Degree project
A Brief Overview of Outer
Billiards on Polygons
Author: Michelle Zunkovic Supervisor: Hans Frisk Examiner: Karl-Olof Lindahl Date: 2015-12-17
Course Code: 2MA11E Subject:Mathematics Level: Bachelor
A Brief Overview of Outer Billiards on Polygons
Michelle Zunkovic
December 17, 2015
Contents
1 Introduction 3 2 Theory 3 2.1 Affine transformations . . . 32.2 Definition of Outer Billiards . . . 4
2.3 Boundedness and unboundedness of orbits . . . 4
2.4 Periodic orbits . . . 6
2.5 The T2 map and necklace dynamics . . . . 6
2.6 Quasi rational billiards . . . 8
3 Some different polygons 8 3.1 Orbits of the 2-gon . . . 8
3.2 Orbits of the triangle . . . 9
3.3 Orbits of the quadrilaterals . . . 10
3.3.1 The square . . . 10
3.3.2 Trapezoids and kites . . . 10
3.4 The regular pentagon . . . 11
3.5 The regular septagon . . . 12
3.6 Structuring all polygons . . . 13
4 Double kite 13 4.1 Background . . . 13
4.2 Method . . . 15
4.3 Result . . . 15
Abstract
Outer billiards were presented by J¨urgen Moser in 1978 as a toy model
of the solar system. It is a geometric construction concerning the motions around a convex shaped space. We are going to bring up the basic ideas with many figures and not focus on the proofs. Explanations how different types of orbits behave are given.
1
Introduction
It was J¨urgen Moser that aroused the interest of outer billiards to the public. In 1978 Moser published an article named ”Is the Solar System Stable?” [1]. He describes the outer billiards as a toy model of the solar system. There are several stability proofs for the solar system but only for limited time. Here the question is what happens with the motion for unlimited time, which is a pure mathematical question and does not necessary have a real world meaning. When studying outer billiards the easiest part is probably to understand the definition. There is still much unknown about the outer billiards and proofs are advanced and will not be the focus in this paper. The basic properties of outer billiards will be discussed and some different examples will be studied as well. In section two the theory will be treated and the definition of an outer billiard, classification of orbits and their motion will be given. In section three special cases of polygons will be brought up and explained, from the easiest case to more complicated ones. In section four we try to find a table that has an unbounded orbit and also try to find this orbit, and at last in section five the conclusions will be given.
2
Theory
In this section we are going to define some relevant concepts used to the study of specific polygons in section three. In particular using vector geometry we describe the motion of periodic orbits.
2.1
Affine transformations
A transformation A that maps points in the set R2 to itself and whose
deter-minant is nonzero is said to be an affine transformation, or an affinity. If X belongs to R2
then A(X) also belongs to R2.
With affinity some properties come, one of them is parallelism. That is, if n and m are two parallel lines, then their transformations A(n) and A(m) are parallel lines as well.
Another property is the ratio for affine transformations. Given three points on a line, p1, p2 and p3, the ratio between the vectors |p1− p2| and |p1− p3|
will be the same as the ratio between |A(p1) − A(p2)| and |A(p1) − A(p3)|.
In the plane you can displace any given vector in one specific direction by a shear transf ormation. With a strain transf ormation you can map the vector
(x, y) to (r1x, r2y). A geometric figure can be transformed into a similar figure
by a similarity transf ormation. Any affinity can be described by the product of strain, shear and similarity transformations and for two given triangles 41
and 42, there is an affine transformation from 41to 42. More about affinities
can be found in [2].
A transformation T is a reflection in the origin if it takes a point (x, y) in the plane and map it into (−x, −y). The transformation T commutes with affine transformations A, in the plane. That is, T A = AT , which means that if you apply T first and then the affine transformation A you will get the same result if you first apply the affine transformation A and then T . Affine transformations are important because if two billiards are connected with an affinity then their motions are qualitatively the same. As we will see below the outer billiard transformation is a reflection in a tangency point.
If a polygon is a lattice polygon then its vertices lie at rational points, that is the coordinates are rational numbers. When discussing the different polygons in section three we will see that it plays a big role whether the polygon is a lattice polygon or can be transformed into one with an affinity.
