3394:
1369:. Most of the results in this article have to do with establishing that the pentagram map is an integrable system, that these tori really exist. The monodromy invariants, discussed below, turn out to be the equations for the tori. The Poisson bracket, discussed below, is a more sophisticated math gadget that sort of encodes the local geometry of the tori. What is nice is that the various objects fit together exactly, and together add up to a proof that this torus motion really exists.
5828:
2750:
3445:
6221:, 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
1730:
881:
5512:
6702:
1723:
6526:
343:
3389:{\displaystyle {\begin{aligned}a_{2k-1}&={\frac {(1-x_{2k+1}x_{2k+2})}{(1-x_{2k-3}x_{2k-2})}}x_{2k+0}\\a_{2k+0}&={\frac {(1-x_{2k-3}x_{2k-2})}{(1-x_{2k+1}x_{2k+2})}}x_{2k-1}\\b_{2k+1}&={\frac {(1-x_{2k-2}x_{2k-1})}{(1-x_{2k+2}x_{2k+3})}}x_{2k+0}\\b_{2k+0}&={\frac {(1-x_{2k+2}x_{2k+3})}{(1-x_{2k-2}x_{2k-1})}}x_{2k-1}\end{aligned}}}
510:
4302:. 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. (One is considering the projective equivalence classes of convex polygons. The fact that the pentagram map visibly shrinks a convex polygon is irrelevant.)
5823:{\displaystyle H(f)=\left(x_{i+1}{\frac {\partial f}{\partial x_{i+1}}}-x_{i-1}{\frac {\partial f}{\partial x_{i-1}}}\right)x_{i}{\frac {\partial }{\partial x_{i}}}+\left(y_{i-1}{\frac {\partial f}{\partial y_{i-1}}}-y_{i+1}{\frac {\partial f}{\partial y_{i+1}}}\right)y_{i}{\frac {\partial }{\partial y_{i}}}}
1246:
so as to try to get yet a better view of these polygons. If you do this process as systematically as possible, you find that you are simply looking at what happens to points in the moduli space. The attempts to zoom in to the picture in the most perceptive possible way lead to the introduction of the
3648:
to the ones shown in the figure. In other words, the figures involved in the relations can be in all possible positions and orientations. The labels on the horizontal edges are simply auxiliary variables introduced to make the formulas simpler. Once a single row of non-horizontal edges is provided,
7045:
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.
4324:
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
1225:
moduli space. The moduli space relevant to the pentagram map is the moduli space of projective equivalence classes of polygons. Each point in this space corresponds to a polygon, except that two polygons which are different views of each other are considered the same. Since the pentagram map is
1039:
looks like a straight line from any perspective. The straight lines are the building blocks for the subject. The pentagram map is defined entirely in terms of points and straight lines. This makes it adapted to projective geometry. If you look at the pentagram map from another point of view
1289:
One can do experiments with the pentagram map, where one looks at how this mapping acts on the moduli space of polygons. One starts with a point and just traces what happens to it as the map is applied over and over again. One sees a surprising thing: These points seem to line up along
2465:
Here is a formula for the pentagram map, expressed in corner coordinates. The equations work more gracefully when one considers the second iterate of the pentagram map, thanks to the canonical labelling scheme discussed above. The second iterate of the pentagram map is the
505:
There is a perfectly natural way to label the vertices of the second iterate of the pentagram map by consecutive integers. For this reason, the second iterate of the pentagram map is more naturally considered as an iteration defined on labeled polygons. See the figure.
4325:
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
3455:
of triangular grid, as shown in the figure. In this interpretation, the corner invariants of a polygon P label the non-horizontal edges of a single row, and then the non-horizontal edges of subsequent rows are labeled by the corner invariants of
1290:
multi-dimensional tori. These invisible tori fill up the moduli space somewhat like the way the layers of an onion fill up the onion itself, or how the individual cards in a deck fill up the deck. The technical statement is that the tori make a
436:-vertices are naturally even integers. A more conventional approach to the labeling would be to label the vertices of P and Q by integers of the same parity. One can arrange this either by adding or subtracting 1 from each of the indices of the
4635:
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
1642:
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, the formula above only makes sense for points in the affine line.
1654:
and then uses the formula above. The result is independent of any choices made in the identification. The inverse cross ratio is used in order to define a coordinate system on the moduli space of polygons, both ordinary and twisted.
6225:
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
6269:. 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
1286:. So, too, with manifolds like the torus. There are higher-dimensional tori as well. You could imagine playing Asteroids in your room, where you can freely go through the walls and ceiling/floor, popping out on the opposite side.
3956:
4088:
1625:
6264:
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
2755:
5446:
4906:
2739:
2639:
2138:-gon having this list of corner invariants. Such a list will not always give rise to an ordinary polygon; there are an additional 8 equations which the list must satisfy for it to give rise to an ordinary
1233:
The reason for considering what the pentagram map does to the moduli space is that it gives more salient features of the map. If you just watch, geometrically, what happens to an individual polygon, say a
6237:. 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
2356:
4991:
2054:
5329:
5077:
6374:
5243:
5160:
2218:
6290:
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
4484:
3735:
2431:
1978:
6806:
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
3802:
1217:
are coordinates for the moduli space of scale equivalence classes of triangles. If you want to index all possible quadrilaterals, either up to scale or not, you would need some additional
1044:, you tilt the paper on which it is drawn) then you are still looking at the pentagram map. This explains the statement that the pentagram map commutes with projective transformations.
6151:
224:
157:
6497:
6071:
6017:
1857:
1469:
995:
which depend on the initial polygon. Here we are taking about the geometric action on the polygons themselves, not on the moduli space of projective equivalence classes of polygons.
4402:
1906:
1791:
4789:
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
7057:
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
6277:(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.
4775:
4723:
1238:, then repeated application shrinks the polygon to a point. To see things more clearly, you might dilate the shrinking family of polygons so that they all have, say, the same
6589:
5884:
1230:
on this particular moduli space. That is, given any point in the moduli space, you can apply the pentagram map to the corresponding polygon and see what new point you get.
502:
or just counterclockwise. In most papers on the subject, some choice is made once and for all at the beginning of the paper and then the formulas are tuned to that choice.
6861:
4273:
4199:
337:
294:
993:
3568:
1361:
of the moduli space. They are only revealed when one does the pentagram map and watches a point move round and round, filling up one of the tori. Roughly speaking, when
1139:
1027:
convert between the various shapes one can see when looking at same object from different points of view. This is why it plays such an important role in old topics like
6672:
7043:
7007:
6971:
2493:
772:
6789:
3521:
1215:
1177:
855:
801:
598:
7049:
Recently, there has been some work on higher-dimensional generalizations of the pentagram map and its connections to
Boussinesq-type partial differential equations
6935:
6900:
3606:
3435:
2132:
2094:
961:
935:
426:
6431:
6404:
6212:
6185:
4630:
4603:
4545:
4518:
4149:
4122:
3639:
2245:
1101:
708:
565:
500:
473:
251:
5940:
5475:
3483:
830:
7207:
6746:
6726:
6692:
6633:
6613:
6550:
5911:
5501:
2533:
2513:
1352:
1332:
1312:
875:
736:
3824:
3962:
1480:
2058:
The corner invariants make sense on the moduli space of twisted polygons. When one defines the corner invariants of a twisted polygon, one obtains a 2
5959:
The monodromy invariants and the invariant bracket combine to establish Arnold–Liouville integrability of the pentagram map on the space of twisted
3807:
are invariant under the pentagram map. This observation is closely related to the 1991 paper of Joseph Zaks concerning the diagonals of a polygon.
4648:. There is also a description of the monodromy invariants in terms of the (ab) coordinates. In these coordinates, the invariants arise as certain
887:
The action of the pentagram map on pentagons and hexagons is similar in spirit to classical configuration theorems in projective geometry such as
7750:
6807:
1262:
video game. Yet another way to describe the torus is to say that it is a computer screen with wrap, both left-to-right and up-to-down. The
718:
is the second iterate of the pentagram map, which acts naturally on labeled hexagons, as described above. This is to say that the hexagons
55:
introduced the pentagram map for a general polygon in a 1992 paper though it seems that the special case, in which the map is defined for
7493:
7456:
5335:
4795:
7134:
6499:
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
1003:
This section is meant to give a non-technical overview for much of the remainder of the article. The context for the pentagram map is
2645:
2545:
6273:
on a torus (both in the twisted and the ordinary case), and they allow one to construct explicit solutions formulas using
Riemann
6507:. Typically one labels two horizontal layers of the tiling and then uses the basic rule to let the labels propagate dynamically.
52:
7664:
Bruckstein, Alfred M.; Shaked, Doron (1997). "On
Projective Invariant Smoothing and Evolutions of Planar Curves and Polygons".
2445:
1908:
to an n-gon. If two n-gons are related by a projective transformation, they get the same coordinates. Sometimes the variables
2257:
4287:
877:
by shifting the labels by 3. See the figure. It seems entirely possible that this fact was also known in the 19th century.
6814:
4778:
4520:
is defined as the sum of all monomials coming from odd admissible sequences composed of k blocks. The monodromy invariant
7823:
6266:
4920:
1983:
5249:
4997:
3451:
The formula for the pentagram map has a convenient interpretation as a certain compatibility rule for labelings on the
6674:
get the same numerical label. The octahedral recurrence applied to an adapted labeling is the same as a recurrence on
6296:
5942:. In practical terms, the fact that the monodromy invariants Poisson-commute means that the corresponding Hamiltonian
5166:
5083:
2161:
6798:
to the two shown. When this labeling is done, the edge-labeling of G satisfies the relations for the pentagram map.
4409:
3668:
2364:
1911:
1242:. If you do this, then typically you will see that the family of polygons gets long and thin. Now you can change the
3662:
It follows directly from the formula for the pentagram map, in terms of corner coordinates, that the two quantities
907:
exponentially fast to a point. This is to say that the diameter of the nth iterate of a convex polygon is less than
7828:
3741:
1024:
379:
440:-vertices. Either choice is equally canonical. An even more conventional choice would be to label the vertices of
1651:
1399:
775:
677:
533:
7091:
6532:
Alternatively, the following construction relates the octahedral recurrence directly to the pentagram map. Let
6086:
5504:
170:
103:
7454:
Schwartz, Richard Evan; Tabachnikov, Sergei (October 2009). "The pentagram integrals for inscribed polygons".
