ab) coordinates. In these coordinates, the invariants arise as certain [[determinants]] of 4-diagonal [[matrix (mathematics)|matrices]].
| |
| <ref name=OST1/><ref name=ST1/>
| |
| | |
| Whenever P has all its vertices on a [[conic section]] (such as a circle) one has
| |
| <math>O_k(P)=E_k(P)</math> for all k.
| |
| <ref name=ST1>{{cite journal|title= The pentagram integrals for inscribed polygons|arxiv= 1004.4311|first1= Richard Evan |last1=Schwartz |first2=Sergei |last2=Tabachnikov|journal=[[Electronic Journal of Combinatorics]] |date=October 2009 }}
| |
| </ref>
| |
| | |
| ===Poisson bracket===
| |
| | |
| A [[Poisson bracket]] is an anti-symmetric [[linear]] operator <math> \{\cdot,\cdot\} </math> on the space of functions which satisfies the [[Derivation (abstract algebra)|Leibniz Identity]] and the [[Jacobi identity]].
| |
| In a 2010 paper,<ref name=OST1/>
| |
| Valentin Ovsienko, Richard Schwartz and Sergei Tabachnikov produced a [[Poisson bracket]] on the space of twisted polygons
| |
| which is invariant under the pentagram map. They also showed that monodromy invariants commute with respect to this
| |
| bracket. This is to say that
| |
| * <math> \{O_i,O_j\}=\{O_i,E_j\}=\{E_i,E_j\}=0 </math>
| |
| for all indices.
| |
| | |
| Here is a description of the invariant Poisson bracket in terms of the variables.
| |
| | |
| : <math> x_1,y_1,x_2,y_2,\ldots\,. </math>
| |
| | |
| : <math>\{x_i,x_{i+1}\} = -x_i\, x_{i+1}</math>
| |
| | |
| : <math>\{x_i, x_{i-1}\} = x_i\, x_{i-1}</math>
| |
| | |
| : <math>\{y_i,y_{i+1}\} = y_i\, y_{i+1}</math>
| |
| | |
| : <math>\{y_i,y_{i-1}\} = -y_i\, y_{i-1}</math>
| |
| | |
| : <math>\{x_i,x_j\} = \{y_i,y_j\} = \{x_i,y_j\} = 0 </math> for all other <math> i,j.</math>
| |
| | |
| There is also a description in terms of the (ab) coordinates, but it is more
| |
| complicated.<ref name=OST1/>
| |
| | |
| Here is an alternate description of the invariant bracket.
| |
| Given any function <math> f </math> on the moduli space, we have the so-called
| |
| [[Hamiltonian vector field]]
| |
| * <math> H(f)=(x_{i+1} \partial f/\partial x_{i+1} - x_{i-1} \partial f/\partial x_{i-1})x_i \partial/\partial x_i +
| |
| (y_{i-1} \partial f/\partial y_{i-1} - y_{i+1} \partial f/\partial y_{i+1}) y_i \partial/\partial y_i </math>
| |
| where a summation over the repeated indices is understood.
| |
| Then
| |
| *<math> H(f) g = \{f,g\} </math>
| |
| The first expression is the [[directional derivative]] of <math> g </math> in the direction of the vector field <math> H(f) </math>.
| |
| In practical terms, the fact that the monodromy invariants Poisson-commute means that the
| |
| corresponding Hamiltonian [[vector fields]] define commuting flows.
| |
| | |
| ==Complete integrability==
| |
| | |
| ===Arnold–Liouville integrability===
| |
| | |
| The monodromy invariants and the invariant bracket combine to establish
| |
| Arnold–Liouville integrability of the pentagram map on the space
| |
| of twisted ''N''-gons.
| |
| <ref name=OST1/>
| |
| The situation is easier to describe for N odd.
| |
| In this case, the two products
| |
| * <math> O_n =x_1\cdots x_n </math>
| |
| * <math> E_n = y_1\cdots y_n </math>
| |
| are [[Casimir invariant]]s for the bracket, meaning (in this context) that
| |
| * <math> \{O_n,f\}=\{E_n,f\} =0 </math>
| |
| for all functions f.
| |
| A Casimir [[level set]] is the set of all points in the space having
| |
| a specified value for both <math> O_n </math> and <math> E_n </math>.