2.2
Definition of Outer Billiards
The motion around an outer billiard is a reflection through a tangency point on a convex billiard table. The table do not need to be regular or have a special shape. It can be a polygon, ellipse or a combination, for example a half circle. In this paper only polygons will be treated.
Choose a starting point x0outside the table, see figure 1. The point now has
two tangency points on the table, and we will choose to go clockwise consistently throughout this article. Reflect x0 in the tangency point γ0, so the distance
between x0 and γ0 will be the same as the distance between the new point
x1 and γ0. The dynamical system takes x0 to T (x0) = x1. Then x1 will be
mapped into x2 in the same way , T stands for tangent map. If a point has
more than one tangency point the system is not defined on that point. For a polygon it is the vertices that are the tangency points. In figure 1 the first two iterations are shown.
The first and most important thing you want to study is the behavior of the orbits. The question is, what is the character of the different orbits around the table we consider?
2.3
Boundedness and unboundedness of orbits
The dynamical system takes a point
x → T (x) → T2(x) → ... → Tn(x), were Tn(x) is the n-fold composition of itself, that is
Tn = T ◦ T ◦ T ◦ ... ◦ T
| {z }
n times
Figure 1: definition of motion.
An orbit of a point x0 ∈ R2 under T is defined as the set {xn}n≥0 where
xn= Tn(x0) and x0= T0(x0).
An orbit can be bounded or unbounded. The latter one is what one could guess, a trajectory that goes to infinity. There are two types of bounded orbits. The periodic ones and the infinite ones. An orbit is said to be periodic with respect to T if xn = x0 for some integer n ≥ 0. It is infinite if you never come back
to the same point but the orbit stays bounded in a certain area. A sufficient condition for boundedness is given in section 2.6.
With an example one can show that it can occur infinite orbits in a bounded region that never visit the same point twice. One of these is to consider
Xn+1= 2Xn (mod 1)
in the binary base where X0∈]0, 1]. For example will 13 = 0.010101... which we
can confirm by rewriting the right hand side to 1 4+ 1 16+ 1 64+ ... and by factoring 14, the sum will be equal to
1 4 · 1 1 − 1 4 = 1 3.
If we compute X1 for X0 = 0.010101..., the only thing that will happen is
the decimal sign will move one step to the right. Since the operation is in mod 1 the integer part will also vanish. Then X1 = 0.101010... and X2 =
0.010101... and we see that X0 = X2, hence it is a periodic expansion. Any
rational number has a periodic decimal expansion in all bases. Periodic orbits will therefore correspond to the rational numbers and the infinite ones to the irrational numbers. The unique thing with the irrationals is that it is no self repeating in its decimal form. Move the decimal point one step to the right and you will never have the same numbers to the right of the decimal point as you started with.
2.4
Periodic orbits
When we now have a better understanding of the motion around outer billiards we can give some properties of periodic orbits. Start with a point x0 and an
arbitrary outer billiard table. Then x0should reflect in the corner xc1 and end
up in the point x1. This map can be described using vectors. Then the first
step,
x1= x0+ 2(xc1− x0) = 2xc1− x0,
will give us the point x1. We do the same procedure,
x2= 2xc2− x1= 2xc2− 2xc1+ x0, (1)
to get the next point. Doing this one more time to then see the pattern, x3= 2xc3− x2= 2xc3− 2xc2+ 2xc1− x0.
For the n :th point,
xn= 2(xcn − xcn−1+ ... + (−1)
n+1x
c1) + (−1)
nx
0. (2)
In order for an orbit to be periodic, with period n, x0= xn. The starting
point must be equal to the n:th iteration. If we look at equation (2) we observe that the right hand side will only be equal to x0 when the first parenthesis is
equal to zero and n is even. So, the conclusion is that all periodic orbits are even.
One exception when an odd period can occur is when (xcn− xcn−1+ ... + (−1)
n+1x
c1) = x0
happens. We understand that this happens only for particular points x0 so in
general the periodic orbits have an even period, see section three.
