Description: Description: Description: Description: Description: 11WBuds1zoom.jpg

Dynamics of Polygons

Description: Description: Description: Description: Description: 11WBuds1zoompt3.jpg

Site Map










Description: Description: Description: Description: Description: 5star.jpg

The web of the regular pentagon (N = 5)


Description: Description: Description: Description: Description: 8starpt2.jpg

The web of the regular octagon ( N = 8)


Three nested projections for N = 8 showing generations 3,4 and 5.



The inner web for N = 11


The First Family for N = 11. The region of interest lies between two regular 22-gons known as S1 and S2.









Description: Description: Description: Description: Description: Compass.jpg


Compass and straightedge construction for the regular heptadecagon (N= 17)





A P2 projection for N= 17





Description: Description: Description: Description: Description: kite.jpg

A Penrose kite which can be flown using Cantor string.


Billiards inside a regular pentagon





Trailing edges for the regular triangle in magenta and the inverse image of these edges in blue. So blue maps to magenta under τ.


The web for the regular triangle



The web for a non-regular triangle is affinely equivalent to the web for a regular triangle.




The web for an ‘elliptical’ pentagon is affinely equivalent to the web for the regular case – which is shown at the top of the page.








A Young Hare by Albrecht Dürer - 1502



The web for the regular heptagon on the left compared to the Dürer heptagon on the right.









































Description: Description: Description: Description: Description: Planets.png



Description: Description: Description: Description: Description: jupiter-saturn.jpg
















































Step-1 orbit of S1 center. It is period 7. The remaining points in S1 take two circuits around Mom, so they are period 14.


























Description: Description: Description: Description: Description: N7chaosNew.jpg

A region above S1 showing small-scale ‘chaos’































8 Generations of Moms for N = 7





























4k + 1 primes

4k + 3 primes















6 Generations of Moms for N = 13




































The First Family for N = 17 showing Mom on the right and Dad on the left. There are 17 such Dads surrounding Mom.

















P2 projection of the S[1] center in

ring 2 for N = 17






























Disquisitiones Arithmeticae

By C. F. Gauss – 1801



   C.F. Gauss (1777-1855)



Neils Able (1802−1829)


Evariste Galois 1811−1832)




A period 12 orbit for N = 6



A P5 projection for N = 16



This is a non-profit site devoted to the history and dynamics of convex polygons. We will define the dynamical structure of a polygon using repeated reflections about the vertices as shown in the two examples below. This is called the ‘outer billiards’ map or Tangent map. (The definition is given below.) There are other maps which yield a similar structure. (See Chronology or Digital Filters for an overview of a wide range of related mappings which occur in electrical circuits, celestial mechanics, complex analysis and quantum mechanics.)


These reflections reveal algebraic and geometric structure which is inherent in the polygon so we call it the ‘genetic structure’ of the polygon. For regular polygons with prime number of vertices, the basic geometry can be derived independently of the Tangent map. Regular polygons typically have structure on all scales but it is not clear what other polygons have this property. The pentagon and octagon above have a simple fractal structure, but most regular polygons have a more complex structure which is often multi-fractal.

Aside from a few simple cases, very little is known about the genetic structure of regular polygons. The vertices of the regular 7-gon cannot be specified algebraically without a cubic equation and the regular 11-gon requires a quintic. In both cases the resulting dynamics are very complex.

At the top of this page are two ‘density’ plots showing the small-scale dynamics of the regular 11-gon. (Click on any image to see  a larger image or go to Images to choose from thumbnails.) These two plots are based on the same data which is the ‘winding number’ of each point in the grid. The ‘winding number’ of a point is a measure of the average rotation per iteration. For example the points in S1 get their name from their ‘Step-1’ periodic orbit which advances just one vertex on each iteration, so the winding number is ω = 1/11. This is the smallest possible winding number for N = 11. In the 3D plot above, S1 cannot be seen because it is in a deep hole at the upper left. The colored plot is from the opposite perspective and S1 is in the foreground colored deep red. The dark blue strip is the ‘shock wave’ that occurs when the sphere of influence of S1 meets that of S2.

The points in the thin hexagon ‘skating rink’ are also periodic so they have constant winding number. The period is 169·2 and ω = 41/169. The neighboring dynamics are very complex and the skating rink is bordered by a multi-fractal landscape of immense complexity. This plot is 1000 by 1000 and each point is iterated for 3000 iterations to estimate the winding numbers, but there are still many regions which are poorly resolved.

In the first plot below, the colors show the 4 invariant regions which surround N = 11. The points of interest here are confined to the inner region. Sometimes a single orbit is dense in an entire invariant region, but is not likely that such orbits exist for N = 11. However such an orbit does exist for N = 5 (which has only one invariant region). The web shown above for N = 5 is generated from a single non-periodic orbit.

              Description: Description: Description: Description: Description: 11S1newpt3.jpg

We may never know the limiting small-scale structure for the regular 11-gon because the complexity tends to increase with each new ‘generation’. The plots above resolve only the first 4 or 5 generations. The 11-gon has four non-trivial scales which define each generation. These scales are typically irrational and non-commensurate and they act across generations, so a 6th generation polygon could have radius Scale[3]2·Scale[5]3/Scale[2]. This type of interaction is well documented with the regular heptagon (N = 7) where just two non-trivial scales provide the potential for endless variation as each generation depends on all the previous generations. The issue not just size, but the corresponding dynamics which determine the size. The theory of multi-fractals provides some insight, but after years of investigations, we see little hope of ever resolving the small-scale dynamics of N = 7 much less N = 11. More than brute force is needed because computational issues show up very early. For N = 11, the overall scale for each new generation is Scale[5] ≈ 0.0422171162 so after 6 generations the dynamics are on the scale of 5.66·10-9 and the 25th generation is smaller than the Plank scale of 1.6·10-35 m. See Winding Numbers, N11Summary and N7Summary.

The primary motivation for this site is the hope that someone will provide answers where we have none. The ‘wish’ list below contains some of the major unanswered questions. On balance, what is not known far exceeds what is known. The Site Map gives an overview of the content of these pages and the Chronology is a brief history of this project. Whenever possible we have provided content in Pdf format which the reader can download. The Software link contains Mathematica code which can be downloaded and pasted into any notebook.  This is ‘working code’ which is designed to be easily modified or optimized. It should run on any version of Mathematica from 5.0 to 8.0. See Tangent Map Algorithms.

The Animation link has video animations in multiple formats which can be run or downloaded. There are also manipulates in the new Computable Data Format from Wolfram. These manipulates run from any browser using a free plug-in. Download the plug-in at and then click on Manipulates link above. There are other software packages such as Maple and Matlab which would do equally well for investigations of this type. The free Geogebra package is excellent for geometric manipulations.

The Pdf on Constructions of Regular Polygons gives a brief history of attempts to construct regular polygons with compass and straightedge – culminating with the discovery by Carl Friedrich Gauss in 1796 that the regular 17-gon is constructible. This implies that the vertices for the regular 17-gon can be described by nested quadratic equations – even though the minimal equation for these vertices is degree 8. It is still not clear what effect this has on the recursive geometry of the regular 17-gon. In that same Pdf we discuss the issue of what regular polygons are constructible using origami. The Animations folder has four video clips showing the dynamics of the regular 17-gon (also known as the heptadecagon).

The algebraic structure of a regular polygon can be described using Galois theory which yields a nested sequence of extension fields of the rationals. Each extension introduces an additional level of complexity. Very little is known about the geometry of these field extensions, but we have found that 4k+1 primes such as 17 have a local recursive structure which appears to be lacking in 4k + 3 primes. This 4k + 1 structure may to be a reflection (no pun intended) of Gauss’s results on quadratic reciprocity because these results partially determine the nature of the extension fields.