6436:
6023:
5969:
1796:
1408:
1294:
of the moduli space. The tori have half the dimension of the moduli space. For instance, the moduli space of
6794:
to every edge. This rule refers to the figure at right and is meant to apply to every configuration that is
6515:
3407:. The 0 is added to align the formulas.) In these coordinates, the pentagram map is a birational mapping of
2444:
The (ab) coordinates bring out the close analogy between twisted polygons and solutions of 3rd order linear
4347:
4093:
are likewise seen, directly from the formula, to be invariant functions. All these products turn out to be
448:
by consecutive integers, but again there are two natural choices for how to align these labellings: Either
5890:
4641:
4314:
1862:
1747:
1259:
892:
643:-gons are equivalent if a projective transformation carries one to the other. The moduli space of twisted
6270:
6238:
4213:
1646:
In the slightly more general set-up below, the cross ratio makes sense for any four collinear points in
4748:
4670:
358:. From a more sophisticated point of view, the pentagram map is defined for a polygon contained in the
68:
7796:
1007:. Projective geometry is the geometry of our vision. When one looks at the top of a glass, which is a
7772:
7471:
7383:
6555:
4653:
4490:
2467:
375:
72:
6694:
in which the same rule as for the octahedral recurrence is applied to every configuration of points
7241:
7058:
6245:
5839:
2062:-periodic bi-infinite sequence of numbers. Taking one period of this sequence identifies a twisted
1715:, as shown in the figure. In this figure, a flag is represented by a thick arrow. Thus, there are 2
1398:. Any finite list of points in the projective line can be moved into the affine line by a suitable
1028:
1004:
888:
391:
363:
6827:
4632:
are counted as monodromy invariants, even though they are not produced by the above construction.
4235:
4161:
1663:
The corner invariants are basic coordinates on the space of twisted polygons. Suppose that P is a
1179:, satisfying the constraints just mentioned, indexes a triangle (up to scale). One might say that
299:
256:
7788:
7762:
7681:
7602:
7579:
7561:
7516:
7461:
7429:
7425:
7399:
7373:
7320:
7302:
7293:
Schwartz, Richard Evan (2008). "Discrete monodromy, pentagrams, and the method of condensation".
7258:
7177:
6261:
4640:
in the corner coordinates. The monodromy invariants are essentially the homogeneous parts of the
2536:
2151:
1222:
1060:
966:
7731:
3526:
1106:
76:
6642:
4097:
with respect to the invariant
Poisson bracket discussed below. At the same time, the functions
7524:
7012:
6976:
6940:
6592:
6222:
6077:
4637:
4094:
2472:
1366:
1362:
1250:
To explain how the pentagram map acts on the moduli space, one must say a few words about the
1227:
1048:
741:
383:
367:
93:
6754:
3488:
1182:
1144:
570:
7780:
7708:
7673:
7638:
7571:
7508:
7391:
7312:
7250:
7216:
7198:
7169:
6905:
6870:
6864:
6615:
to the configuration of 6 points shown in the first figure. Say that an adapted labeling of
6519:
3576:
3410:
2155:
2150:
There is a second set of coordinates for the moduli space of twisted polygons, developed by
2107:
2069:
1647:
1283:
1278:
is another example of a manifold. This is why it took people so long to figure out that the
940:
910:
405:
371:
359:
355:
64:
40:
28:
7701:"Differential Invariant Signatures and Flows in Computer Vision: A Symmetry Group Approach"
7650:
6409:
6382:
6190:
6163:
4608:
4581:
4523:
4496:
4127:
4100:
3612:
2223:
1074:
686:
543:
478:
451:
229:
7646:
7074:
7062:
6511:
6274:
6249:
6227:
5916:
5480:
There is also a description in terms of the (ab) coordinates, but it is more complicated.
5451:
4786:
4782:
4738:
3459:
3452:
1668:
1650:
One just identifies the line containing the points with the projective line by a suitable
1395:
1271:
1036:
1032:
806:
4785:. In a 2010 paper, Valentin Ovsienko, Richard Schwartz and Sergei Tabachnikov produced a
4305:
The recurrence result is subsumed by the complete integrability results discussed below.
7776:
7623:
7475:
7387:
6728:
which obeys the planar octahedral recurrence, one can create a labeling of the edges of
835:
781:
7358:
6821:
6731:
6711:
6677:
6618:
6598:
6535:
5896:
5486:
4578:/2, one recovers the product invariants discussed above. In both cases, the invariants
4206:
2518:
2498:
2248:
2101:
1337:
1317:
1297:
1254:. One way to roughly define the torus is to say that it is the surface of an idealized
1235:
904:
860:
721:
60:
6698:
to the configuration in the first figure. Call this the planar octahedral recurrence.
7817:
7583:
7262:
7181:
7086:
6636:
6500:
6231:
5963:-gons. The situation is easier to describe for N odd. In this case, the two products
4664:
4276:
4210:
4202:
1020:
7792:
7700:
7324:
7221:
7202:
3951:{\displaystyle O_{k}=x_{1}x_{5}x_{9}\cdots x_{2N-3}+x_{3}x_{7}x_{11}\cdots x_{2N-1}}
7685:
7520:
7512:
7403:
6504:
5943:
4649:
4318:
4083:{\displaystyle E_{k}=x_{2}x_{6}x_{10}\cdots x_{2N-2}+x_{4}x_{8}x_{12}\cdots x_{2N}}
1620:{\displaystyle X={\frac {(t_{1}-t_{2})(t_{3}-t_{4})}{(t_{1}-t_{3})(t_{2}-t_{4})}}.}
1243:
1056:
387:
32:
6820:
Here is a description of the geometric action of the
Boussinesq equation. Given a
4914:
Here is a description of the invariant
Poisson bracket in terms of the variables.
7712:
7239:
Zaks, Joseph (1996). "On the products of cross-ratios on diagonals of polygons".
16:
Discrete dynamical system on the moduli space of polygons in the projective plane
7129:
4229:
1635:
1472:
1387:
1016:
350:
On a basic level, one can think of the pentagram map as an operation defined on
20:
7073:
In a 2010 paper Max Glick identified the pentagram map as a special case of a
3444:
342:
7677:
7642:
7395:
7316:
6811:
5483:
Here is an alternate description of the invariant bracket. Given any function
4493:
of the number of single-digit blocks in the sequence. The monodromy invariant
4279:
1218:
351:
7707:. Computational Imaging and Vision. Vol. 1. Springer. pp. 255–306.
7575:
6514:
formalism to find formulas for the iterates of the pentagram map in terms of
6234:
6218:
6157:
4155:
2449:
1291:
1282:
was not flat; on small scales one cannot easily distinguish a sphere from a
609:
56:
44:
4329:, the notion of well separation is meant in the cyclic sense. Thus, 1 and 2
1729:
880:
521:
The pentagram map is also defined on the larger space of twisted polygons.
7784:
4294:
is recurrent: The orbit of almost any equivalence class of convex polygon
6795:
6695:
4337:
3645:
1267:
1068:
1064:
1052:
673:
528:-gon is a bi-infinite sequence of points in the projective plane that is
67:. The pentagram map is similar in spirit to the constructions underlying
48:
36:
6701:
6160:
is the set of all points in the space having a specified value for both
1722:
1059:
is an auxiliary space whose points index other objects. For example, in
7749:
Ovsienko, Valentin; Schwartz, Richard Evan; Tabachnikov, Serge (2009).
7624:"On generalizations of the pentagram map: discretizations of AGD flows"
7357:
Ovsienko, Valentin; Schwartz, Richard Evan; Tabachnikov, Serge (2010).
7254:
7203:"The pentagon in the projective plane, with a comment on Napier's rule"
7173:
6525:
6260:
In a 2011 preprint, Fedor
Soloviev showed that the pentagram map has a
3649:
the remaining rows are uniquely determined by the compatibility rules.
2100:
is the underlying field. Conversely, given almost any (in the sense of
1664:
1012:
711:
97:
51:
of the polygon, and constructs a new polygon from these intersections.
75:. It echoes the rationale and construction underlying a conjecture of
4742:
4151:
are the simplest examples of the monodromy invariants defined below.
1358:
1275:
1008:
628:-gon whose vertices have been listed out repeatedly. Thus, a twisted
7607:
7157:
4547:
is defined the same way, with even replacing odd in the definition.
4209:. Hence, each orbit of the pentagram map acting on this space has a
1793:
To each flag F, we associate the inverse cross ratio of the points
47:
map takes a given polygon, finds the intersections of the shortest
7767:
7566:
7466:
7434:
7378:
7307:
6433:
are the labels of antipodal vertices. A common convention is that
1279:
1263:
1255:
1251:
509:
7699:
Olver, Peter J.; Sapiro, Guillermo; Tannenbaum, Allen R. (1994).
6522:
and Harold Rumsey for the iterates of the octahedral recurrence.
5441:{\displaystyle \{x_{i},x_{j}\}=\{y_{i},y_{j}\}=\{x_{i},y_{j}\}=0}
4901:{\displaystyle \{O_{i},O_{j}\}=\{O_{i},E_{j}\}=\{E_{i},E_{j}\}=0}
4566:
is even, the allowable values of k are 1, 2, ...,
1859:
shown in the figure at left. In this way, one associates numbers
1226:
adapted to projective geometry, as mentioned above, it induces a
6518:. These formulas are similar in spirit to the formulas found by
5833:
where a summation over the repeated indices is understood. Then
1239:
7428:(October 2009). "Elementary Surprises in Projective Geometry".
7009:
is a good model for the time t evolution of the original curve
3644:
These rules are meant to hold for all configurations which are
1035:. Projective geometry is built around the fact that a straight
7552:
Soloviev, Fedor (2011). "Integrability of the Pentagram Map".
4340:
in the corner invariants. This is best illustrated by example
2734:{\displaystyle B(x_{1},\ldots ,x_{2N})=(b_{1},\ldots ,b_{2N})}
2634:{\displaystyle A(x_{1},\ldots ,x_{2N})=(a_{1},\ldots ,a_{2N})}
428:
is slightly problematic, in the sense that the indices of the
1274:
at each point, but somehow is hooked together differently. A
6700:
6524:
3443:
1728:
1721:
1266:
is a classical example of what is known in mathematics as a
879:
508:
432:-vertices are naturally odd integers whereas the indices of
341:
4232:. At the same time, as was already mentioned, the function
7755:
Electronic Research Announcements in Mathematical Sciences
680:
carrying a pentagon to its image under the pentagram map.