2.5
The T
2map and necklace dynamics
For an understanding of the motion far away from the billiard table it is useful to consider the T2 map. The map contains every second point of an orbit. If
we have a set of points that are mapped into each other , {x0, x1, x2...}, then
every second point will correspond to the T2map, {x
0, x2, x4...}.
When x0is mapped into x1it will go through a tangency point xc1 on the
outer billiard table γ. From x1 to x2 the tangency point is xc2. The T
2 map
maps a point in the direction of the vector between the tangency points. The distance vector between x0and x2will be two times the vector V0= xc2− xc1,
see equation (1).
If the point x0is chosen far away from the table, then T2will form a polygon
that lies close to a so called necklace polygon Γ. For the construction of Γ see [7]. If γ is a polygon with n non-parallel sides then Γ will have 2n sides. In figure 2 the T2map for a quadrilateral, with vertices in (0, 0), (0, −1), (1, −2) and (3, 0),
Figure 2: T2 map around billiard table γ.
(a) Periodic orbit with
pe-riod 12 (b) Invension
Figure 3: Necklace dynamics for a triangle billiard.
is shown. The sides in the polygon represent the sides of the quadrilateral and its diagonals. Since we chose a starting point far away from the table we will not see γ inside of Γ in figure 2.
Instead of considering a point that maps around a fixed polygon we can hold the point fixed and map the polygon. This motion is called necklace dynamics, see figure 3. The point x0 should be mapped in the point γ0 for the regular
motion. Now, in the necklace dynamics, the polygon γ will be mapped in the same point γ0. The mapping is an inversion of the polygon in γ0.
An inversion in the origin takes (x, y) and maps it into (−x, −y). The inversion can be accomplished in any point, not just the origin. For the necklace dynamics the polygon moves by inversion around the fixed point.
Draw lines through the point x0 that are parallel with the sides of γ, then
2n secions appear if γ has n non-parallel sides. In every section, i, the polygon will be mapped in the same direction as the vector Vibetween the two tangency
points. That is the same vectors as for the T2 map. In every section there is
a different vector that the polygon follows. When it maps into a new section it also switch direction. The necklace dynamics is interesting of several reasons, partly it is a different veiw of the motion. Also it is a necessary concept in the proof that all quasi rational polygons are bounded. The term quasi rational will be explaned in next subsection.
2.6
Quasi rational billiards
When the polygon maps around a fixed point an orbit around the point occurs. The further the billiard γ lies from the point the closer this orbit will be to the necklace polygon. Every side Si for i = 1, 2, ..., 2n, in the necklace polygon
can be described as Si= tiVi. Where Vi is the vector and ti is the number of
vectors on each side. The necklace polygon will always be closed, even if the orbits are non periodic or unbounded [7]. For lattice polygons tiwill be rational.
Definition 1. Consider a polygon γ and the corresponding necklace polygon Γ which in general has 2n sides. Every side is on the form Si = tiVi, for
i = 1, 2, ..., 2n.. If the ratio ti/tj for any sides i and j belongs to the rational
numbers, then γ is said to be quasi rational.
When a polygon is quasi rational then it implies that all orbits are bounded [3, 7]. Note that this is only an implication and not an equivalence. That is, if all orbits are bounded the table does not need to be quasi rational. All lattice polygons are quasi rational [3]. Since all vertices lies at rational coordinates then the vectors Vi are rational and ti will be rational as well.
When ti/tjis an irrational number the polygon is not quasi rational and we
can not say if it will have unbounded orbits or not.
You can approximate an irrational number with a fraction. To get closer to the number, the nominator and denominator will be larger and larger. This can be explained with a so called continued fraction. Any irrational number can be expressed with an infinite continued fraction, a unique one. [13]
Here is an example of a continuous fraction of the irrational number√5 − 2, √
5 − 2 = 0 + 1
4 + 4+ 11 4+4+...1
,
and this can be written as√5 − 2 = [0; 4, 4, 4, 4, ...].
3
Some different polygons
In this section some different polygons will be considered. It will be explained how the orbits behave and the character of every polygon is discussed.
The letter n will refer to the polygon with n sides. For instance the pentagon can be mentioned as n = 5.