| |
| | |
| Each Casimir level set has an iso-monodromy [[foliation]], namely, a
| |
| decomposition into the common level sets of the remaining monodromy functions.
| |
| 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. That is, the Hamiltonian vector fields impart a
| |
| flat Euclidean structure on the iso-monodromy levels, forcing them
| |
| to be flat tori when they are [[Smooth manifold|smooth]] and [[compact space|compact]] [[manifolds]].
| |
| This happens for almost every level set.
| |
| Since everything in sight is pentagram-invariant, the
| |
| pentagram map, restricted to an iso-monodromy leaf,
| |
| must be a translation. This kind of motion is known as
| |
| [[quasi-periodic motion]].
| |
| This explains the Arnold-Liouville integrability.
| |
| | |
| From the point of view of [[symplectic geometry]], the Poisson
| |
| bracket gives rise to a [[symplectic form]] on each Casimir
| |
| level set.
| |
| | |
| ===Algebro-geometric integrability===
| |
| | |
| In a 2011 preprint,
| |
| <ref name=SOL>{{cite arXiv|title = Integrability of the Pentagram Map|eprint = 1106.3950|author=Soloviev, Fedor |year=2011}}</ref>
| |
| 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 (algebraic geometry)|divisor]]. The spectral curve is determined by the monodromy invariants, and the
| |
| divisor corresponds to a point on a torus—the Jacobi variety of the spectral curve.
| |
| 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).
| |
| Soloviev also obtains the invariant Poisson bracket from the Krichever-Phong
| |
| universal formula.
| |
| | |
| ==Connections to other topics==
| |
| | |
| ===The Octahedral recurrence ===
| |
| | |
| The octahedral recurrence is a dynamical system defined on the
| |
| vertices of the octahedral tiling of space. Each octahedron has
| |
| 6 vertices, and these vertices are labelled in such a way that
| |
| *<math> a_1b_1 + a_2b_2 = a_3b_3 </math>
| |
| Here
| |
| <math> a_i </math> and <math> b_i </math> are the labels
| |
| of antipodal vertices. A common convention is that
| |
| <math> a_2,b_2,a_3,b_3 </math> always lie in a central horizontal plane
| |
| and a_1,b_1 are the top and bottom vertices.
| |
| The octahedral recurrence is closely related to [[Lewis Carroll|C. L. Dodgson's]]
| |
| method of condensation for computing [[determinants]].<ref name=SCH2/>
| |
| Typically one labels two horizontal layers of the tiling and
| |
| then uses the basic rule to let the labels propagate dynamically.
| |
| | |
| Max Glick used the [[cluster algebra]] formalism to find formulas for the iterates
| |
| of the pentagram map in terms of [[alternating sign matrix|alternating sign matrices]].<ref name=GLI/> These formulas
| |
| are similar in spirit to the formulas found by [[David P. Robbins]] and Harold Rumsey for the
| |
| iterates of the octahedral recurrence.
| |
| [[File:penta oct9.svg|border|450px|right]]
| |
| Alternatively, the following construction relates the octahedral recurrence
| |
| directly to the pentagram map.
| |
| <ref name=SCH2/>
| |
| Let <math> T </math> be the octahedral tiling. Let
| |
| <math> \pi: T \to R^2 </math> be the [[linear projection]]
| |
| which maps each octahedron in <math> T </math> to the configuration of
| |
| 6 points shown in the first figure.
| |
| Say that an adapted labeling of <math> T </math> is
| |
| a labeling so that all points in the (infinite) [[inverse image]] of any point
| |
| in <math> G=\pi(T) </math> get the same numerical label.
| |
| The octahedral recurrence applied to an adapted labeling
| |
| is the same as a recurrence on <math> G </math>
| |
| in which the same rule as for the octahedral recurrence
| |
| is applied to every configuration of points [[isometry|congruent]]
| |
| to the configuration in the first figure. Call this the
| |
| planar octahedral recurrence.
| |
| | |
| Given a labeling of <math> G </math> which obeys the
| |
| planar octahedral recurrence, one can create a labeling of the edges of
| |
| <math> G </math> by applying the rule
| |
| * <math> v=AD/BC </math>
| |
| to every edge. This rule refers to the figure at right
| |
| and is meant to apply to every configuration that
| |
| is [[isometry|congruent]] to the two shown.