672:
The pentagram map is the identity on the moduli space of
7601:
Glick, Max (2010). "The Pentagram Map and Y-Patterns".
4313:
The so-called monodromy invariants are a collection of
2351:{\displaystyle V_{i+3}=a_{i}V_{i+2}+b_{i}V_{i+1}+V_{i}}
2154:
and Valentin Ovsienko. One describes a polygon in the
1258:. Another way is that it is the playing field for the
7295:
Journal of Fixed Point Theory and Applications (2008)
7015:
6979:
6943:
6908:
6873:
6830:
6757:
6734:
6714:
6680:
6645:
6621:
6601:
6558:
6538:
6439:
6412:
6385:
6299:
6193:
6166:
6089:
6026:
5972:
5919:
5899:
5842:
5515:
5489:
5454:
5338:
5252:
5169:
5086:
5000:
4923:
4798:
4751:
4673:
4611:
4584:
4526:
4499:
4412:
4350:
4238:
4164:
4130:
4103:
3965:
3827:
3744:
3671:
3615:
3579:
3529:
3491:
3462:
3413:
2753:
2648:
2548:
2521:
2501:
2475:
2433:
serve as coordinates for the moduli space of twisted
2367:
2260:
2226:
2164:
2110:
2072:
1986:
1914:
1865:
1799:
1750:
1483:
1411:
1340:
1320:
1300:
1185:
1147:
1109:
1077:
969:
943:
913:
863:
838:
809:
784:
744:
724:
689:
573:
546:
481:
454:
408:
302:
259:
232:
173:
106:
7732:"Discrete integrable systems in projective geometry"
6241:. This explains the Arnold-Liouville integrability.
1703:
is involved in two flags, and likewise each edge of
1270:. This is a space that looks somewhat like ordinary
6635:is a labeling so that all points in the (infinite)
4224:The pentagram map, when acting on the moduli space
4205:, when f is restricted to the moduli space of real
2247:so that each consecutive triple of vectors spans a
1382:When the field underlying all the constructions is
647:-gons is the set of equivalence classes of twisted
7037:
7001:
6965:
6937:. The envelope of all these chords is a new curve
6929:
6894:
6855:
6783:
6740:
6720:
6686:
6666:
6627:
6607:
6583:
6544:
6491:
6425:
6398:
6368:
6206:
6179:
6145:
6065:
6011:
5934:
5905:
5878:
5822:
5495:
5469:
5440:
5323:
5237:
5154:
5071:
4986:{\displaystyle x_{1},y_{1},x_{2},y_{2},\ldots \,.}
4985:
4900:
4769:
4717:
4624:
4597:
4539:
4512:
4478:
4396:
4298:returns infinitely often to every neighborhood of
4267:
4193:
4143:
4116:
4082:
3950:
3796:
3729:
3633:
3600:
3562:
3515:
3477:
3429:
3388:
2733:
2633:
2527:
2507:
2487:
2425:
2350:
2239:
2212:
2126:
2088:
2049:{\displaystyle x_{1},x_{2},x_{3},x_{4},\ldots \,.}
2048:
1972:
1900:
1851:
1785:
1619:
1463:
1346:
1326:
1306:
1209:
1171:
1133:
1095:
987:
955:
929:
869:
849:
824:
795:
766:
730:
702:
592:
559:
494:
467:
420:
331:
288:
245:
218:
151:
7359:"The Pentagram Map, A Discrete Integrable System"
5324:{\displaystyle \{y_{i},y_{i-1}\}=-y_{i}\,y_{i-1}}
5072:{\displaystyle \{x_{i},x_{i+1}\}=-x_{i}\,x_{i+1}}
4290:to imply that the action of the pentagram map on
6369:{\displaystyle a_{1}b_{1}+a_{2}b_{2}=a_{3}b_{3}}
6080:for the bracket, meaning (in this context) that
5238:{\displaystyle \{y_{i},y_{i+1}\}=y_{i}\,y_{i+1}}
5155:{\displaystyle \{x_{i},x_{i-1}\}=x_{i}\,x_{i-1}}
2213:{\displaystyle \ldots V_{1},V_{2},V_{3},\ldots }
1047:The pentagram map is fruitfully considered as a
7730:Ovsienko, Valentin; Tabachnikov, Serge (2008).
4479:{\displaystyle +x_{1}x_{2}x_{3}x_{7}x_{8}x_{9}}
3730:{\displaystyle O_{N}=x_{1}x_{3}\cdots x_{2N-1}}
2426:{\displaystyle a_{1},b_{1},a_{2},b_{2},\ldots }
2251:having unit volume. This leads to the relation
1973:{\displaystyle x_{1},y_{1},x_{2},y_{2},\ldots }
4777:on the space of functions which satisfies the
7208:Bulletin of the American Mathematical Society
6810:. This equation is a classical example of an
4336:Each odd admissible sequence gives rise to a
3797:{\displaystyle E_{N}=x_{2}x_{4}\cdots x_{2N}}
1471:in the affine line one defines the (inverse)
903:The iterates of the pentagram map shrink any
8:
7751:"Quasiperiodic Motion for the Pentagram Map"
7705:Geometry-Driven Diffusion in Computer Vision
6863:, and real numbers x and t, we consider the
6217:Each Casimir level set has an iso-monodromy
6134:
6115:
6109:
6090:
5873:
5861:
5429:
5403:
5397:
5371:
5365:
5339:
5285:
5253:
5202:
5170:
5119:
5087:
5033:
5001:
4889:
4863:
4857:
4831:
4825:
4799:
4764:
4752:
4321:that are invariant under the pentagram map.
3570:, and so forth. the compatibility rules are
1711:are ordered according to the orientation of
1071:(up to scale) by giving 3 positive numbers,
5503:on the moduli space, we have the so-called
2539:of order 2, and have the following action.
1365:have these invariant tori, they are called
1019:door, one sees a typically non-rectangular
7666:Journal of Mathematical Imaging and Vision
1334:dimensional and the tori in this case are
659:-gons as a sub-variety of co-dimension 8.
7766:
7606:
7565:
7465:
7433:
7377:
7306:
7220:
7020:
7014:
6984:
6978:
6948:
6942:
6907:
6872:
6847:
6829:
6770:
6756:
6733:
6713:
6679:
6644:
6620:
6600:
6575:
6557:
6537:
6483:
6470:
6457:
6444:
6438:
6417:
6411:
6390:
6384:
6360:
6350:
6337:
6327:
6314:
6304:
6298:
6198:
6192:
6171:
6165:
6146:{\displaystyle \{O_{n},f\}=\{E_{n},f\}=0}
6122:
6097:
6088:
6057:
6044:
6031:
6025:
6003:
5990:
5977:
5971:
5918:
5898:
5841:
5811:
5798:
5792:
5768:
5750:
5738:
5716:
5698:
5686:
5665:
5652:
5646:
5622:
5604:
5592:
5570:
5552:
5540:
5514:
5488:
5453:
5423:
5410:
5391:
5378:
5359:
5346:
5337:
5309:
5304:
5298:
5273:
5260:
5251:
5223:
5218:
5212:
5190:
5177:
5168:
5140:
5135:
5129:
5107:
5094:
5085:
5057:
5052:
5046:
5021:
5008:
4999:
4979:
4967:
4954:
4941:
4928:
4922:
4883:
4870:
4851:
4838:
4819:
4806:
4797:
4750:
4700:
4678:
4672:
4616:
4610:
4589:
4583:
4531:
4525:
4504:
4498:
4470:
4460:
4450:
4440:
4430:
4420:
4411:
4388:
4378:
4368:
4358:
4349:
4259:
4249:
4237:
4185:
4175:
4163:
4135:
4129:
4108:
4102:
4071:
4058:
4048:
4038:
4016:
4003:
3993:
3983:
3970:
3964:
3933:
3920:
3910:
3900:
3878:
3865:
3855:
3845:
3832:
3826:
3785:
3772:
3762:
3749:
3743:
3712:
3699:
3689:
3676:
3670:
3614:
3578:
3528:
3490:
3461:
3418:
3412:
3367:
3342:
3323:
3290:
3271:
3255:
3233:
3210:
3185:
3166:
3133:
3114:
3098:
3076:
3053:
3028:
3009:
2976:
2957:
2941:
2919:
2896:
2871:
2852:
2819:
2800:
2784:
2762:
2754:
2752:
2719:
2700:
2678:
2659:
2647:
2619:
2600:
2578:
2559:
2547:
2520:
2500:
2474:
2411:
2398:
2385:
2372:
2366:
2342:
2323:
2313:
2294:
2284:
2265:
2259:
2231:
2225:
2198:
2185:
2172:
2163:
2115:
2109:
2077:
2071:
2042:
2030:
2017:
2004:
1991:
1985:
1958:
1945:
1932:
1919:
1913:
1889:
1870:
1864:
1843:
1830:
1817:
1804:
1798:
1774:
1755:
1749:
1602:
1589:
1573:
1560:
1542:
1529:
1513:
1500:
1490:
1482:
1455:
1442:
1429:
1416:
1410:
1339:
1319:
1299:
1184:
1146:
1108:
1076:
1067:is always 180 degrees. You can specify a
968:
942:
921:
912:
862:
837:
808:
783:
749:
743:
723:
694:
688:
578:
572:
551:
545:
486:
480:
459:
453:
407:
320:
310:
301:
277:
267:
258:
237:
231:
219:{\displaystyle Q_{2},Q_{4},Q_{6},\ldots }
204:
191:
178:
172:
152:{\displaystyle P_{1},P_{3},P_{5},\ldots }
137:
124:
111:
105:
4286:. These two properties combine with the
710:is the identity on the space of labeled
676:. This is to say that there is always a
632:-gon is a generalization of an ordinary
536:That is, some projective transformation
7103:
6973:. When t is extremely small, the curve
6492:{\displaystyle a_{2},b_{2},a_{3},b_{3}}
6066:{\displaystyle E_{n}=y_{1}\cdots y_{n}}
6012:{\displaystyle O_{n}=x_{1}\cdots x_{n}}
4333: − 1 are not well separated.
