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 Department Of Mathematics

A Brief Overview of Outer Billiards on Polygons

Michelle Zunkovic

December 17, 2015

Contents

1 Introduction 3

2 Theory 32.1 Affine transformations . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Definition of Outer Billiards . . . . . . . . . . . . . . . . . . . . . 42.3 Boundedness and unboundedness of orbits . . . . . . . . . . . . . 42.4 Periodic orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 The T 2 map and necklace dynamics . . . . . . . . . . . . . . . . 62.6 Quasi rational billiards . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Some different polygons 83.1 Orbits of the 2-gon . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Orbits of the triangle . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Orbits of the quadrilaterals . . . . . . . . . . . . . . . . . . . . . 10

3.3.1 The square . . . . . . . . . . . . . . . . . . . . . . . . . . 103.3.2 Trapezoids and kites . . . . . . . . . . . . . . . . . . . . . 10

3.4 The regular pentagon . . . . . . . . . . . . . . . . . . . . . . . . 113.5 The regular septagon . . . . . . . . . . . . . . . . . . . . . . . . . 123.6 Structuring all polygons . . . . . . . . . . . . . . . . . . . . . . . 13

4 Double kite 134.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Conclusions 16

2

Abstract

Outer billiards were presented by Jurgen Moser in 1978 as a toy modelof the solar system. It is a geometric construction concerning the motionsaround a convex shaped space. We are going to bring up the basic ideaswith many figures and not focus on the proofs. Explanations how differenttypes of orbits behave are given.

1 Introduction

It was Jurgen 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 areseveral stability proofs for the solar system but only for limited time. Herethe question is what happens with the motion for unlimited time, which is apure mathematical question and does not necessary have a real world meaning.When studying outer billiards the easiest part is probably to understand thedefinition. There is still much unknown about the outer billiards and proofsare advanced and will not be the focus in this paper. The basic properties ofouter billiards will be discussed and some different examples will be studied aswell. In section two the theory will be treated and the definition of an outerbilliard, classification of orbits and their motion will be given. In section threespecial cases of polygons will be brought up and explained, from the easiest caseto more complicated ones. In section four we try to find a table that has anunbounded orbit and also try to find this orbit, and at last in section five theconclusions will be given.

2 Theory

In this section we are going to define some relevant concepts used to the studyof specific polygons in section three. In particular using vector geometry wedescribe 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 Xbelongs to R2 then A(X) also belongs to R2.

With affinity some properties come, one of them is parallelism. That is, ifn and m are two parallel lines, then their transformations A(n) and A(m) areparallel lines as well.

Another property is the ratio for affine transformations. Given three pointson 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 ashear transformation. With a strain transformation you can map the vector

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(x, y) to (r1x, r2y). A geometric figure can be transformed into a similar figureby a similarity transformation. Any affinity can be described by the productof strain, shear and similarity transformations and for two given triangles 41

and 42, there is an affine transformation from 41 to 42. More about affinitiescan be found in [2].

A transformation T is a reflection in the origin if it takes a point (x, y) in theplane and map it into (−x,−y). The transformation T commutes with affinetransformations A, in the plane. That is, TA = AT , which means that if youapply T first and then the affine transformation A you will get the same result ifyou first apply the affine transformation A and then T . Affine transformationsare important because if two billiards are connected with an affinity then theirmotions are qualitatively the same. As we will see below the outer billiardtransformation is a reflection in a tangency point.

If a polygon is a lattice polygon then its vertices lie at rational points, thatis the coordinates are rational numbers. When discussing the different polygonsin section three we will see that it plays a big role whether the polygon is alattice 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 ona convex billiard table. The table do not need to be regular or have a specialshape. 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 x0 outside the table, see figure 1. The point now hastwo tangency points on the table, and we will choose to go clockwise consistentlythroughout this article. Reflect x0 in the tangency point γ0, so the distancebetween x0 and γ0 will be the same as the distance between the new pointx1 and γ0. The dynamical system takes x0 to T (x0) = x1. Then x1 will bemapped into x2 in the same way , T stands for tangent map. If a point hasmore than one tangency point the system is not defined on that point. For apolygon it is the vertices that are the tangency points. In figure 1 the first twoiterations are shown.

The first and most important thing you want to study is the behavior of theorbits. The question is, what is the character of the different orbits around thetable we consider?