To peer thought these layers of complexity we will sometimes use projections which generate ‘cross-sections’of the orbits. On the left below are 8 cyclic mappings of the vertices for N = 17. Each mapping takes the vertices mod k where k is relatively prime to 17, so there φ(17)/2 = 8 distinct mappings where φ(n) is the Euler totient function which counts the number of positive integers less than or equal to n which are relatively prime to n. (The minimal degree of the polynomial used to define the vertices of a regular n-gon is never higher than φ(n)/2)

Note that each vertex remapping has the form R17R17 where R is the real numbers . When the maps are applied to a given orbit, they yield 8 projections as shown on the right. These images are all generated from the same orbit. The first image corresponds to the identity mapping, so it tracks the orbit itself. Note that it is confined to the region surrounding the 17-gon. (This region is one of 7 invariant annuli surrounding the 17-gon. These 7 annuli form what we call the inner star region. See the related plots at the bottom of this page.) The other projections are much larger in scope and some of the plots below are thousands of units wide after just 10,000 iterations. The 17-gon would be invisible on these scales. The initial point is q1 ≈ {-0.6716275988, -0.7997683214660}. See Projections and N17Summary.


We urge readers to contribute their software and results so we may be able to reach a consensus on the major issues. It is inconceivable that there are no errors in these pages and we ask for help in finding them and making corrections.

The most patient reader will thank me for compressing so

much nonsense and falsehood into a few lines.

—Edward Gibbon (1737-1794)

In this area of study, proofs are difficult and most of the results obtained so far were motivated by numerical studies. To avoid the pitfalls of making conjectures based on data alone, we ask the readers to compare their results with the data given here.

Non-regular polygons have their own unique structure, but the geometry is very diverse. At this time there is no known classification scheme that can be used to determine the genetic structure for a given polygon, but there are results in special cases. The Penrose kite shown on the left is an example of a non-regular polygon which can generate unbounded orbits when the coordinate q is irrational. See Penrose Summary and N5Kite Summary.

The mapping which generates these structures is called the Tangent map. It is also known as the outer billiards map or the dual billiards map. Classical billiards is played on a rectangle table. In physics, this is called inner billiards because it involves tracking the path of a billiard ball inside a convex polygon such as the pentagon shown at left. In outer billiards the ball is outside the table. There is a form of projective duality between inner and outer billiards, but the strict duality only applies to smooth curves so there is virtually no correlation between an orbit inside a regular pentagon and an orbit outside.

Definition of theTangent Map τ : Given a convex polygon with a clockwise (or counter-clockwise) orientation, pick a point p outside the polygon and draw the 'tangent line' (supporting line) in the same orientation as the polygon. Extend the line an equal distance on the other side of the point of contact. The endpoint of this line segment is defined to be τ(p), so τ(p) = − p + 2c where c is the point of contact.

Description: Description: Description: Description: Description: tangentmap.jpg

In Euclidean geometry, τ(p) is called the point reflection of p across c. It is more accurately known as the inversion of p with respect to c. Any reflection preserves distance, so τ is a (piecewise) isometry. As a mapping of the plane, it preserves area and this means it can be used to model systems which conserve energy - such as orbital dynamics. Run CDF Orbit Demo.


The Tangent map is not defined on the extended trailing edges of the polygon. These are the magenta edges shown on the left for the regular (equilateral) triangle. Since τ is not defined on these edges, τ2 = τ◦τ is not defined in the inverse image of these edges. These are the blue edges shown here. Points on these blue edges map to the magenta edges, so they also have to be excluded from the domain of τ. Continuing in this way, the 'web' of points which must be excluded at iteration k is


Wk = 

Where W0 is the level 0 web, consisting of the magenta trailing edges. Taking the limit as k→∞ yields the (inverse) web W. (The forward web is defined in a similar fashion using forward edges and τ, and the full web is the union of these two.) The Tangent map software will generate the forward webs to any desired accuracy and a reflection transform will yield the inverse webs when necessary. The complement of this web are points where τ is always defined. For the regular triangle, this web consists of rings of congruent triangles and hexagons as shown on the left. This is what we call the genetic structure of the regular triangle. Any other triangle would have a similar web because every triangle is an affine transformation of the equilateral triangle. Run CDF Web Demo.


An affine transformation T is a linear transformation together with a possible translation so it has the form  


T[{x1,x2}] =  + = Ax + t


where we make no distinction between {e,f} and


The most common affine transformations are rotations, shears and scaling (including negative scalings such as reflections). The Tangent map is an affine transformation so it commutes with any affine transformation T as long as the matrix A is invertible (det(A) not 0).  In Hamiltonian (energy-preserving) dynamics, the translation t is called a ‘kick’ or perturbation so the Tangent map is similar to a kicked Hamiltonian where the kicks are dependent on the vertices of the N-gon. Saturn gets a ‘kick’ from each encounter with Jupiter and it is not clear whether these kicks will eventually destabilize Saturn’s orbit. The ‘kicks’ that the Earth feels are not ‘synchronized’ so they are relatively harmless.


The set of invertible affine transformations is called the affine group. We will assume that all affine transformations are in this group. This means that for a polygon P with web W, T(P) will have web T(W). In a sense, all affinely equivalent polygons have the 'same' web and hence the same genetic structure. On the left below is the web for an arbitrary triangle. It is just a distorted version of the web for the regular triangle so the dynamics are unchanged. The same is true for the pentagon.


The regular pentagon is shown here along with 5 affine transformatios


The first transformation in the graphic above is an example of an 'elliptical' affine transformation. The corresponding web is shown in the side-bar at left. In Mathematica:


T = AffineTransform[{{3,0},{0,2}}] (*scale by 3 in x and 2 in y with no translation*)


All affinely regular pentagons will have similar webs, but 'most' transformations are not affine and the webs will be very different. Except for the triangle, any small change in a vertex of a regular polygon will yield a non-affine version with very different dynamics. An affine transformation is a special case of a linear fractional transformation that maps z to (az+b)/(cz+d). (In complex analysis these are called Mobius transformations.) In Mathematica  T= LinearFractionalTransform[{a, b, c, d}]; For example



In 1525 the Renaissance artist Albrecht Dürer wrote a book called Underweysung der Messung mit dem Zirckel und Richtscheyt (A Course in the Art of Measurement with Compass and Ruler). The book contains illustrations and directions for the construction of geometrical objects, such as the ‘regular’ heptagon shown here. The construction is very simple – first construct an equilateral triangle and then bisect one side to obtain the sides of the heptagon. In his drawing shown below, just the first edge is shown. If the circle has radius 1, the heptagon will have edge length  /2 ≈ .866025 compared to a regular heptagon which has edge length ≈ 0.867767. This was an ancient construction and Dürer knew that it was only approximate, but he did not know that it was impossible to construct a regular heptagon with compass and (unmarked) straightedge. The resulting heptagon looks almost regular, but it is not in the same affine family, so the dynamics are very different. See Polygons of Albrecht Dürer which can be downloaded here or at the Cornell University arxiv site. See also TangentDurer.




We will use N to designate the number of sides (or vertices) of a convex polygon. As indicated above the case of N = 3 holds few surprises, even when irrational coordinates are allowed, but the case of N = 4 is surprisingly complex when at least one of the vertices is irrational. A Penrose Kite is a quadrilateral such as the one shown above. In 2007 Richard Schwartz of Brown University showed that when q=  -2, there are orbits which diverge.We call polygons in this class 'unstable', so this irrational Penrose Kite is unstable. In 2009 Schwartz generalized this proof to include any irrational q value and he conjectured that ‘most’ polygons are unstable. Our numerical evidence supports this conjecture. There is a summary table below which gives results about an assortment of polygons.There are longer versions in Pdf form which can be accessed via the Site Map or the Pdf folder link above.

In his Wikipedia article on the outer billiards map, Schwartz lists the following as the most important open questions:


(i) Show that outer billiards relative to almost every convex polygon has unbounded orbits.


(ii) Show that outer billiards relative to a regular polygon has almost every orbit periodic. The cases of the equilateral triangle and the square are trivial, and S.Tabachnikov answered this for the regular pentagon.


(iii) More broadly, characterize the structure of the set of periodic orbits relative to the typical convex polygon.