1852:{\displaystyle t_{1},t_{2},t_{3},t_{4}}
1707:is involved in two flags. The flags of
1464:{\displaystyle t_{1},t_{2},t_{3},t_{4}}
624:-gon can be interpreted as an ordinary
163:under the pentagram map is the polygon
79:concerning the diagonals of a polygon.
7737:. Banff International Research Station
7595:
7593:
7547:
7545:
7487:
7485:
7352:
7350:
7348:
7346:
7344:
7342:
7340:
7338:
7336:
7334:
7288:
7286:
7284:
7282:
7280:
7278:
7276:
7274:
7272:
7123:
7121:
7119:
7117:
7115:
7113:
7111:
7109:
7107:
6248:, the Poisson bracket gives rise to a
7703:. In Romeny, Bart M. ter Haar (ed.).
7449:
7447:
7445:
7234:
7232:
7193:
7191:
7151:
7149:
6503:method of condensation for computing
5913:in the direction of the vector field
4397:{\displaystyle -x_{1}x_{5}x_{6}x_{7}}
4228:of convex polygons, has an invariant
1394:. The affine line is a subset of the
857:is the labeled hexagon obtained from
774:are equivalent by a label-preserving
655:-gons contains the space of ordinary
253:is the intersection of the diagonals
7:
1901:{\displaystyle x_{1},\ldots ,x_{2n}}
1786:{\displaystyle F_{1},\ldots ,F_{2N}}
59:only, goes back to an 1871 paper of
7457:Electronic Journal of Combinatorics
832:are projectively equivalent, where
5804:
5800:
5761:
5753:
5709:
5701:
5658:
5654:
5615:
5607:
5563:
5555:
14:
7494:"Recurrence of the Pentagram Map"
4770:{\displaystyle \{\cdot ,\cdot \}}
4718:{\displaystyle O_{k}(P)=E_{k}(P)}
4554:is odd, the allowable values of
1373:Coordinates for the moduli space
668:Action on pentagons and hexagons
7492:Schwartz, Richard Evan (2001).
7222:10.1090/S0002-9904-1945-08488-2
7128:Schwartz, Richard Evan (1992).
6584:{\displaystyle \pi :T\to R^{2}}
6256:Algebro-geometric integrability
6156:for all functions f. A Casimir
4558:are 1, 2, ..., (
3440:As grid compatibility relations
2446:ordinary differential equations
778:. More precisely, the hexagons
7513:10.1080/10586458.2001.10504671
7053:Projectively natural evolution
7032:
7026:
6996:
6990:
6960:
6954:
6924:
6912:
6889:
6877:
6840:
6661:
6655:
6595:which maps each octahedron in
6568:
6552:be the octahedral tiling. Let
5955:Arnold–Liouville integrability
5929:
5923:
5852:
5846:
5525:
5519:
4712:
4706:
4690:
4684:
4562: − 1)/2. When
4489:The sign is determined by the
3557:
3554:
3551:
3545:
3539:
3533:
3510:
3507:
3501:
3495:
3472:
3466:
3357:
3310:
3305:
3258:
3200:
3153:
3148:
3101:
3043:
2996:
2991:
2944:
2886:
2839:
2834:
2787:
2728:
2693:
2687:
2652:
2628:
2593:
2587:
2552:
1719:flags associated to an N-gon.
1608:
1582:
1579:
1553:
1548:
1522:
1519:
1493:
1204:
1186:
1166:
1148:
819:
813:
761:
755:
412:
326:
303:
283:
260:
1:
6815:partial differential equation
5879:{\displaystyle H(f)g=\{f,g\}}
3403: + 0 is just 2
2456:Formula for the pentagram map
1063:, the sum of the angles of a
226:as shown in the figure. Here
7713:10.1007/978-94-017-1699-4_11
7631:Journal of Nonlinear Science
6856:{\displaystyle C:R\to R^{2}}
5889:The first expression is the
4268:{\displaystyle f=O_{N}E_{N}}
4194:{\displaystyle f=O_{N}E_{N}}
2441:is not divisible by 3.
2134:one can construct a twisted
651:-gons. The space of twisted
332:{\displaystyle (P_{3}P_{7})}
289:{\displaystyle (P_{1}P_{5})}
7622:Marí Beffa, Gloria (2013).
6281:Connections to other topics
6252:on each Casimir level set.
4667:(such as a circle) one has
4288:Poincaré recurrence theorem
988:{\displaystyle 0<a<1}
620:is the identity, a twisted
7845:
6244:From the point of view of
4663:has all its vertices on a
3658:Corner coordinate products
3563:{\displaystyle A(B(A(P)))}
2448:, normalized to have unit
1134:{\displaystyle x+y+z=180.}
1025:Projective transformations
380:projective transformations
7643:10.1007/s00332-012-9152-3
7554:Duke Mathematical Journal
7396:10.1007/s00220-010-1075-y
7317:10.1007/s11784-008-0079-0
7158:"Ueber das ebene Funfeck"
6667:{\displaystyle G=\pi (T)}
6516:alternating sign matrices
6286:The Octahedral recurrence
2158:by a sequence of vectors
1652:projective transformation
1400:projective transformation
776:projective transformation
678:projective transformation
534:projective transformation
7576:10.1215/00127094-2382228
7424:Schwartz, Richard Evan;
7092:Periodic table of shapes
7038:{\displaystyle C_{0}(x)}
7002:{\displaystyle C_{t}(x)}
6966:{\displaystyle C_{t}(x)}
5946:define commuting flows.
5505:Hamiltonian vector field
2488:{\displaystyle B\circ A}
1630:Most authors consider 1/
1011:, one typically sees an
767:{\displaystyle T^{2}(H)}
7678:10.1023/A:1008226427785
6802:The Boussinesq equation
6784:{\displaystyle v=AD/BC}
3818:is even, the functions
3516:{\displaystyle B(A(P))}
2461:As a birational mapping
1695:is an adjacent line of
1357:The tori are invisible
1221:. This would lead to a
1210:{\displaystyle (x,y,z)}
1172:{\displaystyle (x,y,z)}
1051:on the moduli space of
593:{\displaystyle P_{N+k}}
475:is just clockwise from
7039:
7003:
6967:
6931:
6930:{\displaystyle C(x+t)}
6896:
6895:{\displaystyle C(x-t)}
6857:
6785:
6742:
6722:
6705:
6688:
6668:
6629:
6609:
6585:
6546:
6529:
6493:
6427:
6400:
6370:
6208:
6181:
6147:
6067:
6013:
5950:Complete integrability
5936:
5907:
5891:directional derivative
5880:
5824:
5497:
5471:
5442:
5325:
5239:
5156:
5073:
4987:
4902:
4771:
4719:
4626:
4599:
4541:
4514:
4480:
4398:
4269:
4195:
4145:
4118:
4084:
3952:
3798:
3731:
3635:
3602:
3601:{\displaystyle c=1-ab}
3564:
3517:
3479:
3448:
3431:
3430:{\displaystyle F^{2N}}
3390:
2735:
2635:
2529:
2509:
2489:
2427:
2352:
2241:
2214:
2128:
2127:{\displaystyle F^{2N}}
2090:
2089:{\displaystyle F^{2N}}
2050:
1974:
1902:
1853:
1787:
1733:
1726:
1659:The corner coordinates
1621:
1465:
1405:Given the four points
1348:
1328:
1308:
1211:
1173:
1135:
1097:
1015:. When one looks at a
989:
957:
956:{\displaystyle K>0}
931:
930:{\displaystyle Ka^{n}}
884:
871:
851:
826:
797:
768:
732:
704:
594:
561:
513:
496:
469:
422:
421:{\displaystyle P\to Q}
382:and thereby induces a
347:
333:
290:
247:
220:
153:
7785:10.3934/era.2009.16.1
7530:on September 27, 2011
7162:Mathematische Annalen
7040:
7004:
6968:
6932:
6897:
6858:
6786:
6748:by applying the rule
6743:
6723:
6704:
6689:
6669:
6630:
6610:
6586:
6547:
6528:
6494:
6428:
6426:{\displaystyle b_{i}}
6401:
6399:{\displaystyle a_{i}}
6371:
6271:quasi-periodic motion
6239:quasi-periodic motion
6209:
6207:{\displaystyle E_{n}}
6182:
6180:{\displaystyle O_{n}}
6148:
6068:
6014:
5937:
5908:
5881:
5825:
5498:
5472:
5443:
5326:
5240:
5157:
5074:
4988:
4903:
4772:
4741:is an anti-symmetric
4720:
4627:
4625:{\displaystyle E_{N}}
4600:
4598:{\displaystyle O_{N}}
4542:
4540:{\displaystyle E_{k}}
4515:
4513:{\displaystyle O_{k}}
4481:
4406:123789 gives rise to
4399:
4270:
4196:
4146:
4144:{\displaystyle E_{k}}
4119:
4117:{\displaystyle O_{k}}
4085:
3953:
3799:
3732:
3636:
3634:{\displaystyle wx=yz}
3603:
3565:
3518:
3480:
3447:
3432:
3391:
2736:
2636:
2530:
2510:
2490:
2428:
2353:
2242:
2240:{\displaystyle R^{3}}
2215:
2129:
2091:
2066:-gon with a point in
2051:
1980:are used in place of
1975:
1903:
1854:
1788:
1732:
1725:
1622:
1466:
1349:
1329:
1309:
1212:
1174:
1136:
1098:
1096:{\displaystyle x,y,z}
999:Motivating discussion
990:
958:
932:
899:Exponential shrinking
883:
872:
852:
827:
798:
769:
733:
705:
703:{\displaystyle T^{2}}
663:Elementary properties
595:
562:
560:{\displaystyle P_{k}}
512:
497:
495:{\displaystyle P_{k}}
470:
468:{\displaystyle Q_{k}}
423:
345:
334:
291:
248:
246:{\displaystyle Q_{4}}
221:
154:
83:Definition of the map
7013:
6977:
6941:
6906:
6871:
6828:
6755:
6732:
6712:
6708:Given a labeling of
6678:
6643:
6619:
6599:
6556:
6536:
6437:
6410:
6383:
6297:
6191:
6164:
6087:
6024:
5970:
5935:{\displaystyle H(f)}
5917:
5897:
5840:
5513:
5487:
5470:{\displaystyle i,j.}
5452:
5336:
5250:
5167:
5084:
4998:
4921:
4796:
4749:
4671:
4609:
4582:
4524:
4497:
4410:
4348:
4309:Monodromy invariants
4236:
4162:
4128:
4101:
3963:
3825:
3742:
3669:
3653:Invariant structures
3613:
3577:
3527:
3489:
3478:{\displaystyle A(P)}
3460:
3411:
2751:
2646:
2546:
2519:
2499:
2473:
2365:
2258:
2224:
2162:
2108:
2070:
1984:
1912:
1863:
1797:
1748:
1481:
1409:
1338:
1318:
1298:
1183:
1145:
1107:
1075:
967:
941:
911:
861:
836:
825:{\displaystyle T(H)}
807:
782:
742:
722:
687:
571:
544:
479:
452:
406:
398:Labeling conventions
374:. The pentagram map
370:are in sufficiently
300:
257:
230:
171:
104:
63:and a 1945 paper of
7824:Projective geometry
7777:2009arXiv0901.1585O
7476:2010arXiv1004.4311S
7388:2010CMaPh.299..409O
7242:Geometriae Dedicata
7156:A. Clebsch (1871).