| |
| [[File:penta oct10.svg|border|450px|right]]
| |
| When this labeling is done, the edge-labeling
| |
| of G satisfies the relations for the pentagram map.
| |
| | |
| ===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 classical [[Boussinesq equation]]{{disambiguation needed|date=November 2012}}.<ref name=SCH2/><ref name=OST1/> This equation is a classical example of an
| |
| [[integrable]] [[partial differential equation]].
| |
| | |
| Here is a description of the geometric action of the Boussinesq equation. | |
| Given a [[locally convex]] curve <math> C:R->R^2 </math>, and real numbers x and t, we consider the [[chord (geometry)|chord]]
| |
| connecting <math> C(x-t) </math> to <math> C(x+t) </math>. The envelop of all these chords is a new curve
| |
| <math> C_t(x) </math>. When t is extremely small, the curve <math> C_t(x) </math> is a good model for the time t evolution of the original curve <math> C_0(x) </math> under the Boussinesq equation. This geometric description makes it fairly
| |
| obvious that the B-equation is the continuous limit of the pentagram map.
| |
| At the same time, the pentagram invariant bracket is a discretization of a well known invariant Poisson bracket associated to the Boussinesq equation.
| |
| <ref name=OST1/>
| |
| | |
| Recently, there has been some work on higher dimensional generalizations of the pentagram map and its connections
| |
| to Boussinesq-type partial differential equations
| |
| <ref name=GMB>{{cite journal|url=http://www.math.wisc.edu/~maribeff/pentagrammap1.pdf |title=On Generalizations of the Pentagram Map: Discretizal of AGD Flows |format=pdf|first1=Gloria Marỉ|last1= Beffa |place=Madison, Wisconsin |publisher=University of Wisconsin}}</ref>
| |
| | |
| ===Projectively natural evolution===
| |
| | |
| The pentagram map and the Boussinesq equation are examples of
| |
| projectively natural geometric evolution equations. Such equations arise
| |
| in diverse fields of mathematics, such as [[projective geometry]] and [[computer vision]].
| |
| <ref>{{cite journal|title=On Projective Invariant Smoothing and Evolutions of Planar Curves and Polygons|first1=Alfred M. |last1=Bruckstein |first2=Doron |last2=Shaked |url= http://www.springerlink.com/content/u364060228p51548/fulltext.pdf |format=pdf |journal=Journal of Mathematical Imaging and Vision
| |
| |volume=7 |number=3 |pages=225–240 |doi=10.1023/A:1008226427785 |accessdate= 2010-02-12}}</ref>
| |
| <ref>{{cite journal|title = Differential Invariant Signatures and Flows in Computer Vision: A Symmetry Group Approach|author= Peter J. Olver; Guillermo Sapiro; Allen Tannenbaum; MINNESOTA UNIV MINNEAPOLIS DEPT OF MATHEMATICS|url= http://www.stormingmedia.us/71/7169/A716954.html|accessdate= 2010-02-12}}</ref>
| |
| | |
| ===Cluster algebras===
| |
| | |
| In a 2010 paper
| |
| <ref name=GLI>*{{cite arXiv|title= The Pentagram Map and Y-Patterns|author=Glick, Max |journal=Advances in Mathematics |date=2010 |eprint=arXiv:1005.0598v2 }}</ref>
| |
| Max Glick identified the pentagram map as a special case of a
| |
| [[cluster algebra]].