 Below is a list of further questions which we hope to answer with the help of the community (or a Deity).


(iv) For a given polygon what are the admissible step sequences ? (This is one for the Deity.)


(v) Which polygons have unbounded orbits  ? ( These are called 'unstable' polygons.)


(vi) Why is it true for 4k+ 1 prime N-gons that the Dads and Moms survive the turmoil and why does the ratio of consecutive Dad (& Mom) periods approach N + 1?


(vii) Which polygons have structure on all scales ?  Are there a well-defined class of non-regular convex polygons with this property ?


(viii) For regular N-gons, does the small-scale complexity tend to increase with each new generation ?  Does it increase with N ?


(ix) What is the limiting domain structure for the regular hendecagon,N = 11 ? What is the structure at the Plank scale ? Will we ever be able to take a 3D stroll through the N = 11 landscape in real time ?  (In the words of Richard Schwartz “A case such as n = 11 seems beyond the

reach of current technology. The orbit structure seems unbelievably complex.”)


(x) Every regular n-gon has a corresponding number field which is an algebraic extension of the rationals Q. This number field is the cyclotomic field Kn =Q(z) where z is an nth root of unity.To what degree does this field determine the dynamics of the polygon ?


(xi) In the Digital Filter map, it appears that for all odd integers N, the angular parameter θ = 2*Pi/2N 'shadows' the dynamics of the Tangent Map for a regular 2N-gon  (and corresponding N -gon). What dynamics are modeled by θ =2*Pi/N ? We know the answer only for N = 5 and N = 7.


The Tangent Map still makes sense in the limiting case when the polygon becomes a convex closed curve as illustrated below. This is the front cover of The Mathematical Intelligencer Volume 1 Number 2 from 1978. This drawing was from a featured article by Jurgen Moser (1929-1999)  called "Is The Solar System Stable ?"


Description: Description: Description: Description: Description: Intelligencer.jpg




Moser presented an historical perspective of the 1963 KAM theorem - named after the three contributors: V.I Arnold, A.N.Kolgomorov and J. Moser. This theorem was a major breakthrough in the question of stability for classical mechanics - which includes the orbital dynamics of the solar system.


Moser's contribution to the KAM Theorem was the Twist Theorem which shows that a 'smooth' Hamiltonian (conservative) system could survive periodic perturbations as long as there were no 'major' resonances such as that found between Jupiter and Saturn. Five orbits of Jupiter are a very close match for two of Saturn and this 5:2 resonance could create instabilities over long time periods. Uranus and Neptune also have a resonance which is nearly 2:1. This leaves open the question of stability for the solar system. See Torus Manipulate.


For a sufficiently smooth 'convex' curve, the Tangent Map becomes a Twist Map in the sense that points tend to follow a simple angular rotation (the twist) - which depends only on the distance from the origin. The mapping below shows some orbits of a perturbed twist map.


Each curve is a different set of initial conditions. Moving away from the origin, the perturbations increase and eventually the system breaks down and points diverge. Even in the 'stable' zone there are 'resonant' rational orbits such as the 6:1 resonance shown here. A planet or asteroid in this region might have a stable orbit if the initial conditions put it inside one of the 'islands', but in between the islands are regions of local instability.



In 1972 J. Moser stated that the Twist Theorem could be used to prove that the Tangent Map for 'smooth' curves is 'stable' in that all orbits are bounded, as long as the curve is sufficiently smooth (6-times continuously differentiable). This was proven by Douady in 1982. A convex polygon is not even 1-times continuously differentiable because derivatives do not exist at the vertices, so the KAM theorem would be expected to fail. Moser raised this question in his 1978 article - "Are there orbits for a convex polygon which diverge under the Tangent Map ? This became known as the Moser-Numann question because B. H. Neumann had earlier stated the problem in the context of outer billiards. This predicted divergence has only recently been confirmed by Richard Schwartz with the Penrose kite.


The Regular Heptagon: N = 7


We will use the regular heptagon to introduce some of the basic concepts. For 'most' N-gons and 'most' initial points, the orbits under τ are periodic. Above is an example of a period 7 orbit with a regular heptagon at the origin. The larger polygons are regular 14-gons. In our genetic language they are called 'Dads' and the generating heptagon is 'Mom'. So here we have a Mom surrounded by 7 Dads. This type of symmetry is typical for regular polygons. The positions and sizes of the smaller progeny are also generic. These represent 'resonances' of the mapping τ. The major resonances are clearly the Dads whose centers are period 7.


In between these larger resonances, the orbits can be very complex, but all orbits in this region are bounded by the ring of Dads. Outside of this ring, there are endless concentric rings of Dads with dynamics which mimic this region. So for regular polygons, it is sufficient to study the dynamics of this inner 'star' region. By symmetry we can also restrict our study to one of the Mom-Dad pairs with the corresponding progeny. Below is the canonical first generation family for N = 7. We call it the First Family. The parameters of this family can be derived independently of the Tangent map because they depend only on linear combinations of the vertices of Mom. There is much evidence that the structures seen here are actually a natural extension of the geometry of the polygon.




The regular heptagon has three ‘step-regions’ and each of these regions has a ‘canonical’ occupant. These occupants are ‘resonances’ of the mapping τ. They are regular 14-gons and their names are S1, S2 and S3. The names come from their ‘step-sequences’. The step sequence of an orbit is the number of vertices advanced on each iteration. The orbit of the center of S1 is period 7 and constant 1-step so the step sequence is {1,1,1,1,1,1,1} which we shorten to just {1}. The remaining points in S1 have the same step sequence but they need to make two revolutions around Mom, so they are period 14. The situation is the same for S2. The center is period 7 with step sequence {2} and the remaining points have period 14. Dad is also known as S3 because his orbit around Mom is constant step-3. There are no step-4 orbits in the inner star region, but they do exist outside the ring of Dads.


Dad has 6 step regions but just the first 5 are shown above. The 6th region corresponds to another symmetric Dad off the screen on the right. The S2 region is shared by Dad and Mom. It should make sense that a step-2 orbit of Mom would correspond to a step-4 orbit around Dad, because Dad is composed of two copies of Mom. All of Dad’s odd-step regions have canonical occupants which are 7-gons (female) and his even regions have 14-gons (male). We will sometimes use indices for the step sequences and refer to S1 and S2 as S[1] and S[2].


There is a ‘conjugacy’ for prime regular polygons which says that the local dynamics are ‘unchanged’ if the origin is shifted from Mom to Dad. This is illustrated on the left below where the origin is unmarked. If we put Dad at the origin, Mom is now 5-step relative to Dad, but the progeny have step sequences which are essentially unchanged. To illustrate this, on the right is the orbit of DS3 around Mom. Her period is 14 and the step sequence is constant {32}.(The points in N-gons such as DS3 all map together, so the center has the same period as the remaining points.) Note that she visits Dad on every third iteration so she is step-3 relative to Dad. Her orbit ‘unfolds’ from a {3,2} to {3} as we pass from Mom’s world to Dad’s. In this process the 7 Dad’s act as one. Algebraically, this conjugacy is a reflection of the fact that the cyclotomic polynomials  Φ7(X) and Φ14(-X) are  identical. This is true for all regular N-gons with N odd.

                                    Description: Description: Description: Description: Description: 7plot354.wmf                  


For a regular polygon with N sides there are Floor[N/2] scales which determine the position and size of the progeny. See FirstFamilies.