7130:"The Pentagram Map"
7059:projective geometry
6808:Boussinesq equation
6510:Max Glick used the
6246:symplectic geometry
4344:1567 gives rise to
2537:birational mappings
1029:perspective drawing
1005:projective geometry
893:Desargues's theorem
532:-periodic modulo a
392:equivalence classes
7426:Tabachnikov, Serge
7255:10.1007/BF00160619
7174:10.1007/bf01455078
7035:
6999:
6963:
6927:
6892:
6853:
6781:
6738:
6718:
6706:
6684:
6664:
6625:
6605:
6581:
6542:
6530:
6489:
6423:
6396:
6366:
6262:Lax representation
6204:
6177:
6143:
6078:Casimir invariants
6063:
6009:
5932:
5903:
5876:
5820:
5493:
5467:
5438:
5321:
5235:
5152:
5069:
4983:
4898:
4767:
4715:
4622:
4595:
4537:
4510:
4476:
4394:
4265:
4191:
4141:
4114:
4095:Casimir invariants
4080:
3948:
3794:
3727:
3631:
3598:
3560:
3513:
3475:
3449:
3427:
3399:(Note: the index 2
3386:
3384:
2731:
2631:
2525:
2505:
2485:
2423:
2348:
2237:
2210:
2152:Sergei Tabachnikov
2124:
2086:
2046:
1970:
1898:
1849:
1783:
1734:
1727:
1638:, and that is why
1617:
1461:
1390:is just a copy of
1367:integrable systems
1344:
1324:
1304:
1223:higher-dimensional
1207:
1169:
1131:
1093:
1061:Euclidean geometry
1031:and new ones like
985:
953:
927:
885:
867:
850:{\displaystyle H'}
847:
822:
796:{\displaystyle H'}
793:
764:
728:
700:
590:
557:
514:
492:
465:
418:
366:provided that the
348:
329:
286:
243:
216:
149:
88:Basic construction
69:Desargues' theorem
7829:Dynamical systems
7560:(15): 2815–2853.
7501:Experimental Math
7135:Experimental Math
6741:{\displaystyle G}
6721:{\displaystyle G}
6687:{\displaystyle G}
6628:{\displaystyle T}
6608:{\displaystyle T}
6593:linear projection
6545:{\displaystyle T}
6223:coordinate charts
5906:{\displaystyle g}
5818:
5781:
5729:
5672:
5635:
5583:
5496:{\displaystyle f}
4911:for all indices.
4638:rational function
3361:
3204:
3047:
2890:
2528:{\displaystyle B}
2508:{\displaystyle A}
2437:-gons as long as
1744:-gon, with flags
1699:. Each vertex of
1612:
1363:dynamical systems
1347:{\displaystyle 3}
1327:{\displaystyle 6}
1307:{\displaystyle 7}
870:{\displaystyle H}
731:{\displaystyle H}
92:Suppose that the
73:Poncelet's porism
7836:
7810:
7808:
7807:
7801:
7795:. Archived from
7770:
7745:
7743:
7742:
7736:
7717:
7716:
7696:
7690:
7689:
7661:
7655:
7654:
7628:
7619:
7613:
7612:
7610:
7597:
7588:
7587:
7569:
7549:
7540:
7539:
7537:
7535:
7529:
7523:. Archived from
7498:
7489:
7480:
7479:
7469:
7451:
7440:
7439:
7437:
7421:
7415:
7414:
7412:
7410:
7381:
7366:Comm. Math. Phys
7363:
7354:
7329:
7328:
7310:
7290:
7267:
7266:
7236:
7227:
7226:
7224:
7195:
7186:
7185:
7153:
7144:
7143:
7125:
7069:Cluster algebras
7044:
7042:
7041:
7036:
7025:
7024:
7008:
7006:
7005:
7000:
6989:
6988:
6972:
6970:
6969:
6964:
6953:
6952:
6936:
6934:
6933:
6928:
6901:
6899:
6898:
6893:
6862:
6860:
6859:
6854:
6852:
6851:
6790:
6788:
6787:
6782:
6774:
6747:
6745:
6744:
6739:
6727:
6725:
6724:
6719:
6693:
6691:
6690:
6685:
6673:
6671:
6670:
6665:
6639:of any point in
6634:
6632:
6631:
6626:
6614:
6612:
6611:
6606:
6590:
6588:
6587:
6582:
6580:
6579:
6551:
6549:
6548:
6543:
6520:David P. Robbins
6498:
6496:
6495:
6490:
6488:
6487:
6475:
6474:
6462:
6461:
6449:
6448:
6432:
6430:
6429:
6424:
6422:
6421:
6405:
6403:
6402:
6397:
6395:
6394:
6375:
6373:
6372:
6367:
6365:
6364:
6355:
6354:
6342:
6341:
6332:
6331:
6319:
6318:
6309:
6308:
6213:
6211:
6210:
6205:
6203:
6202:
6186:
6184:
6183:
6178:
6176:
6175:
6152:
6150:
6149:
6144:
6127:
6126:
6102:
6101:
6072:
6070:
6069:
6064:
6062:
6061:
6049:
6048:
6036:
6035:
6018:
6016:
6015:
6010:
6008:
6007:
5995:
5994:
5982:
5981:
5941:
5939:
5938:
5933:
5912:
5910:
5909:
5904:
5885:
5883:
5882:
5877:
5829:
5827:
5826:
5821:
5819:
5817:
5816:
5815:
5799:
5797:
5796:
5787:
5783:
5782:
5780:
5779:
5778:
5759:
5751:
5749:
5748:
5730:
5728:
5727:
5726:
5707:
5699:
5697:
5696:
5673:
5671:
5670:
5669:
5653:
5651:
5650:
5641:
5637:
5636:
5634:
5633:
5632:
5613:
5605:
5603:
5602:
5584:
5582:
5581:
5580:
5561:
5553:
5551:
5550:
5502:
5500:
5499:
5494:
5476:
5474:
5473:
5468:
5447:
5445:
5444:
5439:
5428:
5427:
5415:
5414:
5396:
5395:
5383:
5382:
5364:
5363:
5351:
5350:
5330:
5328:
5327:
5322:
5320:
5319:
5303:
5302:
5284:
5283:
5265:
5264:
5244:
5242:
5241:
5236:
5234:
5233:
5217:
5216:
5201:
5200:
5182:
5181:
5161:
5159:
5158:
5153:
5151:
5150:
5134:
5133:
5118:
5117:
5099:
5098:
5078:
5076:
5075:
5070:
5068:
5067:
5051:
5050:
5032:
5031:
5013:
5012:
4992:
4990:
4989:
4984:
4972:
4971:
4959:
4958:
4946:
4945:
4933:
4932:
4907:
4905:
4904:
4899:
4888:
4887:
4875:
4874:
4856:
4855:
4843:
4842:
4824:
4823:
4811:
4810:
4779:Leibniz Identity
4776:
4774:
4773:
4768:
4724:
4722:
4721:
4716:
4705:
4704:
4683:
4682:
4631:
4629:
4628:
4623:
4621:
4620:
4604:
4602:
4601:
4596:
4594:
4593:
4546:
4544:
4543:
4538:
4536:
4535:
4519:
4517:
4516:
4511:
4509:
4508:
4485:
4483:
4482:
4477:
4475:
4474:
4465:
4464:
4455:
4454:
4445:
4444:
4435:
4434:
4425:
4424:
4403:
4401:
4400:
4395:
4393:
4392:
4383:
4382:
4373:
4372:
4363:
4362:
4274:
4272:
4271:
4266:
4264:
4263:
4254:
4253:
4200:
4198:
4197:
4192:
4190:
4189:
4180:
4179:
4158:of the function
4150:
4148:
4147:
4142:
4140:
4139:
4123:
4121:
4120:
4115:
4113:
4112:
4089:
4087:
4086:
4081:
4079:
4078:
4063:
4062:
4053:
4052:
4043:
4042:
4030:
4029:
4008:
4007:
3998:
3997:
3988:
3987:
3975:
3974:
3957:
3955:
3954:
3949:
3947:
3946:
3925:
3924:
3915:
3914:
3905:
3904:
3892:
3891:
3870:
3869:
3860:
3859:
3850:
3849:
3837:
3836:
3803:
3801:
3800:
3795:
3793:
3792:
3777:
3776:
3767:
3766:
3754:
3753:
3736:
3734:
3733:
3728:
3726:
3725:
3704:
3703:
3694:
3693:
3681:
3680:
3640:
3638:
3637:
3632:
3607:
3605:
3604:
3599:
3569:
3567:
3566:
3561:
3522:
3520:
3519:
3514:
3484:
3482:
3481:
3476:
3436:
3434:
3433:
3428:
3426:
3425:
3395:
3393:
3392:
3387:
3385:
3381:
3380:
3362:
3360:
3356:
3355:
3337:
3336:
3308:
3304:
3303:
3285:
3284:
3256:
3247:
3246:
3224:
3223:
3205:
3203:
3199:
3198:
3180:
3179:
3151:
3147:
3146:
3128:
3127:
3099:
3090:
3089:
3067:
3066:
3048:
3046:
3042:
3041:
3023:
3022:
2994:
2990:
2989:
2971:
2970:
2942:
2933:
2932:
2910:
2909:
2891:
2889:
2885:
2884:
2866:
2865:
2837:
2833:
2832:
2814:
2813:
2785:
2776:
2775:
2740:
2738:
2737:
2732:
2727:
2726:
2705:
2704:
2686:
2685:
2664:
2663:
2640:
2638:
2637:
2632:
2627:
2626:
2605:
2604:
2586:
2585:
2564:
2563:
2534:
2532:
2531:
2526:
2514:
2512:
2511:
2506:
2494:
2492:
2491:
2486:
2432:
2430:
2429:
2424:
2416:
2415:
2403:
2402:
2390:
2389:
2377:
2376:
2361:The coordinates
2357:
2355:
2354:
2349:
2347:
2346:
2334:
2333:
2318:
2317:
2305:
2304:
2289:
2288:
2276:
2275:
2246:
2244:
2243:
2238:
2236:
2235:
2219:
2217:
2216:
2211:
2203:
2202:
2190:
2189:
2177:
2176:
2156:projective plane
2146:(ab) coordinates
2133:
2131:
2130:
2125:
2123:
2122:
2095:
2093:
2092:
2087:
2085:
2084:
2055:
2053:
2052:
2047:
2035:
2034:
2022:
2021:
2009:
2008:
1996:
1995:
1979:
1977:
1976:
1971:
1963:
1962:
1950:
1949:
1937:
1936:
1924:
1923:
1907:
1905:
1904:
1899:
1897:
1896:
1875:
1874:
1858:
1856:
1855:
1850:
1848:
1847:
1835:
1834:
1822:
1821:
1809:
1808:
1792:
1790:
1789:
1784:
1782:
1781:
1760:
1759:
1648:projective space
1626:
1624:
1623:
1618:
1613:
1611:
1607:
1606:
1594:
1593:
1578:
1577:
1565:
1564:
1551:
1547:
1546:
1534:
1533:
1518:
1517:
1505:
1504:
1491:
1470:
1468:
1467:
1462:
1460:
1459:
1447:
1446:
1434:
1433:
1421:
1420:
1353:
1351:
1350:
1345:
1333:
1331:
1330:
1325:
1313:
1311:
1310:
1305:
1216:
1214:
1213:
1208:
1178:
1176:
1175:
1170:
1140:
1138:
1137:
1132:
1102:
1100:
1099:
1094:
994:
992:
991:
986:
962:
960:
959:
954:
936:
934:
933:
928:
926:
925:
889:Pascal's theorem
876:
874:
873:
868:
856:
854:
853:
848:
846:
831:
829:
828:
823:
802:
800:
799:
794:
792:
773:
771:
770:
765:
754:
753:
737:
735:
734:
729:
709:
707:
706:
701:
699:
698:
599:
597:
596:
591:
589:
588:
566:
564:
563:
558:
556:
555:
517:Twisted polygons
501:
499:
498:
493:
491:
490:
474:
472:
471:
466:
464:
463:
427:
425:
424:
419:
372:general position
360:projective plane
354:polygons in the
338:
336:
335:
330:
325:
324:
315:
314:
295:
293:
292:
287:
282:
281:
272:
271:
252:
250:
249:
244:
242:
241:
225:
223:
222:
217:
209:
208:
196:
195:
183:
182:
158:
156:
155:
150:
142:
141:
129:
128:
116:
115:
65:Theodore Motzkin
53:Richard Schwartz
41:projective plane
29:dynamical system
7844:
7843:
7839:
7838:
7837:
7835:
7834:
7833:
7814:
7813:
7805:
7803:
7799:
7748:
7740:
7738:
7734:
7729:
7726:
7724:Further reading
7721:
7720:
7698:
7697:
7693:
7663:
7662:
7658:
7626:
7621:
7620:
7616:
7600:
7598:
7591:
7551:
7550:
7543:
7533:
7531:
7527:
7496:
7491:
7490:
7483:
7453:
7452:
7443:
7423:
7422:
7418:
7408:
7406:
7361:
7356:
7355:
7332:
7292:
7291:
7270:
7238:
7237:
7230:
7215:(12): 985–989.
7197:
7196:
7189:
7155:
7154:
7147:
7127:
7126:
7105:
7100:
7083:
7075:cluster algebra
7071:
7063:computer vision
7055:
7016:
7011:
7010:
6980:
6975:
6974:
6944:
6939:
6938:
6904:
6903:
6869:
6868:
6843:
6826:
6825:
6804:
6753:
6752:
6730:
6729:
6710:
6709:
6676:
6675:
6641:
6640:
6617:
6616:
6597:
6596:
6571:
6554:
6553:
6534:
6533:
6512:cluster algebra
6501:C. L. Dodgson's
6479:
6466:
6453:
6440:
6435:
6434:
6413:
6408:
6407:
6386:
6381:
6380:
6356:
6346:
6333:
6323:
6310:
6300:
6295:
6294:
6288:
6283:
6275:theta functions
6258:
6250:symplectic form
6194:
6189:
6188:
6167:
6162:
6161:
6118:
6093:
6085:
6084:
6053:
6040:
6027:
6022:
6021:
5999:
5986:
5973:
5968:
5967:
5957:
5952:
5915:
5914:
5895:
5894:
5838:
5837:
5807:
5803:
5788:
5764:
5760:
5752:
5734:
5712:
5708:
5700:
5682:
5681:
5677:
5661:
5657:
5642:
5618:
5614:
5606:
5588:
5566:
5562:
5554:
5536:
5535:
5531:
5511:
5510:
5485:
5484:
5450:
5449:
5419:
5406:
5387:
5374:
5355:
5342:
5334:
5333:
5305:
5294:
5269:
5256:
5248:
5247:
5219:
5208:
5186:
5173:
5165:
5164:
5136:
5125:
5103:
5090:
5082:
5081:
5053:
5042:
5017:
5004:
4996:
4995:
4963:
4950:
4937:
4924:
4919:
4918:
4879:
4866:
4847:
4834:
4815:
4802:
4794:
4793:
4787:Poisson bracket
4783:Jacobi identity
4747:
4746:
4739:Poisson bracket
4735:
4733:Poisson bracket
4696:
4674:
4669:
4668:
4612:
4607:
4606:
4585:
4580:
4579:
4527:
4522:
4521:
4500:
4495:
4494:
4466:
4456:
4446:
4436:
4426:
4416:
4408:
4407:
4384:
4374:
4364:
4354:
4346:
4345:
4311:
4255:
4245:
4234:
4233:
4222:
4207:convex polygons
4181:
4171:
4160:
4159:
4131:
4126:
4125:
4104:
4099:
4098:
4067:
4054:
4044:
4034:
4012:
3999:
3989:
3979:
3966:
3961:
3960:
3929:
3916:
3906:
3896:
3874:
3861:
3851:
3841:
3828:
3823:
3822:
3781:
3768:
3758:
3745:
3740:
3739:
3708:
3695:
3685:
3672:
3667:
3666:
3660:
3655:
3611:
3610:
3575:
3574:
3525:
3524:
3487:
3486:
3458:
3457:
3442:
3414:
3409:
3408:
3383:
3382:
3363:
3338:
3319:
3309:
3286:
3267:
3257:
3248:
3229:
3226:
3225:
3206:
3181:
3162:
3152:
3129:
3110:
3100:
3091:
3072:
3069:
3068:
3049:
3024:
3005:
2995:
2972:
2953:
2943:
2934:
2915:
2912:
2911:
2892:
2867:
2848:
2838:
2815:
2796:
2786:
2777:
2758:
2749:
2748:
2715:
2696:
2674:
2655:
2644:
2643:
2615:
2596:
2574:
2555:
2544:
2543:
2517:
2516:
2497:
2496:
2471:
2470:
2463:
2458:
2407:
2394:
2381:
2368:
2363:
2362:
2338:
2319:
2309:
2290:
2280:
2261:
2256:
2255:
2227:
2222:
2221:
2194:
2181:
2168:
2160:
2159:
2148:
2111:
2106:
2105:
2073:
2068:
2067:
2026:
2013:
2000:
1987:
1982:
1981:
1954:
1941:
1928:
1915:
1910:
1909:
1885:
1866:
1861:
1860:
1839:
1826:
1813:
1800:
1795:
1794:
1770:
1751:
1746:
1745:
1687:is a vertex of
1661:
1598:
1585:
1569:
1556:
1552:
1538:
1525:
1509:
1496:
1492:
1479:
1478:
1451:
1438:
1425:
1412:
1407:
1406:
1396:projective line
1380:
1375:
1336:
1335:
1316:
1315:
1296:
1295:
1272:Euclidean space
1181:
1180:
1143:
1142:
1141:So, each point
1105:
1104:
1073:
1072:
1033:computer vision
1001:
965:
964:
939:
938:
917:
909:
908:
901:
859:
858:
839:
834:
833:
805:
804:
785:
780:
779:
745:
740:
739:
720:
719:
690:
685:
684:
670:
665:
612:of the twisted
574:
569:
568:
547:
542:
541:
519:
482:
477:
476:
455:
450:
449:
404:
403:
400:
316:
306:
298:
297:
273:
263:
255:
254:
233:
228:
227:
200:
187:
174:
169:
168:
133:
120:
107:
102:
101:
100:P are given by
90:
85:
77:Branko Grünbaum
17:
12:
11:
5:
7842:
7840:
7832:
7831:
7826:
7816:
7815:
7812:
7811:
7746:
7725:
7722:
7719:
7718:
7691:
7672:(3): 225–240.
7656:
7637:(2): 303–334.
7614:
7589:
7541:
7507:(4): 519–528.
7481:
7441:
7416:
7372:(2): 409–446.
7330:
7301:(2): 379–409.
7268:
7249:(2): 145–151.
7228:
7187:
7168:(3): 476–489.