2.3 Boundedness and unboundedness of orbits

The dynamical system takes a point

x→ T (x)→ T 2(x)→ ...→ Tn(x),

were Tn(x) is the n-fold composition of itself, that is

Tn = T ◦ T ◦ T ◦ ... ◦ T︸ ︷︷ ︸n times

.

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Figure 1: definition of motion.

An orbit of a point x0 ∈ R2 under T is defined as the set {xn}n≥0 wherexn = Tn(x0) and x0 = T 0(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. Theperiodic ones and the infinite ones. An orbit is said to be periodic with respectto T if xn = x0 for some integer n ≥ 0. It is infinite if you never come backto the same point but the orbit stays bounded in a certain area. A sufficientcondition for boundedness is given in section 2.6.

With an example one can show that it can occur infinite orbits in a boundedregion 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− 14

=1

3.

If we compute X1 for X0 = 0.010101..., the only thing that will happen isthe decimal sign will move one step to the right. Since the operation is inmod 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. Anyrational number has a periodic decimal expansion in all bases. Periodic orbitswill therefore correspond to the rational numbers and the infinite ones to theirrational numbers. The unique thing with the irrationals is that it is no selfrepeating in its decimal form. Move the decimal point one step to the right andyou will never have the same numbers to the right of the decimal point as youstarted with.

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2.4 Periodic orbits

When we now have a better understanding of the motion around outer billiardswe can give some properties of periodic orbits. Start with a point x0 and anarbitrary outer billiard table. Then x0 should reflect in the corner xc1 and endup in the point x1. This map can be described using vectors. Then the firststep,

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+1xc1) + (−1)nx0. (2)

In order for an orbit to be periodic, with period n, x0 = xn. The startingpoint must be equal to the n:th iteration. If we look at equation (2) we observethat the right hand side will only be equal to x0 when the first parenthesis isequal to zero and n is even. So, the conclusion is that all periodic orbits areeven.

One exception when an odd period can occur is when

(xcn − xcn−1 + ...+ (−1)n+1xc1) = x0

happens. We understand that this happens only for particular points x0 so ingeneral the periodic orbits have an even period, see section three.

2.5 The T 2 map and necklace dynamics

For an understanding of the motion far away from the billiard table it is usefulto consider the T 2 map. The map contains every second point of an orbit. Ifwe have a set of points that are mapped into each other , {x0,x1,x2...}, thenevery second point will correspond to the T 2 map, {x0,x2,x4...}.

When x0 is mapped into x1 it will go through a tangency point xc1 on theouter billiard table γ. From x1 to x2 the tangency point is xc2 . The T 2 mapmaps a point in the direction of the vector between the tangency points. Thedistance vector between x0 and x2 will be two times the vector V0 = xc2−xc1 ,see equation (1).

If the point x0 is chosen far away from the table, then T 2 will form a polygonthat 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 2the T 2 map for a quadrilateral, with vertices in (0, 0), (0,−1), (1,−2) and (3, 0),

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Figure 2: T 2 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 andits diagonals. Since we chose a starting point far away from the table we willnot see γ inside of Γ in figure 2.

Instead of considering a point that maps around a fixed polygon we can holdthe 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 regularmotion. Now, in the necklace dynamics, the polygon γ will be mapped in thesame 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). Theinversion can be accomplished in any point, not just the origin. For the necklacedynamics the polygon moves by inversion around the fixed point.

Draw lines through the point x0 that are parallel with the sides of γ, then2n secions appear if γ has n non-parallel sides. In every section, i, the polygonwill be mapped in the same direction as the vector Vi between the two tangencypoints. That is the same vectors as for the T 2 map. In every section there isa different vector that the polygon follows. When it maps into a new section italso 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 theproof that all quasi rational polygons are bounded. The term quasi rationalwill be explaned in next subsection.

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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 tothe necklace polygon. Every side Si for i = 1, 2, ..., 2n, in the necklace polygoncan be described as Si = tiVi. Where Vi is the vector and ti is the number ofvectors on each side. The necklace polygon will always be closed, even if theorbits are non periodic or unbounded [7]. For lattice polygons ti will be rational.

Definition 1. Consider a polygon γ and the corresponding necklace polygonΓ which in general has 2n sides. Every side is on the form Si = tiVi, fori = 1, 2, ..., 2n.. If the ratio ti/tj for any sides i and j belongs to the rationalnumbers, 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, ifall orbits are bounded the table does not need to be quasi rational. All latticepolygons are quasi rational [3]. Since all vertices lies at rational coordinatesthen the vectors Vi are rational and ti will be rational as well.