Scale[1] = 1.0

M[0]  radius = Scale[1]

D[0] radius ≈ 1.9498558243 (rDad)

Scale[2] ≈ .3840429432

DS3 radius = Scale[2]

S1 radius = rDad·GenScale (same as D[1])

Scale[3] ≈ 1099162641 (GenScale)

M[1]  radius = Scale[3]

S2 radius = rDad·GenScale/Scale[2]






Each generation provides the seeds for a new generation. M[1] fits symmetrically on an edge of D[0] and her relationship with D[1] is an exact scaled copy of the M[0] and D[0] relationship. This determines Scale[3], which is also called GenScale[7]. This provides the potential for self-similarity - but the regular heptagon does not support perfect self-similarity. The only known regular polygons with perfect generation self-similarity are the pentagon and octagon. The heptagon appears to support a 'mod-2' self-similarity where the even and odd generations are self-similar to each other. We examine the heptagon in much more detail in N7Summary, but note that 7 is a 4k+3 prime while 5 is a 4k+1 prime. Our numerical evidence supports the following:


4k+1 Conjecture: Supppose M is a regular N-gon with N odd centered at the origin with a vertex at {0,1}. Define:


(i) GenScale[N] = (1-Cos[Pi/N])/Cos[Pi/N]   (*this is how generations scale under τ *)

(ii) GenStar[N] ={-Cot[Pi/N]*(1+Cos[Pi/N]), -Cos[Pi/N]}  (*the point of convergence *)


Suppose N is prime of the form N = 4k+1 for k a positive integer. Then there will be infinite sequences of regular N-gons M[j] (the Mom's) and regular 2N-gons D[j] (the Dads) converging to GenStar. M[j] will have radius r[M[j]] = GenScale j and D[j] will have height h[D[j]] = (1 + GenScale)GenScale j for j a non negative integer.  The center of M[k] is (1- r[M[j])GenStar and center of D[j] = (1+ h[Dad[j]])GenStar. The periods of these centers have ratios which approach N+1. 


We will call polygons of this form 'super-symmetric'. It is easy to show that N = 5 (shown above) satisfies the conjecture, and it may be the only case where the self-similarity begins with the first generation. The heptagon is not super-symmetric but it appears to share some of the properties of the 4k+1 primes - namely the sequences of Dads and Moms along with their families. The even generations appear to satisfy the N + 1 rule. See the table on the side bar. Starting with N = 11, the 4k+3 prime polygons appear to have no canonical family structure past Mom[2]- there is not even a Dad[2]. This makes for very complex dynamics.


N = 13 and N = 17 are 'twin' 4k+1 prime polygons and both appear to have the expected strings of Dads and Moms. The corresponding families show signs of  even and odd alternation, starting with generation 2. This is similar to N = 7, but the first generation does not reappear. This generation self-similarity does not imply that the dynamics elsewhere are 'well-behaved'.


From the perspective of number theory, there is an asymmetry between 4k+3 and 4k+ 1 primes. This distinction is a fundamental part of the law of quadratic reciprocity as conjectured by Leonhard Euler and Adrien-Marie Legendre and then proven by Carl Friedrich Gauss. It is an easy matter to show that the set of 4k+3 primes is infinite. However the proof that the 4k+ 1 class is infinite relies on nontrivial results about reciprocity. When Gauss died in 1855, Gustav Dirichlet succeeded him at the university in Gttingen and he used the newly emerging tools of analytic number theory to settle the general question about generating primes from arithmetic progressions. His theorems also imply that the resulting classes have equal distributions, so in the limit there are an ‘equal’ number of 4k + 1 and 4k + 3 primes.


On the left below is the central star region for N = 17, and on the right is an enlargement showing the 7 invariant rings . (For prime polygons the number of invariant rings appears to be Floor[N/2]-1 and the first canonical occupant of the outer ring is DS[Floor[N/2]] which in this case is DS[8] (the center of DS[8] is shown below in green) . Since it is step-8 relative to Dad, it is a 2N-gon. If this was a 4k+3 prime, Floor[N/2] would be odd and this last occupant would be an N-gon. This shift in dynamics seems to destroy the subsequent family structure, but some of this structure is preserved in the case of N = 7.



Each of these rings (or annuli) have their own dynamics determined by the corresponding step sequences. For N = 17 the steps are bounded above by 9 and below by 1, but steps of size 9 occur only outside of the star region. The allowable steps in each of the inner star rings are given in the table below. Note that the last 5 rings differ only in the distribution of 7's and 8's. If the table says '(6)' it means that 6 only appears as an isolated term so {..,6,6,...}never occurs.


Star Ring











7, (8)

7, 8

7, 8

(7), 8

(7) , 8

Typical distribution







Moving outwards, the ring of Dads is constant step-8 and outside this ring, the steps are only 8 and 9. There are endless rings of Dads at equal intervals. Three of these rings can be seen on the left below. The center to center spacing is twice the distance from the origin to Dad's center, so it is |2*cS[8][[1]]| ≈  21.58 (where [[1]] signifies the first co-ordinate). These rings of Dads have step sequences which increment by {89}, so the first ring is {8}, then {889}, then {88989}, yielding a limiting sequence of {89} and limiting winding number of 1/2.


In between the rings are invariant regions which have dynamics ‘identical’ to the star region (after rotations are filtered out). That means that the two digits {8,9} code the same information as the 8 digits of the inner star. Each invariant region contains two copies of each annulus from the inner star. Shown below in magenta is the second inner star annulus and its two 'clones' in the region between Ring1 and Ring 2. This region is known collectively as the Ring 1 region. There is one canonical ‘Mom’ and hence one S[1] center in each such region and the P2 projection in the side-bar at left shows the dynamics in Ring 2.





As indicated above, N = 17 may not be a 'typical' 4k+1 prime because it is one of only 5 known Fermat primes (3, 5, 17, 257, 65537). These are primes of the form . It was known since antiquity that 'prime' regular polygons with 3 and 5 sides are constructible, but in 1796, Carl Friedrich Gauss showed that N = 17 is also constructible and five years later he proved that any Fermat prime must be constructible. He also conjectured that these were the only constructible primes and that was later proven to be true.


Gauss published his work in Disquisitiones Arithmeticae in 1801. Gauss systematically developed the theory of modular arithmetic and in the last section of the book, he applied this theory to the task of finding equations for the vertices of regular polygons. He used modular arithmetic to partition the vertices so that they defined nested equations.This is now part of Galois theory where each equation defines a field extension of the previous. With N = 5 shown below in the complex plane, the task is easy. The 4 non-trivial vertices can be grouped into conjugate pairs: s1= z + z4 and s2= z2 + z3 , where z = cos(2π/5) + isin(2π/5). Note that s1 + s2 = − 1 and s1s2 = − 1 so they are roots of x2 + x −1 = 0. The solutions of this equation are  ± 1)/2. Since s1 = 2cos(2π/5) (the complex terms cancel) this must be the positive root, so cos(2π/5) =  −1)/4 .


This shows that the regular pentagon is constructible because square roots can be constructed using compass and straightedge. The Greeks could derive this formula for cos(2π/5) from the Pythagorean Theorem, but the method devised by Gauss works for all regular polygons (although only a small portion are constructible). This approach also has implications for solving equations in general. In 1821, a young Norwegian named Neils Able used the results of Gauss and Joseph-Louis Lagrange to show that there is no general formula for the roots of a quintic and a few years later Evariste Gaolis extended these results to polynomial equations in general. (Gauss, Able and Galois made these discoveries before their 20th birthdays.)


Every regular polygon has its own unique Galois group and all of these groups are abelian and hence soluble– which means that in theory, Gauss’s procedure can always be carried out, but only the Fermat primes will yield nested quadratic equations. For example  N= 19 will yield two cubics and one quadratic for a total degree of 18, but the degree of cos(2π/19) will always be half of this.


The vertices of any regular n-gon are the (complex) solutions to Xn = 1. This is called the nth cyclotomic equation. The corresponding polynomial Xn -1 is always divisible by X-1, and for prime n-gons, this is the only divisor, so the minimal cyclotomic polynomial, Φn(X), is degree n-1. In general the degree of Φn(X) is the Euler totient function φ(n). (This explains why it is common to use the symbol Φn(X) for this class of polynomials.)


For the pentagon above there are 5 solutions to the cyclotomic equation. These solutions can be written as zk = cos(2πk/n) + isin(2πk/n) for k = 0,1,2,3,4. The trivial solution z = 1 is written as z5 in the example above to emphasize that all the roots can be found in terms of a single (primitive) root, which in this case is z = cos(2π/5) + isin(2π/5). Because the vertices are defined by a polynomial with rational coefficients, they must be algebraic and not transcendental. This is a little strange since it is common to use trigonometric functions to describe these vertices and ‘most’ trigonometric expressions are not algebraic. In Mathematica, Element[Cos[2*Pi/11],Algebraics] yields True but Element[Cos[2],Algebraics] yields False.