7145:
7102:
7101:
7099:
7096:
7095:
7094:
7089:
7082:
7079:
7070:
7067:
7054:
7051:
7034:
7031:
7028:
7023:
7019:
6998:
6995:
6992:
6987:
6983:
6962:
6959:
6956:
6951:
6947:
6926:
6923:
6920:
6917:
6914:
6911:
6891:
6888:
6885:
6882:
6879:
6876:
6850:
6846:
6842:
6839:
6836:
6833:
6822:locally convex
6803:
6800:
6792:
6791:
6780:
6777:
6773:
6769:
6766:
6763:
6760:
6737:
6717:
6683:
6663:
6660:
6657:
6654:
6651:
6648:
6624:
6604:
6578:
6574:
6570:
6567:
6564:
6561:
6541:
6486:
6482:
6478:
6473:
6469:
6465:
6460:
6456:
6452:
6447:
6443:
6420:
6416:
6393:
6389:
6377:
6376:
6363:
6359:
6353:
6349:
6345:
6340:
6336:
6330:
6326:
6322:
6317:
6313:
6307:
6303:
6287:
6284:
6282:
6279:
6257:
6254:
6201:
6197:
6174:
6170:
6154:
6153:
6142:
6139:
6136:
6133:
6130:
6125:
6121:
6117:
6114:
6111:
6108:
6105:
6100:
6096:
6092:
6074:
6073:
6060:
6056:
6052:
6047:
6043:
6039:
6034:
6030:
6019:
6006:
6002:
5998:
5993:
5989:
5985:
5980:
5976:
5956:
5953:
5951:
5948:
5931:
5928:
5925:
5922:
5902:
5887:
5886:
5875:
5872:
5869:
5866:
5863:
5860:
5857:
5854:
5851:
5848:
5845:
5831:
5830:
5814:
5810:
5806:
5802:
5795:
5791:
5786:
5777:
5774:
5771:
5767:
5763:
5758:
5755:
5747:
5744:
5741:
5737:
5733:
5725:
5722:
5719:
5715:
5711:
5706:
5703:
5695:
5692:
5689:
5685:
5680:
5676:
5668:
5664:
5660:
5656:
5649:
5645:
5640:
5631:
5628:
5625:
5621:
5617:
5612:
5609:
5601:
5598:
5595:
5591:
5587:
5579:
5576:
5573:
5569:
5565:
5560:
5557:
5549:
5546:
5543:
5539:
5534:
5530:
5527:
5524:
5521:
5518:
5492:
5478:
5477:
5466:
5463:
5460:
5457:
5448:for all other
5437:
5434:
5431:
5426:
5422:
5418:
5413:
5409:
5405:
5402:
5399:
5394:
5390:
5386:
5381:
5377:
5373:
5370:
5367:
5362:
5358:
5354:
5349:
5345:
5341:
5331:
5318:
5315:
5312:
5308:
5301:
5297:
5293:
5290:
5287:
5282:
5279:
5276:
5272:
5268:
5263:
5259:
5255:
5245:
5232:
5229:
5226:
5222:
5215:
5211:
5207:
5204:
5199:
5196:
5193:
5189:
5185:
5180:
5176:
5172:
5162:
5149:
5146:
5143:
5139:
5132:
5128:
5124:
5121:
5116:
5113:
5110:
5106:
5102:
5097:
5093:
5089:
5079:
5066:
5063:
5060:
5056:
5049:
5045:
5041:
5038:
5035:
5030:
5027:
5024:
5020:
5016:
5011:
5007:
5003:
4993:
4982:
4978:
4975:
4970:
4966:
4962:
4957:
4953:
4949:
4944:
4940:
4936:
4931:
4927:
4909:
4908:
4897:
4894:
4891:
4886:
4882:
4878:
4873:
4869:
4865:
4862:
4859:
4854:
4850:
4846:
4841:
4837:
4833:
4830:
4827:
4822:
4818:
4814:
4809:
4805:
4801:
4766:
4763:
4760:
4757:
4754:
4734:
4731:
4714:
4711:
4708:
4703:
4699:
4695:
4692:
4689:
4686:
4681:
4677:
4652:of 4-diagonal
4619:
4615:
4592:
4588:
4534:
4530:
4507:
4503:
4487:
4486:
4473:
4469:
4463:
4459:
4453:
4449:
4443:
4439:
4433:
4429:
4423:
4419:
4415:
4404:
4391:
4387:
4381:
4377:
4371:
4367:
4361:
4357:
4353:
4310:
4307:
4262:
4258:
4252:
4248:
4244:
4241:
4221:
4218:
4188:
4184:
4178:
4174:
4170:
4167:
4138:
4134:
4111:
4107:
4091:
4090:
4077:
4074:
4070:
4066:
4061:
4057:
4051:
4047:
4041:
4037:
4033:
4028:
4025:
4022:
4019:
4015:
4011:
4006:
4002:
3996:
3992:
3986:
3982:
3978:
3973:
3969:
3958:
3945:
3942:
3939:
3936:
3932:
3928:
3923:
3919:
3913:
3909:
3903:
3899:
3895:
3890:
3887:
3884:
3881:
3877:
3873:
3868:
3864:
3858:
3854:
3848:
3844:
3840:
3835:
3831:
3814: = 2
3805:
3804:
3791:
3788:
3784:
3780:
3775:
3771:
3765:
3761:
3757:
3752:
3748:
3737:
3724:
3721:
3718:
3715:
3711:
3707:
3702:
3698:
3692:
3688:
3684:
3679:
3675:
3659:
3656:
3654:
3651:
3642:
3641:
3630:
3627:
3624:
3621:
3618:
3608:
3597:
3594:
3591:
3588:
3585:
3582:
3559:
3556:
3553:
3550:
3547:
3544:
3541:
3538:
3535:
3532:
3512:
3509:
3506:
3503:
3500:
3497:
3494:
3474:
3471:
3468:
3465:
3441:
3438:
3424:
3421:
3417:
3397:
3396:
3379:
3376:
3373:
3370:
3366:
3359:
3354:
3351:
3348:
3345:
3341:
3335:
3332:
3329:
3326:
3322:
3318:
3315:
3312:
3307:
3302:
3299:
3296:
3293:
3289:
3283:
3280:
3277:
3274:
3270:
3266:
3263:
3260:
3254:
3251:
3249:
3245:
3242:
3239:
3236:
3232:
3228:
3227:
3222:
3219:
3216:
3213:
3209:
3202:
3197:
3194:
3191:
3188:
3184:
3178:
3175:
3172:
3169:
3165:
3161:
3158:
3155:
3150:
3145:
3142:
3139:
3136:
3132:
3126:
3123:
3120:
3117:
3113:
3109:
3106:
3103:
3097:
3094:
3092:
3088:
3085:
3082:
3079:
3075:
3071:
3070:
3065:
3062:
3059:
3056:
3052:
3045:
3040:
3037:
3034:
3031:
3027:
3021:
3018:
3015:
3012:
3008:
3004:
3001:
2998:
2993:
2988:
2985:
2982:
2979:
2975:
2969:
2966:
2963:
2960:
2956:
2952:
2949:
2946:
2940:
2937:
2935:
2931:
2928:
2925:
2922:
2918:
2914:
2913:
2908:
2905:
2902:
2899:
2895:
2888:
2883:
2880:
2877:
2874:
2870:
2864:
2861:
2858:
2855:
2851:
2847:
2844:
2841:
2836:
2831:
2828:
2825:
2822:
2818:
2812:
2809:
2806:
2803:
2799:
2795:
2792:
2789:
2783:
2780:
2778:
2774:
2771:
2768:
2765:
2761:
2757:
2756:
2742:
2741:
2730:
2725:
2722:
2718:
2714:
2711:
2708:
2703:
2699:
2695:
2692:
2689:
2684:
2681:
2677:
2673:
2670:
2667:
2662:
2658:
2654:
2651:
2641:
2630:
2625:
2622:
2618:
2614:
2611:
2608:
2603:
2599:
2595:
2592:
2589:
2584:
2581:
2577:
2573:
2570:
2567:
2562:
2558:
2554:
2551:
2524:
2504:
2484:
2481:
2478:
2462:
2459:
2457:
2454:
2422:
2419:
2414:
2410:
2406:
2401:
2397:
2393:
2388:
2384:
2380:
2375:
2371:
2359:
2358:
2345:
2341:
2337:
2332:
2329:
2326:
2322:
2316:
2312:
2308:
2303:
2300:
2297:
2293:
2287:
2283:
2279:
2274:
2271:
2268:
2264:
2249:parallelepiped
2234:
2230:
2209:
2206:
2201:
2197:
2193:
2188:
2184:
2180:
2175:
2171:
2167:
2147:
2144:
2121:
2118:
2114:
2102:measure theory
2083:
2080:
2076:
2045:
2041:
2038:
2033:
2029:
2025:
2020:
2016:
2012:
2007:
2003:
1999:
1994:
1990:
1969:
1966:
1961:
1957:
1953:
1948:
1944:
1940:
1935:
1931:
1927:
1922:
1918:
1895:
1892:
1888:
1884:
1881:
1878:
1873:
1869:
1846:
1842:
1838:
1833:
1829:
1825:
1820:
1816:
1812:
1807:
1803:
1780:
1777:
1773:
1769:
1766:
1763:
1758:
1754:
1660:
1657:
1628:
1627:
1616:
1610:
1605:
1601:
1597:
1592:
1588:
1584:
1581:
1576:
1572:
1568:
1563:
1559:
1555:
1550:
1545:
1541:
1537:
1532:
1528:
1524:
1521:
1516:
1512:
1508:
1503:
1499:
1495:
1489:
1486:
1458:
1454:
1450:
1445:
1441:
1437:
1432:
1428:
1424:
1419:
1415:
1379:
1376:
1374:
1371:
1343:
1323:
1303:
1247:moduli space.