When ti/tj is an irrational number the polygon is not quasi rational and wecan not say if it will have unbounded orbits or not.

You can approximate an irrational number with a fraction. To get closer tothe number, the nominator and denominator will be larger and larger. This canbe explained with a so called continued fraction. Any irrational number can beexpressed 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 + 14+ 1

4+ 14+...

,

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 explainedhow 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 pentagoncan 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 aline segment and not a polygon. Then there are only two tangent points whichlie on the end point of a straight line, and the motion will go from one pointto the other. The straight line going through both points is called the collapseline. A point on this line does not have an unique tangent point on the dual

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billiard table. By definition a point x0 should go through its tangent point andtwice the distance, and if this tangent point is not unique then the system isundefined and the system collapses.

There will only exist one collapse line of the dual billiard table of a 2-gon anda point will never reach this line if it is not starting on it. Choose an arbitrarypoint x0, not lying on the line, for example under the line. When the pointmaps 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 willgo on without risk of landing on the line.

All orbits around the 2-gon are unbounded. No matter what starting pointyou choose the orbit will go to infinity. Since the motion is a reflection in thecollapse line, the distance from the 2-gon will increase for every iteration. Forsome points close to the polygon it will take two iterations before the distancestarts to increase. In equation (1) it is shown that for the second iteration itis a movement twice the distance between the tangency points. And since thetangency points here always will be the same, the iteration will go further fromthe 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 theirreflections are drawn, tiles shaped as triangles and hexagons occur as shownin figure 4a. Points belonging to the triangles have the period 12, 24, 36.... Ifyou 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 beexactly in the middle the period will be 6, 18, 30..., in other words twice as longperiods 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 othertriangle.

Therefore the periods and motions are the same for all triangles, no matterhow they look. In figure 4b and 4c you see two different triangles, and theperiodic tiles look very similar to the onces in figure 4a. The periodic tiles arealmost the same, just transformed. In figure 4b it shows a triangle that has allsides of different length and in 4c an isosceles triangle with a 150◦ angle. Forboth 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◦ thetriangle will be closer and closer to the 2-gon. But will the motion be similarto the 2-gons? When the angel increase the triangular and hexagonal regionswill be stretched out. But the structure will always be the same. So not untilthe angle becomes 180◦, unbounded orbits will occur. All triangles belong tothe same affinity class so we only have to study lattice triangles. This is not thesame case as for the quadrilaterals.

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(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 willstart by looking at some examples of lattice quadrilaterals and then see whathappens 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. Theyare all periodic with period 4k, for k = 1, 2, 3, .... Drawing the collapse lineswe see the different tiles and all of them are squares. In figure 5b you see theperiods for different values of k.

We can also take a look how the T 2 map will look like. Since we are startingfar away from the table γ, orbits will always go through every second vertex ofthe square. Thus, the map Γ have four sides as well. But the form of Γ will bea square where the sides has the same directions as the diagonals in the squareγ. See figure 5c.

(a) collapse lines (b) periods 4k(c) The T 2 map, Γ

Figure 5: Tiles and T 2 for the square

3.3.2 Trapezoids and kites

A trapezoid is a quadrilateral that has two parallel sides.The trapezoid is notan affine transformation of a square. We know that for a quasi rational billiard

10

Figure 6: Unbounded orbit for the Penrose kite

table, all motion is bounded. But this does not hold the other way around. Ifall 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 motionsare bounded [6].

A quadrilateral were the two adjacent sides have the same length is calleda kite, no sides are parallel. A kite with exactly one irrational coordinate isnot an affine transformation of a square. There is no such matrix that can mapthis kite into a square. The interesting thing with the kites is that unboundedmovement can occur. It was first shown in the the special case when one vertexlies 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 RichardSchwartz 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 ofthe Penrose kite is plotted in figure 6. Here it is 10 000 points plotted and thebiggest y-value around y = 90. The motion will go on out from the kite, in aspiral fashion.

3.4 The regular pentagon

The regular pentagon is the the first polygon with star regions and inside thembounded orbits occur. If you start on an infinite orbit and plot the points youcan see a star region. Wherever you start inside this region, you will never leaveit.

In figure 7 the first star region is shown. A unique thing with the pentagonis that it only needs one orbit to fill the whole star region. All other polygons

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Figure 7: Star region of pentagon

with star regions need multiple orbits to fill the space. The pentagon and allother regular polygons for n = 5, 7, 8..., have the star region structure. Theseregions 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 ringwith 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 2nsides. By drawing the collapse lines and their reflections you can see how thestructures will look like. And the tiles can only be of at most type 2n since itis only n collapse lines.