The fact that these vertices are algebraic is of little consolation as the number of vertices increases. By N = 11, the minimal cyclotomic polynomial Φ11(X) is degree 10 and the minimal polynomial for cos(2π/11) is a quintic:


MinimalPolynomial[Cos[2Pi/11]] yields 1+ 6x -12x2-32x3+16x4+32x5 = 0


This is the first prime polygon which requires a quintic to define the vertices. (Note that sin(2π/11) does not have to be computed independently because the vertices are assumed to lie on the unit circle. This is fortunate because the minimal polynomial for sin(2π/11) is degree 10. This implies that the algebraic recipe for finding the sine is the same as the construction: first find the cosine and then use the quadratic relationship between sine and cosine.) Since the minimal polynomial for cos(2π/7) is a cubic, it would be necesssary to trisect an angle and to construct a regular 7-gon and it would be necessasry to quinsect an angle – that is, divide an arbitraty angle into 5 equal parts - to construct a regular 11-gon.


If we ask Mathematica to simplify or expand the cyclotomic equation for a prime N, it would return algebraic roots for N = 5, but trigonometic values for N = 7 and beyond: ComplexExpand/@Roots[z^5==1,z] yields the qudratic form for the roots of the regular pentagon, but ComplexExpand/@Roots[z^7==1,z] yields trigonometric roots for the regular heptagon. Mathematica can be forced to find these roots in purely algebraic form by solving the corresponding cubic minimal polynomial, but for N = 11 this option does not exist.


Starting with N = 11, there are no practical choices for the vertices of a prime polygon other than trigonometric forms, even though the vertices must have an algebraic form. In Constructions of Regular Polygons we discuss the Galois extensions for N = 11. In terms of the Tangent map, N = 11 is much more complex that N = 7. These are both 4k+3 primes but N = 7 still retains some of the recursive structure found in 4k + 1 primes. None of this structure has been found with N = 11.


Cyclotomic theory can be applied to non-prime regular polygons as well.  As indicated above, the minimal degree of the cyclotomic polynomial is always φ(n) where φ(n) is the Euler totient function. The (minimal) degree of cos(2π/n) is always φ(n)/2. When n is prime, φ(n) = n – 1 and in this case the degree of sin(2π/n) is always φ(n).


For n = 6 (shown on the left) , φ(n) = 2, so cos(2π/6) is degree 1 which implies that it is rational (in fact equal to 1/2), while sin(2π/6) is quadratic (/2). For n = 12, φ(n) = 4 and the tables are turned: cos(2π/12) is /2 while sin(2π/12) = 1/2. This shows that the degree of sin(2π/n) is not always φ(n). The formula for the degree of sin(2π/n) can be found in Constructions of Regular Polygons.


As indicated earlier all regular polygons share a predictable large scale structure consisting of concentric rings of ‘Dads’ and these rings guarantee that that the small scale structure is similar at any distance from the origin. For non-regular polygons these rings tend to break down and there is very little that is known about the structure on any scale.


In the table below we summarize what is known about the special cases. Most of these are regular polygons, where we have some knowledge of the dynamics. The only non-regular cases are the Penrose kite, a lattice polygon and a ‘woven’ polygon formed by nesting two regular polygons. The last example is N = 281 which is a 4k+1 prime polygon. For regular N-gons with large N, the dynamics approach that of a circle or ellipse.


For odd N, there is a infinite family of regular 2kN-gons, and of this family only N and 2N seem to be conjugate. In terms of compass and straightedge constructions, it is trivial to bisect the generating angle of one family member to get the next, but the Tangent map dynamics appear to have little correlation. For example the dynamics of N = 10 and N = 20 are quite different. It is possible that each family may hold surprises dynamically and algebraically. There may also be non-trivial correlations in the dynamics depending on the divisors.


The situation is worse when N is already even. In this case every family member seems to be unique. For example we have found almost no correlation between N = 4, N = 8 and N = 16 . This ‘powers-of-2’ family is almost totally unexplored. How is the simple fractal structure of the regular octagon destroyed when the generating angle is bisected ?


We plead with users to ‘adopt’ a polygon or a ‘family’ and share their results with us. The software packages cover the four main cases: Nodd, NTwiceEven, NTwiceOdd and Non-regular but the basic algorithms are the same in all packages.


The posthumous message written on Richard Feynman’s blackboard at Cal Tech was “What I cannot create I do not understand”.


Summary of Results


Below is a summary of dynamics for various polygons. There are more detailed summaries in Pdf format which can be accessed via the Site Map or the PDF folder.The table below sometimes refers to the 'winding number',  of an orbit. This is also called the rotation number or the 'twist'. For a given orbit, it measures the mean rotation around Mom on a scale from 0 to 1, with 1 being a full rotation, so a canonical step-3 periodic orbit for a regular N-gon would have winding number ω = 3/N. It should be clear that for regular polygons, ω is bounded above by 1/2 and bounded below by 1/N.


The first example below is a line segment which is not technically a polygon - but the Tangent map still applies and points diverge uniformly as can be confirmed with a few sketches.



N = 2

(Not a



  Description: Description: Description: Description: Description: 2arrows.jpg

• All orbits diverge



N = 3



    Description: Description: Description: Description: Description: 3web2.jpg

Endless rings of Moms and hexagon Dads.

• Ring k has 3k Dads and 12k Moms and these are also the periods if we plot centers of the Dads. Shown here are the 12 Mom's in ring 1. They map to each other in the order given.

• Step sequences of Dads are (1),(211),(21211),etc

with limiting sequence (21) so ω→.5

• Non-regular triangles have similar structure.



  N = 4




    Description: Description: Description: Description: Description: 4rectangle2nd ring.jpg

• 'web' structure is similar to the triangle - but with no distinction between Moms and Dads.

• Ring k has 4k Dads and these are also the periods. There is no period doubling. Shown here is the second ring with period 8 and step sequence (2,1)

• Step sequences are (1), (21),(221), etc with limiting sequence (2) and ω→.5

• All trapezoids have similar structure.


N = 4



     Description: Description: Description: Description: Description: kite2.jpg


Below is the level 600 web


Description: Description: Description: Description: Description: kiteweborbit.jpg


• A Penrose Kite has vertices {{0,1},{q,0},{0,-1},{-1,0}}. When q is irrational, R.E. Schwartz  has shown that there are unbounded orbits.

• The Kite given here has q =  − 2.  The initial point p is {(1−q)/2 ,1}and it has an unbounded orbit. The first 6 points in that orbit are shown here. Note that they all lie on a lattice of  horizontal lines of the form y = k where k is an odd integer.

• The points in the orbit of p are woven through the web like thread in a fabric. They form almost perfect Cantor string for the kite. The arrows here point to the threads. Some of the prominent regions are marked with their periods. All of these regions have period doubling , so these are the periods of the centers.

• The web is intricate but not fractal - so there is no obvious signs pointing to the complexity.

• The winding number  (ω) of any unbounded orbit must approach the 'horizon' value of .5. Tracking ω(p) shows considerable local variability on top of the trend toward .5 (Since p is in exact 'radical' form, Mathematica computes its orbit in this same form with no round-off :

 Τ500000000(p) ={−5730+ (1−q)/2 −6688q , −4417} ).

N = 5


The star region below shows the

location of a non-periodic point p

with orbit dense in the star region.


    Description: Description: Description: Description: Description: 5starpt2.jpg

•First non-trivial star region inside ring of 5 Dads.

•Decagon and pentagon periods satisfy:

dn = 3dn−1+ 2pn−1   &  pn = 6dn−1 + 2pn−1 with d1=5 and p1=10

• dn/dn−1 →6 and decagons are dense so fractal dimension is Ln[6]/Ln[1/GenScale[5]] ≈   1.241

•The point p = {c5[[1]], c4[[2]]} has a dense non-periodic orbit with ω→ .25. The plot on the left is 50,000 points in this orbit.Note perfect self-similarity.