3.1
Orbits of the 2-gon
The simplest table to study outer billiards around is the 2-gon, which only is a line segment and not a polygon. Then there are only two tangent points which lie on the end point of a straight line, and the motion will go from one point to the other. The straight line going through both points is called the collapse line. A point on this line does not have an unique tangent point on the dual
billiard table. By definition a point x0should go through its tangent point and
twice the distance, and if this tangent point is not unique then the system is undefined and the system collapses.
There will only exist one collapse line of the dual billiard table of a 2-gon and a point will never reach this line if it is not starting on it. Choose an arbitrary point x0, not lying on the line, for example under the line. When the point
maps to another point x1, it will end above the line. Since the vector from x0
to the tangency point will be reflected in the line to reach x1, the system will
go on without risk of landing on the line.
All orbits around the 2-gon are unbounded. No matter what starting point you choose the orbit will go to infinity. Since the motion is a reflection in the collapse line, the distance from the 2-gon will increase for every iteration. For some points close to the polygon it will take two iterations before the distance starts to increase. In equation (1) it is shown that for the second iteration it is a movement twice the distance between the tangency points. And since the tangency points here always will be the same, the iteration will go further from the 2-gon in every second iteration.
3.2
Orbits of the triangle
For the equilateral triangle every orbit is periodic. If the collapse lines and their reflections are drawn, tiles shaped as triangles and hexagons occur as shown in figure 4a. Points belonging to the triangles have the period 12, 24, 36.... If you choose a point just in the middle of a hexagon the period will be 3, 9, 15.... That is not the case for the whole hexagon, if the starting point is chosen to be exactly in the middle the period will be 6, 18, 30..., in other words twice as long periods if the point is not chosen in the middle [1].
We know that the outer billiard system commutes with affine transforma-tions and that every triangle can, using an affinity, be transformed into an other triangle.
Therefore the periods and motions are the same for all triangles, no matter how they look. In figure 4b and 4c you see two different triangles, and the periodic tiles look very similar to the onces in figure 4a. The periodic tiles are almost the same, just transformed. In figure 4b it shows a triangle that has all sides of different length and in 4c an isosceles triangle with a 150◦ angle. For both these triangles we see that the structure around the table is very similar. The periods will be the same. If you increase the angle to be near 180◦ the triangle will be closer and closer to the 2-gon. But will the motion be similar to the 2-gons? When the angel increase the triangular and hexagonal regions will be stretched out. But the structure will always be the same. So not until the angle becomes 180◦, unbounded orbits will occur. All triangles belong to the same affinity class so we only have to study lattice triangles. This is not the same case as for the quadrilaterals.
(a) regular triangle (b) nonregular triangle (c) isosceles triangle
Figure 4: Three different triangular billiards, the motion is qualitative the same.
3.3
Orbits of the quadrilaterals
If the quadrilateral is a lattice polygon, then all orbits are bounded. We will start by looking at some examples of lattice quadrilaterals and then see what happens if it is not lattice.
3.3.1 The square
All orbits of the square are as well behaved as the ones for the triangle. They are all periodic with period 4k, for k = 1, 2, 3, .... Drawing the collapse lines we see the different tiles and all of them are squares. In figure 5b you see the periods for different values of k.
We can also take a look how the T2map will look like. Since we are starting
far away from the table γ, orbits will always go through every second vertex of the square. Thus, the map Γ have four sides as well. But the form of Γ will be a square where the sides has the same directions as the diagonals in the square γ. See figure 5c.
(a) collapse lines (b) periods 4k (c) The T
2
map, Γ
Figure 5: Tiles and T2 for the square
3.3.2 Trapezoids and kites
A trapezoid is a quadrilateral that has two parallel sides.The trapezoid is not an affine transformation of a square. We know that for a quasi rational billiard
Figure 6: Unbounded orbit for the Penrose kite
table, all motion is bounded. But this does not hold the other way around. If all motion is bounded this does not imply that the polygon is quasi rational. One of these cases is the trapezoid. It is not quasi rational but still all motions are bounded [6].