1236:convex polygon
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1130:
1127:
1124:
1121:
1118:
1115:
1112:
1092:
1089:
1086:
1083:
1080:
1000:
997:
984:
981:
978:
975:
972:
952:
949:
946:
937:for constants
924:
920:
916:
905:convex polygon
900:
897:
866:
845:
842:
821:
818:
815:
812:
791:
788:
763:
760:
757:
752:
748:
727:
697:
693:
669:
666:
664:
661:
608:is called the
587:
584:
581:
577:
554:
550:
518:
515:
489:
485:
462:
458:
417:
414:
411:
399:
396:
390:of projective
328:
323:
319:
313:
309:
305:
285:
280:
276:
270:
266:
262:
240:
236:
215:
212:
207:
203:
199:
194:
190:
186:
181:
177:
167:with vertices
148:
145:
140:
136:
132:
127:
123:
119:
114:
110:
89:
86:
84:
81:
61:Alfred Clebsch
27:is a discrete
15:
13:
10:
9:
6:
4:
3:
2:
7841:
7830:
7827:
7825:
7822:
7821:
7819:
7802:on 2011-09-30
7798:
7794:
7790:
7786:
7782:
7778:
7774:
7769:
7764:
7760:
7756:
7752:
7747:
7733:
7728:
7727:
7723:
7714:
7710:
7706:
7702:
7695:
7692:
7687:
7683:
7679:
7675:
7671:
7667:
7660:
7657:
7652:
7648:
7644:
7640:
7636:
7632:
7625:
7618:
7615:
7609:
7604:
7596:
7594:
7590:
7585:
7581:
7577:
7573:
7568:
7563:
7559:
7555:
7548:
7546:
7542:
7526:
7522:
7518:
7514:
7510:
7506:
7502:
7495:
7488:
7486:
7482:
7477:
7473:
7468:
7463:
7459:
7458:
7450:
7448:
7446:
7442:
7436:
7431:
7427:
7420:
7417:
7405:
7401:
7397:
7393:
7389:
7385:
7380:
7375:
7371:
7367:
7360:
7353:
7351:
7349:
7347:
7345:
7343:
7341:
7339:
7337:
7335:
7331:
7326:
7322:
7318:
7314:
7309:
7304:
7300:
7296:
7289:
7287:
7285:
7283:
7281:
7279:
7277:
7275:
7273:
7269:
7264:
7260:
7256:
7252:
7248:
7244:
7243:
7235:
7233:
7229:
7223:
7218:
7214:
7210:
7209:
7204:
7200:
7194:
7192:
7188:
7183:
7179:
7175:
7171:
7167:
7163:
7159:
7152:
7150:
7146:
7141:
7137:
7136:
7131:
7124:
7122:
7120:
7118:
7116:
7114:
7112:
7110:
7108:
7104:
7097:
7093:
7090:
7088:
7087:Combinatorics
7085:
7084:
7080:
7078:
7076:
7068:
7066:
7064:
7060:
7052:
7050:
7047:
7029:
7021:
7017:
6993:
6985:
6981:
6957:
6949:
6945:
6921:
6918:
6915:
6909:
6886:
6883:
6880:
6874:
6866:
6848:
6844:
6837:
6834:
6831:
6823:
6818:
6816:
6813:
6809:
6801:
6799:
6797:
6778:
6775:
6771:
6767:
6764:
6761:
6758:
6751:
6750:
6749:
6735:
6715:
6703:
6699:
6697:
6681:
6658:
6652:
6649:
6646:
6638:
6637:inverse image
6622:
6602:
6594:
6576:
6572:
6565:
6562:
6559:
6539:
6527:
6523:
6521:
6517:
6513:
6508:
6506:
6502:
6484:
6480:
6476:
6471:
6467:
6463:
6458:
6454:
6450:
6445:
6441:
6418:
6414:
6391:
6387:
6361:
6357:
6351:
6347:
6343:
6338:
6334:
6328:
6324:
6320:
6315:
6311:
6305:
6301:
6293:
6292:
6291:
6285:
6280:
6278:
6276:
6272:
6268:
6263:
6255:
6253:
6251:
6247:
6242:
6240:
6236:
6233:
6229:
6224:
6220:
6215:
6199:
6195:
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4725:for all
4709:
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4665:conic section
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2018:
2014:
2010:
2005:
2001:
1997:
1992:
1988:
1967:
1964:
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1955:
1951:
1946:
1942:
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1355:
1354:dimensional.
1341:
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1021:quadrilateral
1018:
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947:
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695:
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681:
679:
675:
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662:
660:
658:
654:
650:
646:
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637:
635:
631:
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623:
619:
615:
611:
607:
603:
585:
582:
579:
575:
552:
548:
539:
535:
531:
527:
522:
516:
511:
507:
503:
487:
483:
460:
456:
447:
443:
439:
435:
431:
415:
409:
397:
395:
394:of polygons.
393:
389:
385:
381:
377:
373:
369:
365:
361:
357:
353:
344:
340:
339:, and so on.
321:
317:
311:
307:
278:
274:
268:
264:
238:
234:
213:
210:
205:
201:
197:
192:
188:
184:
179:
175:
166:
162:
159:The image of
146:
143:
138:
134:
130:
125:
121:
117:
112:
108:
99:
95:
87:
82:
80:
78:
74:
70:
66:
62:
58:
54:
50:
46:
42:
38:
34:
30:
26:
25:pentagram map
22:
7804:. Retrieved
7797:the original
7758:
7754:
7739:. Retrieved
7704:
7694:
7669:
7665:
7659:
7634:
7630:
7617:
7557:
7553:
7532:. Retrieved
7525:the original
7504:
7500:
7455:
7419:
7407:. Retrieved
7369:
7365:
7298:
7294:
7246:
7240:
7212:
7206:
7165:
7161:
7139:
7133:
7072:
7056:
7048:
6819:
6805:
6793:
6707:
6531:
6509:
6505:determinants
6378:
6289:
6259:
6243:
6216:
6155:
6075:
5960:
5958:
5888:
5832:
5482:
5479:
4913:
4910:
4736:
4726:
4660:
4658:
4650:determinants
4645:
4634:
4575:
4571:
4567:
4563:
4559:
4555:
4551:
4549:
4488:
4335:
4330:
4326:
4323:
4319:moduli space
4312:
4304:
4299:
4295:
4291:
4283:
4225:
4223:
4153:
4092:
3815:
3811:
3809:
3806:
3661:
3643:
3450:
3404:
3400:
3398:
2743:
2464:
2443:
2438:
2434:
2360:
2149:
2139:
2135:
2097:
2063:
2059:
2057:
1741:
1737:
1735:
1716:
1712:
1708:
1704:
1700:
1696:
1692:
1688:
1684:
1680:
1676:
1672:
1662:
1645:
1639:
1631:
1629:
1404:
1391:
1383:
1381:
1356:
1288:
1249:
1244:aspect ratio
1232:
1057:moduli space
1046:
1041:
1002:
902:
895:and others.
886:
715:
682:
671:
656:
652:
648:
644:
640:
639:Two twisted
638:
633:
629:
625:
621:
617:
613:
605:
601:
537:
529:
525:
523:
520:
504:
445:
441:
437:
433:
429:
401:
388:moduli space
349:
164:
160:
91:
33:moduli space
24:
18:
7608:1005.0598v2
7199:Th. Motzkin
6867:connecting
4230:volume form
4220:Volume form
2495:. The maps
2468:composition
2104:) point in
1675:is a pair (
1636:cross-ratio
1473:cross ratio
1388:affine line
1378:Cross-ratio
1017:rectangular
616:-gon. When
21:mathematics
7818:Categories
7806:2011-06-28
7741:2010-02-12
6812:integrable
4280:level sets
4156:level sets
1634:to be the
1219:parameters
1103:such that
604:. The map
524:A twisted
7768:0901.1585
7584:119586878
7567:1106.3950
7467:1004.4311
7435:0910.1952
7379:0810.5605
7308:0709.1264
7263:123626706
7182:122093180
6884:−
6841:→
6796:congruent
6696:congruent
6653:π
6569:→
6560:π
6235:manifolds
6219:foliation
6158:level set
6051:⋯
5997:⋯
5805:∂
5801:∂
5762:∂
5754:∂
5732:−
5721:−
5710:∂
5702:∂
5691:−
5659:∂
5655:∂
5627:−
5616:∂
5608:∂
5597:−
5586:−
5564:∂
5556:∂
5314:−
5292:−
5278:−
5145:−
5112:−
5040:−
4977:…
4762:⋅
4756:⋅
4745:operator
4659:Whenever
4570:/2. When
4352:−
4315:functions
4065:⋯
4024:−
4010:⋯
3941:−
3927:⋯
3886:−
3872:⋯
3779:⋯
3720:−
3706:⋯
3646:congruent
3590:−
3375:−
3350:−
3331:−
3317:−
3265:−
3160:−
3141:−
3122:−
3108:−
3061:−
3003:−
2984:−
2965:−
2951:−
2879:−
2860:−
2846:−
2794:−
2770:−
2710:…
2669:…
2610:…
2569:…
2480:∘
2450:Wronskian
2421:…
2208:…
2166:…
2040:…
1968:…
1880:…
1765:…
1683:), where
1596:−
1567:−
1536:−
1507:−
1314:-gons is
1292:foliation
1260:Asteroids
674:pentagons
610:monodromy
413:→
214:…
147:…
57:pentagons
49:diagonals
45:pentagram
7793:10821671
7534:June 30,
7409:June 26,
7325:17099073
7201:(1945).
7142:: 90–95.
7081:See also
4781:and the
4654:matrices
4644:of
4338:monomial
1268:manifold
1069:triangle
1065:triangle
1053:polygons
844:′
790:′
712:hexagons
683:The map
600:for all
540:carries
402:The map
376:commutes
368:vertices
94:vertices
37:polygons
7773:Bibcode
7761:: 1–8.
7686:2262433
7651:3041627
7521:4454793
7472:Bibcode
7404:2616239
7384:Bibcode
6591:be the
6267:divisor
6232:compact
4317:on the
4277:compact
4214:closure
4211:compact
4203:compact
1665:polygon
1359:subsets
1228:mapping
1049:mapping
1013:ellipse
714:. Here
386:on the
384:mapping
362:over a
98:polygon
96:of the
39:in the
31:on the
7791:
7684:
7649:
7582:
7519:
7402:
7323:
7261:
7180:
6824:curve
6228:smooth
4743:linear
4491:parity
2744:where
2142:-gon.
2096:where
1740:be an
1386:, the
1276:sphere
1009:circle
636:-gon.
352:convex
43:. The
23:, the
7800:(pdf)
7789:S2CID
7763:arXiv
7735:(PDF)
7682:S2CID
7627:(PDF)
7603:arXiv
7580:S2CID
7562:arXiv
7528:(PDF)
7517:S2CID
7497:(PDF)
7462:arXiv
7430:arXiv
7400:S2CID
7374:arXiv
7362:(PDF)
7321:S2CID
7303:arXiv
7259:S2CID
7178:S2CID
7098:Notes
6865:chord
6379:Here
4642:trace
4550:When
3810:When
3453:edges
1284:plane
1280:Earth
1264:torus
1256:donut
1252:torus
378:with
364:field
356:plane
7536:2011
7411:2011
7061:and
6406:and
6230:and
6187:and
6076:are
4605:and
4275:has
4201:are
4154:The
4124:and
2535:are
2515:and
1736:Let
1691:and
1669:flag
1667:. A
1240:area
1129:180.
1055:. A
1042:i.e.
1037:line
980:<
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