•Bounding Dads have step sequences (2), (322), (32322),..,→(32) with ω→.5

N = 6


    Description: Description: Description: Description: Description: 6web.jpg

Domain structure is identical to N = 3 with any hexagon as Mom and the adjacent triangle as S1.

• As with every 'twice-odd' regular N-gon, the canonical Dad orbits decompose into two groups with N/2 in each group. So the first ring of Dads is 6 hexagons - but linked together as 3 and 3.

• kth ring had 6k hexagons and odd rings have decomposition and period doubling.

• Dad center periods are 3k for odd rings and 6k for even.

• Step sequences of Dads are (2),(32),(332),..→(3)


N = 7


Generation 1(with right-side Dad)

    Description: Description: Description: Description: Description: 7Gen1p3.jpg



  Generation 2 - Portal Generation


Description: Description: Description: Description: Description: N7Gen2Helenpt2.jpg

• First prime N-gon with multiple scales. It is not super-symmetric but it retains some of the properties of 4k+1 primes, so it is a 'hybrid'.

Odd generations are self-similar.

• Even (Portal) generations are self-similar.

• Ratios of periods Mom[k+2]/Mom[k]→200 which factors as 8 and 25 for transitions from odd to even and back.  The value of 8 matches the N+1 rule for super-symmetric primes.

• Central star decomposes into two invariant regions – inner with step sequences of 1’s and 2’s and outer with 2’s and 3’s. Step 4 occurs only outside star region. All prime N-gons have similar decomposition.

• Small scale chaos in qstar region and Star[2] region

•Bounding Dads have step sequence (3),  (334), (33434),..→(34) with limiting ω = .5


N = 8




•Only octagons – no period doubling

•Periods of  Dad[k]/Dad[k-1] →9 and they are dense so fractal dimension is Ln[9]/Ln[1/GenScale[8]] ≈ 1.246

•Dense non-periodic orbit with ω→.25

•S2 orbit decomposes into two period 4 orbits – each with ω = .25. All S2[k] = Dad[k] orbits have same ω.


N = 9


 Description: Description: Description: Description: Description: 9gen1&2.jpg

The small rectangle above outlines a portion of the second generation which is shown below. There are 'islands' of chaos amid perfect self-similarity. The tiny rectangle around the S2[3] bud is enlarged on the right.


Description: Description: Description: Description: Description: 9Gen2combined.jpg

• First generation canonical except that S3 has 12 sides composed of two interwoven hexagons at different radii, and Helen3 has extended edges to form a non-regular hexagon.

•Moms and Dads exist on all scales and ratio of periods Mom[k]/Mom[k-1]→10 (but not dense).

•Second generation is dominated by 'Portal Moms' similar to those of N = 7. In between these Portal Moms are regions with small scale chaos. One of these regions is shown here.

•The chaotic region surrounding the S2[3] bud is called the Small Hadron Collider. The gap between the central S2[3] bud and the three octagons is determined by a sequence of (virtual) buds of S2[3] so it is 2r[GenScale[9]0 + GenScale[9]1 + ...] where r = rDad·GenScale[9]4/Scale[1] is the radius of the first bud. (r ≈.000077)


N = 10


The central star region showing all of the outer ring and half of the inner ring.

   Description: Description: Description: Description: Description: 10inner&outerstar.jpg

• Domain structure is identical to N = 5 but the 10 Dads form two groups of 5 and the 10 S2’s form two groups of 5. This is typical for ‘twice-odds’.

• The decomposition of the Dads persists for odd rings a has no effect on the outer star region, but the decomposition of the S2’s creates two invariant inner star regions – one of which is shown here. Together they define the inner star. The 10 pentagon ‘Moms’ patrol the border between the inner and outer stars.

N = 11





Description: Description: Description: Description: Description: 11Gen1all.jpg


Description: Description: Description: Description: Description: 11Gen2d.jpg

• The second 4k+ 3 prime N-gon

• Normal first generation but no evidence of Dads past Dad[1] or Moms past Mom[2].

• Second generation shown here has some small Mom[2]’s on edges of Dad[1], but no Dad[2]’s. Mom[1] is almost devoid of  canonical buds.

•Dad[1] and most family members are surrounded by ‘halos’ of complex dynamics as the normal bud-forming process breaks down.

•No obvious self-similarity but small invariant ‘islands’ exist on a scale between generations 3 and 4.


N = 12




•Complex geometry due to the factors of 12, but perfect fractal structure with GenScale[12] scaling.

•Ratio of  periods of Dad[k]/Dad[k-1] →27 so the fractal dimension is Ln[27]/Ln[1/GenScale[12]] ≈1.251

•The six-sided S2’s are determined by triplets of virtual Dad[1]’s, as shown here.

•S4 is canonical with buds of size S1.

•S3 is non-regular octagon with center at {0, }

• S1=Mom[1] is surrounded by a ‘halo’ which is period 24 vs. period 12 for Mom[1] (no doubling). This halo is unique among all N-gons studied.

N = 13


Second Generation

Description: Description: Description: Description: Description: 13Gen2b.jpg





•The second ‘super-symmetric’ prime polygon so at GenStar, ratio of periods of Dad[k+1]/Dad[k] → 14, and same for Moms. Ratios alternate high and low, so there is some even-odd differentiation.

•Dynamics around Mom[1] are characterized by dense halo of non-canonical buds.  There are protected pockets at GenStar and under Dad[1] for 3rd generation.

N= 14




         Description: Description: Description: Description: Description: 7weaveindices.jpg



Description: Description: Description: Description: Description: 7weaveDads.jpg



Description: Description: Description: Description: Description: 14weaveindex91dist.jpg

• A woven N-gon consists of a canonical Mom at  radius 1 and a secondary Mom at radius between h0 (height of Mom) and 1 + GenScale[N]. This covers the full range of convex proportions.

• Show here is index .91 for N = 7

•Normal ring of 14 Dads is now subdivided - the orbit of the 7 large Dads 'sees' only the canonical N = 7 Mom and the 7 small Dads 'see'  only the secondary Mom. (For index 1 this is the canonical N = 14 case.)

• Star region is no longer invariant.

• Rings of Dads undergo periodic oscillations. In the first cycle the secondary Dads grow and become dominant. The first Big Crunch at ring 11 sees the canonical Dads almost disappear. Inter-ring spacing (and periods) are same as N = 7, so ring 11 is at radial distance of about 91.6. (The exact parameters of the Crunches are easy to calculate but they do not generally correspond to ring centers.)

• Second Big Crunch at about ring 22 involves Moms. 

• Third Big Crunch at ring 32 is a close repetition of the first so complete cycle is about 22 rings.

• Dynamics are very complex. Many orbits diverge rapidly at first and then fluctuate widely in distance. The plot show here is the first 900 million points for an inner star point. The obit faltered at ring 26 which has a very small dynamical gap. Every ring is unique. Most local points are trapped by rotations, but small tunnels do exist. These are reminiscent of the gravitational assist channels which exist throughout the solar system.






      Description: Description: Description: Description: Description: 4nonregulargridpt2 .jpg

 Level 400 web:


     Description: Description: Description: Description: Description: 4lattice.jpg

• A lattice polygon has vertices with integer co-ordinates. The lattice polygon shown here has vertices  {{2,0},{1,-2},{-3,-1},{1,1}}

• Any polygon with rational vertices can be re-written as a lattice polygon.

• The only regular lattice polygon is N = 4. Every other regular polygon has at least one irrational coordinate so there is no grid size that would make it a lattice polygon.


• The orbit of any lattice point will have lattice co-ordinates. Shown above is the orbit of {4,1} which is period 6. This is a period-doubling orbit and the center point at {-3,2} has period 3.


• The web shows that there are 'rings' of  period doubling orbits  surrounding Mom. The large inner ring is the period 3 ring, The next two rings have centers with periods 7 and 9.

