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Dynamics of Polygons

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Site Map










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The web of the regular pentagon (N = 5)


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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.









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



A P5 projection for N= 17





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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 Drer - 1502


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









































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























































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A region above S1 for N = 7 showing the break-down of self-similar structure.
















































































8 Generations of Ms for N = 7





























4k + 1 primes

4k + 3 primes















6 Generations of Ms for N = 13








































The First Family for N = 17 showing M on the right and D on the left. There are 17 such Ds surrounding M.

















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 appears to be inherent in the polygon so it could be called the genetic structure of the polygon. For regular polygons, the basic geometry can be derived independently of any mapping. See First Families of Regular Polygons. 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. There is more about regular polygons below and at Outer Billiards on Regular Polygons.

Aside from a few simple cases, very little is known about the dynamics 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 1692 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.

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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. N = 11 has four non-trivial scales which define each generation. These scales are non-commensurate and they act across generations, so a 6th generation polygon could have radius Scale[3]2Scale[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 in scale. The generations of First Families provide some structure but there are regions for N = 7 and N = 11 where all the canonical structure breaks down and the dynamics become unpredictable. 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 5.6610-9 and the 25th generation is smaller than the Plank scale of 1.610-35 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. For your safety each manipulate will ask for permission to load the Mathematica kernel before running. Right click to see your options. 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 Gausss 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-sectionsof 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.

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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.


Given a convex n-gon P with vertices {c1, c2,..,cn} which we assume to be numbered clockwise, the corresponding edges E = {E1,E2, ..,En} have both a forward and trailing extension. (Just the trailing extensions are shown below in blue.) Let Ef = Efk be the union of the forward extensions and Et = Etk be the union of the trailing extensions. Each extension is assumed to be an open ray.

The map τ is not defined on P or on the trailing edges, so the level-0 web is defined to be W0 = E Et. This is the exceptional (or singular) set of τ. Since W0 is connected, the complement of W0 external to P consists of n disjoint open (convex) sets which are known as level-0 tiles. These tiles are also called atoms because all the subsequent dynamics are determine by repeated action of τ on this level-0 partition of the space external to P. These primitive tiles define the domain of τ relative to each vertex so they can be labeled by the indices 1,2,..n. These labels will be the first elements of the corner sequence of a point in these regions. As the web progresses, each level-k tile will have a corner sequence of length k+1.

The union of the level-0 tiles is the domain of τ which is abbreviated Dom(τ). Dom(τ2) is Dom(τ) τ-1 (W0). The union of W0 and τ-1 (W0) is called the level-1 web, W1. In general the level k (forward) web is defined to be:

(i) Wk = where W0 = E Et

The inverse web is defined in a similar fashion using τ and the extended forward edges:

(ii) Wik =where Wi0 = E Ef

At each iteration, Wk and Wik partition the plane into disjoint open convex regions which are the level-k tiles. Below is W1 for the polygon P, showing the level-1 tiles and the corner sequences. On the side bar at left is the web for a regular triangle. Run CDF Web Demo (right click to see options).


The analysis of these corner sequences is part of symbolic dynamics. For any convex piecewise isometry such as τ, these corner sequences have polynomial complexity and the degree of that polynomial is a common measure of the complexity of the dynamics. The first differences of these corner sequences yields step-sequences which in turn can be used to define winding numbers - the average rotation per iteration. When the polygon P is regular, there is much added structure to the limiting web W. This will be discussed below.


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 Saturns 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 Drer 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 Drer 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 Drer which can be downloaded here or at the Cornell University arxiv site.


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 Ds and Ms survive the turmoil and why does the ratio of consecutive D (& M) 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 ?"

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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 Tangent Map on Regular Polygons


In 1989 F. Vivaldi and A. Shaidenko showed that all orbits for a regular polygon must be bounded. This follows from the fact that the regular polygons have exceptional sets (webs) which are symmetric with respect to τ and τ-1. Above we defined Wk to be the (forward) exceptional set for τk , with limiting web Wf = The limiting inverse web Wi is defined in a similar fashion by applying τ to the forward edges, and we define W to be the union of these two webs. For a regular polygon the limiting forward web is clearly identical to the limiting inverse web, but they differ on each iteration and it is useful to make the distinction.


Since W is the union of a countable number of lines or line segments, it has zero (Lebsegue) measure so the complement Wc has full measure and τ is defined almost everywhere. At every iteration of the web there are unbounded tiles but for regular polygons the limiting tiles have bounded measure and the tiles with this maximal measure are call the D tiles. These D tiles also have the largest number of possible sides which is 2N for N odd and N for N even. (No tile can have more than 2N sides because all iterations of an extended edge are parallel, so there are never more than 2N directions to choose from. For N even, there are just N directions.)

Examples: Below are large-scale webs for the regular heptagon (N = 7) and the regular tetradecagon (N = 14) showing rings of maximal D tiles. Since the region between rings is invariant, the dynamics are bounded. The region inside the first ring of Ds is called the central star region, and this region can serve as template for the global dynamics.

We will use the regular heptagon (N = 7) to introduce some of the basic concepts. Below is a vector plot of the central star region, showing the first ring of D tiles and the associated family of related tiles. The central N = 7 is also known as M, so D and M serve as patriarch and matriarch of this family.