A quadrilateral were the two adjacent sides have the same length is called a kite, no sides are parallel. A kite with exactly one irrational coordinate is not an affine transformation of a square. There is no such matrix that can map this kite into a square. The interesting thing with the kites is that unbounded movement can occur. It was first shown in the the special case when one vertex lies at (√5 − 2, 0) called the Penrose kite, that unbounded orbits occurred. The other corners are (−1, 0), (0, 1) and (0, −1). Then it was shown by Richard Schwartz that all kites with one irrational coordinate has unbounded orbits[12]. Unbounded orbits for this type of kites can be found by starting at the point (1 − q, 1) where q is the right irrational coordinate. One unbounded orbit of the Penrose kite is plotted in figure 6. Here it is 10 000 points plotted and the biggest y-value around y = 90. The motion will go on out from the kite, in a spiral fashion.
3.4
The regular pentagon
The regular pentagon is the the first polygon with star regions and inside them bounded orbits occur. If you start on an infinite orbit and plot the points you can see a star region. Wherever you start inside this region, you will never leave it.
In figure 7 the first star region is shown. A unique thing with the pentagon is that it only needs one orbit to fill the whole star region. All other polygons
Figure 7: Star region of pentagon
with star regions need multiple orbits to fill the space. The pentagon and all other regular polygons for n = 5, 7, 8..., have the star region structure. These regions are separated with rings of 2n-tiles. This is called the global structure. Part of these tiles is seen in figure 7, outside the inner star region there is a ring with 10-gons. In figure 8 the ring is clearly shown and the tiles are 14-gons. The tiles structuring the area around the table will never have more than 2n sides. By drawing the collapse lines and their reflections you can see how the structures will look like. And the tiles can only be of at most type 2n since it is only n collapse lines.
3.5
The regular septagon
The global structure of the septagon is very similar to the pentagon. It has its star regions and the 2n tiles creating a ring between them. Here it is no longer possible to fill a whole star region with just one orbit as it is for the pentagon. In figure 8 you can see part of the two first star regions and the first ring of tiles n = 14. To create these figures Mathematica has been used. The infinite orbits are not particularly easy to find, therefor unfortunately there is only one orbit in each star region.
The global structure for the septagon is similar to the structure for other regular polygons, but looking at the fine structure it behaves different from all other polygons studied. In [9] Hughes thinks that the 11-gon behaves similar as the septagon. For a n-gon, tiles with n and 2n vertices occur. This is not the case for the regular septagon in the fine structure. Very small tiles, some around three thousandths of the septagons side length, occur in a mysterious way. Their periods are very different from all others, one type of them have period 57848 and these were found by R. Schwartz [4]. A careful study has been
Figure 8: Star regions of septagon
done in [9].
3.6
Structuring all polygons
By subdividing all the polygons in to groups we can give them some structure. Starting with the smallest group and that will be regular n-gons for n = 3, 4, 6 connected to lattice polygons by an affine transformation. These polygons have only periodic orbits. Since they are lattice polygons they are also quasi rational. Among the quasi rational polygons, we also find those with periodic motion and the regular polygons. The question if there is a polygon that is neither a lattice polygon nor regular but quasi rational, is yet open.
Then there are the bounded ones. All quasi rational polygons are bounded but there are also polygons that are bounded but not quasi rational, for example a trapezoid.
The last group are polygons that are not bounded, that has orbits that goes to infinity. Here we have the irrational kite.
The line segment when n = 2 is not included since it is not a polygon. In figure 9 a Venn diagram of all the polygons is shown.
4
Double kite
4.1
Background
Almost every orbit of any polygon is bounded, either periodic or infinite. Most regular polygons have star regions and tile rings that separate the regions. When starting in one star region you will never leave it. By changing the polygons to non-regular ones, these polygon rings get deformed and can open up. This
means it will be holes that an orbit can go trough and leave the star region it started in.
It is proven by R.Schwartz that every kite with an irrational coordinate has unbounded orbits. Is there some other billiard table that it is especially easy to find unbounded orbits in? One candidate that now will be investigated is what we call the double kite. A totally unsymmetrical polygon with two irrational coordinates. The goal is to find an orbit that is unbounded.