| |
| | |
| ==See also==
| |
| * [[Combinatorics]]
| |
| * [[Periodic table of shapes]]
| |
| | |
| ==Notes==
| |
| {{reflist}}
| |
| | |
| ==References==
| |
| *{{cite journal|url=http://www.math.wisc.edu/~maribeff/pentagrammap1.pdf |title=On Generalizations of the Pentagram Map: Discretizal of AGD Flows |format=pdf|first1=Gloria Marỉ| last1= Beffa|place=Madison, Wisconsin |publisher=University of Wisconsin}}
| |
| *{{cite journal|title=On Projective Invariant Smoothing and Evolutions of Planar Curves and Polygons|first1=Alfred M. |last1=Bruckstein |first2=Doron |last2=Shaked |url= http://www.springerlink.com/content/u364060228p51548/fulltext.pdf |format=pdf |journal=Journal of Mathematical Imaging and Vision
| |
| |volume=7 |number=3 |pages=225–240 |doi=10.1023/A:1008226427785 |accessdate= 2010-02-12}}
| |
| *{{cite arXiv|title= The Pentagram Map and Y-Patterns|author=Glick, Max |journal=Advances in Mathematics |date=2010 |eprint=arXiv:1005.0598v2 }}
| |
| *{{cite journal|doi=10.1090/S0002-9904-1945-08488-2|title=The pentagon in the projective plane, with a comment on Napier's rule|journal=Bull. Amer. Math. Soc.|volume=51|issue=12|year=1945|pages=985–989|author=Motzkin, Theodore|authorlink=Theodore Motzkin}}
| |
| *{{cite journal|title = Differential Invariant Signatures and Flows in Computer Vision: A Symmetry Group Approach|first1= Peter J. |last1=Olver |first2=Guillermo |last2=Sapiro |first3=Allen |last3=Tannenbaum |publisher=Minnesota University Minneapolis Department of Mathematics|url= http://www.stormingmedia.us/71/7169/A716954.html|year=1993|accessdate= 2010-02-12}}
| |
| *{{cite journal|title = Discrete integrable systems in projective geometry|first1=Valentin |last1=Ovsienko |first2=Serge |last2=Tabachnikov |url= http://www.birs.ca/workshops/2008/08rit125/report08rit125.pdf|accessdate= 2010-02-12}}
| |
| *{{cite journal|title = The Pentagram Map, A Discrete Integrable System|first1=Valentin |last1=Ovsienko |first2=Richard Evan |last2=Schwartz |first3=Serge |last3=Tabachnikov |url=http://math.univ-lyon1.fr/~ovsienko/Publis/Penta.pdf |format=pdf |journal=Comm. Math. Phys. 299 |year=2010 |issue=2 |pages=409–446 |accessdate=June 26, 2011}}
| |
| *{{cite journal|title =Quasiperiodic Motion for the Pentagram Map |url =http://aimsciences.org/journals/pdfs.jsp?paperID=4031&mode=full|format=pdf |first1=Valentin |last1=Ovsienko |first2=Richard Evan |last2=Schwartz |first3=Serge |last3=Tabachnikov|journal=Electron. Res. Announc. Math. Sci. |volume=16 |year=2009 |pages=1–8}}
| |
| *{{cite journal|title =The Pentagram Map|url =https://eudml.org/doc/228847|author =Schwartz, Richard Evan |journal=Journal of Experimental Math |year=1992 |volume=1|pages=90–95}}
| |
| *{{cite journal|title =Recurrence of the Pentagram Map|url =http://www.expmath.org/expmath/volumes/10/10.4/Schwartz.pdf|author =Schwartz, Richard Evan |journal=Journal of Experimental Math |format=pdf |year=2001 |volume=10.4 |pages=519–528 |accessdate=June 30, 2011}}
| |
| *{{cite journal|title = Discrete monodromy, pentagrams, and the method of condensation|author=Schwartz, Richard Evan |journal=
| |
| Journal of Fixed Point Theory and Applications |year=2008|url=http://www.springerlink.com/content/627311749037p274/|accessdate= 2010-02-12}}
| |
| *{{cite journal|title= The pentagram integrals for inscribed polygons|arxiv= 1004.4311|first1=Richard Evan |last1=Schwartz |first2=Serge |last2=Tabachnikov|journal=Electronic Journal of Combinatorics }}
| |
| *{{cite arXiv|title = Elementary Surprises in Projective Geometry|first1=Richard Evan |last1=Schwartz |first2=Serge |last2=Tabachnikov |eprint=0910.1952 |date=2009}}
| |
| *{{cite arXiv|title = Integrability of the Pentagram Map|eprint = 1106.3950|author=Soloviev, Fedor |year=2011}}
| |
| *{{cite journal|title = On the products of cross-ratios on diagonals of polygons|author= Zaks, Joseph|url=http://www.springerlink.com/content/p592345k82444x61/|journal=[[Geometriae Dedicata]] |volume=60 |number=2 |pages=145–151 |doi=10.1007/BF00160619 |accessdate= 2010-02-12}}
| |
| | |
| [[Category:Projective geometry]]
| |
| [[Category:Dynamical systems]]
| |