3.5 The regular septagon

The global structure of the septagon is very similar to the pentagon. It has itsstar regions and the 2n tiles creating a ring between them. Here it is no longerpossible 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 tilesn = 14. To create these figures Mathematica has been used. The infinite orbitsare not particularly easy to find, therefor unfortunately there is only one orbitin each star region.

The global structure for the septagon is similar to the structure for otherregular polygons, but looking at the fine structure it behaves different from allother polygons studied. In [9] Hughes thinks that the 11-gon behaves similaras the septagon. For a n-gon, tiles with n and 2n vertices occur. This is notthe case for the regular septagon in the fine structure. Very small tiles, somearound three thousandths of the septagons side length, occur in a mysteriousway. Their periods are very different from all others, one type of them haveperiod 57848 and these were found by R. Schwartz [4]. A careful study has been

12

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, 6connected to lattice polygons by an affine transformation. These polygons haveonly periodic orbits. Since they are lattice polygons they are also quasi rational.

Among the quasi rational polygons, we also find those with periodic motionand the regular polygons. The question if there is a polygon that is neither alattice polygon nor regular but quasi rational, is yet open.

Then there are the bounded ones. All quasi rational polygons are boundedbut there are also polygons that are bounded but not quasi rational, for examplea trapezoid.

The last group are polygons that are not bounded, that has orbits that goesto infinity. Here we have the irrational kite.

The line segment when n = 2 is not included since it is not a polygon. Infigure 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. Mostregular polygons have star regions and tile rings that separate the regions. Whenstarting in one star region you will never leave it. By changing the polygonsto non-regular ones, these polygon rings get deformed and can open up. This

13

Figure 9: Structuring all polygons

14

means it will be holes that an orbit can go trough and leave the star region itstarted in.

It is proven by R.Schwartz that every kite with an irrational coordinate hasunbounded orbits. Is there some other billiard table that it is especially easy tofind unbounded orbits in? One candidate that now will be investigated is whatwe call the double kite. A totally unsymmetrical polygon with two irrationalcoordinates. 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 basedon testing carefully selected starting points. Mathematica will be the programused to do the testing. To plot n = 5 and n = 7 infinite orbits, two sides ofthe polygon is extended and their intersection is numerically determined. Bychoosing this intersection to be the starting point infinite orbits will arise. Thesame 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 = 5and n = 7, we extend two sides of the double kite and numerically determinethe 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 stillonly periodic orbits occurs. The height of the Penrose kite is 1, just as the ycoordinate 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 inits expansion exept from the integer part. In other words its continued fractionexpansion 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 numbersis 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 repeatingthen we can not say if it is periodic or not. When we now trying to find anunbounded orbit, we can only conjecture that it is unbounded. For the first107 iterations it behaves similarly to the unbounded orbit in the Penrose kitein figure 6.

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5 Conclusions

We have now discussed the most fundamental properties of outer billiards. Mostpolygons behave in a similar way and then we have those that stand out of thecrowd for example the septagon. Then we have the question, are there anyother polygons, besides the kites, that have unbounded orbits? There are manyquestions whether there really are unbounded orbits or not. Some of the smalltiles that were found in the septagon had a very long period, can one draw anyconnections to the solar system!?

Since we are not able to execute any proofs we can not guarantee thatunbounded orbits have been found for our double kite. Nor we can say that thecontinued fraction has something to do with the unbounded orbit. It would beinteresting to consider the relation between continued fractions and unboundedorbits more. For further studies the proof of unbounded orbits of kites shouldbe considered. Then one can hope to make a hypothesis whether the expansionof 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 Syntheses, Soc. Mathematique 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 polygonalouter billiards, S.Tabachnikov, (2013).

[9] http://dynamicsofpolygons.org/PDFs/N7Summary.pdf (Accessed: 19 May2015).

[10] http://dynamicsofpolygons.org/PDFs/N5Summary.pdf(Accessed: 19 May2015).

[11] http://dynamicsofpolygons.org/PDFs/LatticePolygons.pdf (Accessed: 19May 2015).

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[12] http://dynamicsofpolygons.org/PDFs/PenroseSummary.pdf (Accessed: 19May 2015).

[13] http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/cfINTRO.html (Accessed: 22 May 2015).

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