• The region indicated by the arrow has period 154 which means it spirals out and returns, but unlike the Penrose Kite, these orbits must be bounded and periodic.


N = 281


Description: Description: Description: Description: Description: 281scale.jpg



Description: Description: Description: Description: Description: 281innerring.jpg





• As N increases, the star region grows while the scale shrinks. GenScale[281]≈.000062 so Mom[1] is microscopic and her local dynamics have little or no effect on the global dynamics- which are dominated by simple rotations about Mom – like a twist map on the unit circle with minimal perturbations. The amount of twist increases ‘smoothly’ with the distance from the origin. Dad is S[140] with maximal twist (for the inner star ) at ω = 140/281.


• Shown on the top left is the inner ring which appears to be invariant. Its largest occupant is S[93].


•Dad is surrounded by the ‘canonical’ outer-star ring which is invariant. Its largest occupant is DS[140] who plays the role of a 'shepherd' satellite.  The vertical line from vertex 1 of Dad bisects DS[140]. If N was 4k+3, the shepherd would be an odd-step Helen. The general formula for the shepherd is DS[Floor[N/2]].



Adler R.,  Kitchens B., Tresser C., Dynamics of piecewise affine maps of the torus, Watson Research Center, preprint, 1998. cf. MR 2002f:37075

Alperin R.C., A mathematical theory of origami constructions and numbers, New York Journal of Mathematics, Vol 6 (2000), pp 119-133


Alperin R.C., Trisections and totally real origami, American Mathematical Monthly, Vol 112, No. 3 (March 2005), pp 200-2011


Arnold V.I. , Proof of a theorem by A. N .Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian. - Usp. Mat. Nauk. SSSR 18, no. 5 (1963)

Ashwin P., Elliptical behavior in the sawtooth standard map, Physics Letters A 1997 232:409-416


Ashwin P, Deane J., Fu X-C., Dynamics of a bandpass sigma-delta modulator as a piecewise isometry. Proceedings of the IEEE International Symposium on Circuits and Systems, Sydney Australia,2001:III-811-III-814


Ashwin P., Goetz A. , Polygonal invariant curves for a planar piecewise isometry. Transactions of the AMS, 2006  350:373-390


Ashwin P., Deane J., Fu X-C. , Properties of the invariant disk packing in a model sigma-delta bandpass modulator. International Journal of Bifurcations and Chaos 2003  13:631-641


Ashwin P., Goetz A,. Polygonal invariant curves for a planar piecewise isometry. Transactions of the American Mathematical Society 2005 358:373-390


Ashwin P., Goetz A., Invariant curves and explosion of periodic islands in systems of piecewise rotations, SIAM Journal on Applied Dynamical Systems 2005; 4: 437.


Ashwin P., Non-smooth invariant circles in digital overflow oscillations. Proceedings of NDE96: 4th International Workshop on Nonlinear Dynamics of Electronic Systems (Seville, Spain), 1996.


Ashwin P, Chambers W, Petrov G, Lossless digital filter overflow oscillations; approximation of invariant fractals, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 7 (1997), 2603–2610.


Auckly D., Cleveland J., Real origami and impossible paper folding, American Mathematical Monthly, Vol 102, No. 3, (March 1995), pp. 215-226


Baez J.C., Division algebras and quantum theory, arXiv:1101,5690


Baez J. C., The octonions,  Bulletin of the American MathematicalSociety, Vol. 39, page 145-2002. Paper and additional bibliography at


Ball W.W., Coxeter, H.M.S, Mathematical Recreations and Essays, Dover Publications, 1987

Birkhoff G.D, “Dynamical Systems”. AMS Colloquium Publ., Vol 9, AMS, Providence, 1927

Bold B. , Famous Problems of Geometry and How to Solve Them, Dover Publications, 1982


Boyland P, Dual billiards, twist maps, and impact oscillators, Nonlinearity 9 (1996) 1411-1438


Breslin S., DeAngelis, V.,The minimal polynomials of sin(2π/p) and cos(2π/p), Mathematics Magazine, April 2004    

Kahng  B. , Singularities of two-dimensional invertible piecewise isometric dynamics. Chaos: An Interdisciplinary Journal of Nonlinear Science 2009; 19: 023115.


Cassaigne J. Hubert P, and Tabachnikov S, Complexity and growth for polygonal billiards.


Chua L.O.,Lin T. Chaos in digital filters. IEEE Transactions on Circuits and Systems 1988: 35:648-658


Chua L.O.,Lin T. Chaos and fractals from third order digital filters. International Journal of Circuit Theory and Applications, 1990 18:241-255


Chua L.O.,  Lin T. Fractal pattern of second-order non-linear digital filters: a new symbolic analysis. Int. J. Cir. Theor. Appl. 18 (1990), 541–550. MR1054774 (91d:58149)


Crowe, D.W. , Albrecht Dürer and the regular pentagon, in Fivefold Symmetry by Istvan Hargittai, World Scientific, 1994

Davies A.C. , Nonlinear oscillations and chaos from digital filter overflow. Philosophical Transactions of the Royal Society of London Series A- Mathematical Physical and Engineering Sciences 1995 353:85-99


D. Dolgopyat, B. Fayad, Unbounded orbits for semicircular outer billiard. Ann. Henri Poincare 10  357–375. MR2511890 (2010d:37076)

DogruF. and Tabachnikov S. , Dual billiards, Math Intelligencer vol.27 No. 4 (2005) 18–25

Douady R. , These de 3-eme cycle, Universite de Paris 7, 1982

Dürer, Albrecht, Underweysung der Messung mit dem Zirckel und Richtscheyt, Nuremberg, 1525

Ebert P.M, Mazo J.E., Taylor M.G., Overflow oscillations in digital filters, The Bell System Technical Journal - November 1969


Feely O., Fitzgerald D., Bandpass sigma-delta modulation - an analysis from the perspective of nonlinear dynamics.  Proceedings of the IEEE International Symposium on Circuits and Systems, Atlanta, USA,1996 3:146-149


Gauss, Carl Friedrich, Disquisitiones Arithmeticae, Springer Verlag, Berlin, 1986 (Translated by  Arthur A. Clark, revised by William Waterhouse)


Gauss, Carl Friedrich; Maser, Hermann (translator into German) (1965), Untersuchungen über höhere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition), New York: Chelsea


Genin D., Regular and chaotic dynamics of outer billiards, Penn State Ph.D. thesis (2005)


Genin D. , Research announcement: boundedness of orbits for trapezoidal outer billiards. Electronic Research Announc. Math. Sci. 15 , 71–78. MR2457051 (2009k:37036)

Geretschlager R., Euclidean constructions and the geometry of origami, Mathematics Magazine, 68, No. 5 Dec. 1995 pp357-371

Gleason A. , Angle trisection, the heptagon and the triskaidecagon, American Mathematical Monthly, Vol 95,  No. 3 ,1988,  pg. 185-194

Goetz  A. , Dynamics of piecewise isometries. PhD. Thesis, University of Chicago, 1996. MR2694628


Goetz A. , Dynamics of a piecewise rotation. Discrete and Cont. Dyn. Sys. 4 (1998), 593–608. MR1641165 (2000f:37009)


Goldstein C. Schappacher, N., Schwermer, J. (editors) , The Shaping of Arithemtic after C.F. Gauss’s Disquisitiones Arithmeticae, Springer Verlag, 2007. Pdf available here


Gutkin E,  Tabachnikov S. , Complexity of piecewise convex transformations in two dimensions, with applications to polygonal billiards on surfaces of constant curvature. Mosc. Math. J. 6 (2006), 673–701. MR2291158 (2008f:37080)

Gutkin E, and Simanyi N., Dual polygonal billiard and necklace dynamics, Comm. Math. Phys. 143 (1991) 431–450


Gutkin E., Katok, A., Caustics for inner and outer billiards, Commun. Math. Phys. 173,101-133 (1995)