4.2
Method
Since a proof is out of reach the method to find unbounded orbits will be based on testing carefully selected starting points. Mathematica will be the program used to do the testing. To plot n = 5 and n = 7 infinite orbits, two sides of the polygon is extended and their intersection is numerically determined. By choosing this intersection to be the starting point infinite orbits will arise. The same technique is used for the double kite.
4.3
Result
The double kite with coordinates (−1, 0), (0, −1), (√5 − 2, 0) and (0,√1
3) is first
to be considered. Using the same method as for finding infinite orbits for n = 5 and n = 7, we extend two sides of the double kite and numerically determine the intersection. This only generates periodic orbits.
For the Penrose kite the first unbounded orbit that was found has its start-ing point at (1 − q, 1). This point is also tested for our double kite but still only periodic orbits occurs. The height of the Penrose kite is 1, just as the y coordinate in the starting point, so the coordinate (3 −√5,√1
3) were also tested
without any different result.
An interesting discovery can be made if we take a look at the irrational numbers and their continued fractions. The number √5 − 2 has only fours in its expansion exept from the integer part. In other words its continued fraction expansion is [0; 4, 4, 4, ...], as described in section 2.6. For the number√1
3, this is
equal to [0; 1, 1, 2, 1, 2, 1, 2, ...]. If we instead replace √1
3 with a number that has
the same pattern as√5 − 2, in other words [0; a, a, a, ...]. One of these numbers is √1
2, which only has two:s in its expansion. When starting in (1 − q, 1) for
q =√5 − 2 and the upper coordinate as (0,√1
2) then it looks like unbounded
motion appears.
It is not so easy to find out if an orbit is periodic. If a coordinate repeat itself then the orbit is periodic and we know this. But if there is no repeating then we can not say if it is periodic or not. When we now trying to find an unbounded orbit, we can only conjecture that it is unbounded. For the first 107 iterations it behaves similarly to the unbounded orbit in the Penrose kite
5
Conclusions
We have now discussed the most fundamental properties of outer billiards. Most polygons behave in a similar way and then we have those that stand out of the crowd for example the septagon. Then we have the question, are there any other polygons, besides the kites, that have unbounded orbits? There are many questions whether there really are unbounded orbits or not. Some of the small tiles that were found in the septagon had a very long period, can one draw any connections to the solar system!?
Since we are not able to execute any proofs we can not guarantee that unbounded orbits have been found for our double kite. Nor we can say that the continued fraction has something to do with the unbounded orbit. It would be interesting to consider the relation between continued fractions and unbounded orbits more. For further studies the proof of unbounded orbits of kites should be considered. Then one can hope to make a hypothesis whether the expansion of the irrational coordinate plays any role to the orbits or not.
References
[1] J. Moser, Is the solar system stable?, Math. Int., 1, 65-71, 1978. [2] J. N. Cederberg, A Course in Modern Geometries, Springer, 2000.
[3] S.Tabachnikov, Panoramas et Synth`eses, Soc. Math`ematique de France, 1995.
[4] R. Schwartz, Outer biliards on kites, 2010.
[5] F.Dogru, S.Tabachnikov, Dual Billiard, Math. Int. vol 27, No 4, PP 18-25, 2005.
[6] D. Genin, Research announcement: boundedness of orbits for trape-zoidal outer billiards. Electronic Research Announc. Math. Sci. 15 , 7178. MR2457051 (2009k:37036), 2008.
[7] Dual Polygonal Billiards and Neckalace Dynamics, E.Gutkin and N.Simanyi, Commun. Math. Phys. 143, 431-449 (1992).
[8] A proof of Cutler’s theorem on the existence of periodic orbits in polygonal outer billiards, S.Tabachnikov, (2013).
[9] http://dynamicsofpolygons.org/PDFs/N7Summary.pdf (Accessed: 19 May 2015).
[10] http://dynamicsofpolygons.org/PDFs/N5Summary.pdf(Accessed: 19 May 2015).
[11] http://dynamicsofpolygons.org/PDFs/LatticePolygons.pdf (Accessed: 19 May 2015).
[12] http://dynamicsofpolygons.org/PDFs/PenroseSummary.pdf (Accessed: 19 May 2015).
[13] http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/ cfINTRO.html (Accessed: 22 May 2015).
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