Harper P.G.,Proc. Phys. Soc. , London (1955) A 68, 874


Haug F, Bienert M, Schleich W.P., Motional stability of the quantum kicked rotor: A fidelity approach. Physical Review (2005) A: 71:043803-1- 043803-11


Hooper Patrick, Renormalization of polygon exchange maps arising from corner percolation, arXiv:1105.6137, Feb,2012


Hughes G.H., Henon mapping with Pascal, BYTE, The Small System Journal, Vol 11, No 13. (Dec.1986)  161-178


Hughes G.H.,  Probing the strange attractors of chaos (A.K. Dewdney), Scientific American, July 1987, pg 110-111


Hughes G.H. ,The Polygons of Albrecht Durer – 1525, arXiv:1205.0080


Hughes G.H. , Outer billiards, digital filters and kicked Hamiltonians, arXiv:1206.5223


Jackson L.B, Kaiser J.F., MacDonald H.S. , An approach to the implementation of digital filters, IEEE transactions, Audio and Electro acoustics, AV-16, No 2 (1968) 413-421


Julia G. , Sur l’iteration des fonctions rationnelles, Journal de Math Pure et Appl (1918) 8:47-245

Kean M.,Interval exchange transformations. Mathematische Zeitung 141 (1975), 25–31. MR0357739 (50 #10207)

Kolmogorov  A.N. , Preservation of conditionally periodic movements with small change in the Hamiltonian function,  Doklady Academii Nauk (Report of the Academy of Sciences) SSSR vol. 98 (1954): 527-530.

Kolodziej, The antibilliard outside a polygon, Bull. Polish Acad Sci.Math. 37 (1989) 163–168

Klamkin M.S, Chrestenson H.E., Polygon imbedded in a lattice, Advanced Problems and Solutions, American Mathematical Monthly, Vol 70, No.4, April 1963, pp 447-448

Kunihiko, K, Takahama, T.,Origami for the connoisseur, Japan Publications, Tokyo (1987)

Lapidus M.L., Niemeyer R.G. ,Towards the Koch snowflake fractal billiard. arXiv:0912.3948 – Dec 2009

Lowenstein J.H., Aperiodic orbits of piecewise rational rotations of convex polygons with recursive tiling, Dynamical Systems: An International Journal Volume 22, Issue 1, 2007, Pages 25 - 63


Lowenstein J. H., Kouptsov K. L. and Vivaldi . F, Recursive tiling and geometry of piecewise rotations by π/7, Nonlinearity 17 1–25 MR2039048 (2005)f:37182)


Lowenstein J.H., Poggiaspalla G.,Vivaldi F., Sticky orbits in a kicked oscillator model, Dynamical Systems, Vol 20, No.4 (2005) 413-451


Masur H., Interval exchange transformations and measured foliations. Ann. Math. 115(1982), 169–200. MR0644018 (83e:28012)


Masur H., Hausdorff dimension of the set of nonergodic foliations of a quadratic differential.Duke Math. J. 66(1992), 387–442. MR1167101 (93f:30045)

Masur H., Ergodic theory of translation surfaces. In Handbook of dynamical systems.Vol. 1B, pp. 527–547. Elsevier B. V., Amsterdam, 2006. MR2186247 (2006i:37012)

Masur H , Tabachnikov S. Rational billiards and flat structures. In Handbook of dynamical systems, Vol. 1A, pp. 1015–1089. North-Holland, Amsterdam, 2002. MR1928530 (2003j:37002)

Mather J., Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207, 169-207(1991).

Mather J., Variational construction of orbits of twist diffeomorphisms, J. Amer. Math. Soc., 4, 207-263(1991).

Moore F.L., Robinson J.C., Bharucha C.F., Sundaram B, Raizen M.G. Atom optics realization of the quantum delta-kicked rotor. Physical Review Letters 1995 75:4598-4601


Moser J.K., On invariant curves of area-preserving mappings of an annulus, Nachr. Akads. Wiss, Gottingen, Math. Phys., K1, (1962)

Moser J. K., Stable and Random Motions in Dynamical Systems, Ann. of Math. Stud. 77, 1973 (Reprinted  as Princeton Landmarks in Mathematics, 2000, Princeton University Press.)


Moser J.K., Is the Solar System Stable ?  The Mathematical Intelligencer, (1978) Vol. 1, No. 2: 65-71


Moser J. K., Siegel C.L. Lectures on Celestial Mechanics, Springer-Verlag,(1971) Berlin


Moser J.K., Recent developments in the theory of Hamiltonian systems, SIAM Review Vol 28 No.4, Dec. 1986


Neumann B.H., Sharing ham and eggs, Manchester Mathematics Colloquium, 25 Jan 1959 in Iota, the Manchester University Mathematics students’ journal

Peitgen H.O., Richter P.H. ,The Beauty of Fractals, Springer- Verlag, Berlin, 1986

Penofsky, E. The Life and Art of Albrecht Dürer, Princeton University Press, 1943

Pfeifer G., On the confromal mapping of curvilinar angles, Trans. Amer. Math. Soc. 18 (1917)185-198

Poincare  J.H.,  (1892-99) Les methodes nouvilles de la mechanique celeste,  Paris: Gauthier-Villars

Ribenboim, Paoulo, Algebraic Numbers, Wiley Interscience, 1972

Richmonds H.W., Quarterly Journal of Mathematics, 1893, vol 26, pg. 206

Schwartz R.E. , Outer billiards, quarter turn compositions and polytope exchange transformations., Nov. 28,2011

Schwartz R.E. , Unbounded orbits for outer billiards, Journal of Modern Dynamics 3 (2007)

Schwartz R.E. , Outer billiards on kites, Annals of Mathematics Studies, 171 (2009), Princetom Univeristy Press, ISBN 978-0-691-14249-4

Schwartz R.E., Outer billiards and the pinwheel map. arXiv:1004.3025.

Schwartz R.E., Outer billiards, arithmetic graphs, and the octagon. arXiv:1006.2782.

Schwartz R.E., Obtuse triangular billiards. II. One hundred degrees worth of periodic trajectories. Experiment. Math. 18(2009), 137–171. MR2549685 (2010g:37060)

Scott A.J., Holmes C.A.,Milburn,G.J. Hamiltonian mappings and circle packing phase spaces. Physica D 155 (2001) 34-50. MR1837203

Siegel C. L. , Iteration of Analytic Functions, Ann of Math (1942) 42: 607-612

Siegel C.L., Lectures on the Geometry of Numbers. Springer, Berlin, 1980. MR1020761 (91d:11070)

Smillie J , Weiss B. ,Veech’s dichotomy and the lattice property. Ergodic Theory Dynam. Systems 28(2008), 1959–1972. MR2465608 (2009m:37092)

Tabachnikov S. , On the dual billiard problem. Adv. Math. 115 (1995), no. 2, 221–249. MR1354670

Tabachnikov S., Billiards. Panoramas et Syntheses, Vol. 1. Societe Mathematique de France, Paris, (1995). MR1328336 (96c:58134)


Tabachnikov S., Geometry and billiards, Student Math. Library, 30. Amer. Math. Soc., Providence,RI, 2005. MR2168892 (2006h:51001)


Tabachnikov S., A proof of Culter’s theorem on the existence of periodic orbits in polygonal outer billiards. Geom. Dedicata 129 (2007), 83–87. MR2353984 (2008m:37062)


van der Waerden B.L., Algebra (vol I), Springer Verlag, Berlin, 2003 (from the 7th edition, originally translated in 1966 from the German text Moderne Algebra (1930-31.)


Veech W. A. , Interval exchange transformations. J. D’Analyse Math. 33 (1978), 222–272. MR0516048 (80e:28034)


Veech W. A., Teichmuller curves in moduli space, Eisenstein series and an application to triangular billiards. Invent. Math. 97(1989), 553–583. MR1005006 (91h:58083a)

Vivaldi F. ,  Shaidenko A., Global stability of a class of discontinuous dual billiards. Comm. Math. Phys. 110 , 625–640. MR895220 (89c:58067)