For 'most' N-gons and 'most' initial points, the orbits under τ are periodic. The plot above shows the period 7 orbit of the center of the D tiles. In this vector plot we show just the major resonances of the Tangent map and these constitute the First Family for N = 7. The non-redundant members of this family are shown below:



The three star points determine the bounds of the First Family. Star[3] is called GenStar because in some contexts, there are infinite families of tiles converging to this point. D is also called S[3] because the step-sequence of its orbit is constant {3}. What this means is that the Tangent map orbit skips 3 vertices on each iteration. This can be seen in the orbit plot above. Likewise the orbit of S1 and S[2], skip 1 and 2 vertices on each iteration and the centers are also period 7. DS1, DS2 and DS3 are really sub-tiles of D, and their orbits would skip 1,2, and 3 vertices, if they were relative to the star points of D.


Below is this same region showing the detailed web structure which is very complex. Since φ(7)/2 = 3, N = 7 is classified as a cubic polygon along with N = 9. Typically the algebraic complexity of a regular polygon is directly reflected in the dynamical complexity. The quadratic polygons, N = 5,8,10 and 12, all have relatively simple dynamical structure with webs which have a well-defined fractal dimension. For N = 7 and beyond, the web structure is probably multi-fractal.


M and D are called an M-D pair because their dynamics are linked (see below). The D tile shares the same side length as M, but it has 14-sides so it contains two copies of M. DS[2] and DS[1] above are also an M-D pair and these pairings continue at well-defined scale reduction so that they form an infinite sequence converging to GenStar. If these pairs encompassed all the essential dynamics of N = 7, then N = 7 would have a simple fractal structure just like the quadratic polygons. But this is apparently not the case.


Star Points and Scaling for a Regular Polygon


As indicated above, N = 7 does support chains of generations converging to the GenStar point (and an infinite number of related points). These chains begin with M and D which are the initial M[0] and D]0] which are matriarch and patriarch of the 1st Generation. Likewise M[1] and D[1] will be matriarch and patriarch of the 2nd generation which is scaled to fit on a single edge of D[0] as shown above for DS[1] and DS[2]. The scale that would accomplish this is GenScale. The star points and scales of any regular polygons are defined as follows:


Definition: (i) For a regular N-gon with N-even , the star points are given by Star[N_]:= Table[{-Cot[k*Pi/N],-1},{k, HalfN, 1,-1}].

When N is odd, the star points are Star[N_]:= Table[{-Cot[Pi*(N - 2k)/(2N)],-1},{k, 1, HalfN}].

(For the star points, we assume that the standard position of a regular N-gon is centered at the origin with height (apothem) 1 and bottom edge horizontal. HalfN is defined to be the largest integer less than N/2)


(ii) Every star point defines a scale as follows: scale[k] = star[1][[1]]/star[k][[1]] = (-s/2)/star[k][[1]]

(Where star[k][[1]] is the horizontal coordinate of star[k] and s is the side length of N)


Therefore every regular polygon has HalfN star points with star[1] a vertex of N and star[HalfN] known as GenStar[N]. The scales are strictly decreasing with scale[1] = 1 and scale[HalfN] known as GenScale[N].


Example: The 6 star points of N = 14 can be defined using the {14,6} star polygon shown here on the left. The second image shows the perfect fit of the First Family. Since this {14,6} star polygon consists of the extended edges of N = 14, it forms the level-1 singularity set for this star region. The third image shows the level-3 web which is based on the {14,6} star polygon, so it can be regarded as a level-3 {14,6} star polygon. This star region is invariant under τ and can serve as a template for the global dynamics.

Even though N = 14 has 6 scales, symmetry reduces the essential scales to 3, and these 3 are proportional to the 3 scales of N = 7. This is called the TwiceOdd Lemma in First Families of Regular Polygons and it is generic for N and 2N pairs with N odd - so they appear to have conjugate dynamics. Algebraically, this conjugacy is a reflection of the fact that the cyclotomic polynomials ΦN(X) and Φ2N(-X) are identical when N is odd.


Because of the conjugacy between N = 14 and N = 7, the local dynamics would be unchanged if the origin is shifted from M to D. This is illustrated below with an unmarked origin. If D is at the origin, M is now 5-step relative to D, but the progeny have step sequences which are essentially unchanged. For example on the right is the orbit of DS[3] around M. The period is 14 and the step sequence is constant {32}. Note that the orbit visits D on every third iteration so it is step-3 relative to D. This orbit unfolds from a {3,2} to {3} as we pass from Ms world to Ds. In this process the 7 Ds act as one.


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Example: The First Family of N = 11 showing the five star points in magenta. Star[5] is GenStar[11] and the corresponding scale is GenScale[11] ≈ 0.0422171162264.


Even though N = 11 does have DS[1] and DS[2] in the correct positions to be M[1] and D[1] there is no evidence that this chain continues. There is a small M[2] but no matching D[2]. (See the plots at the top of the page.) As indicated earlier, N= 11 is the first quintic polynomial and the dynamics are more complex than any polygon studied in depth.


Even for N = 7, the generation issue is far from simple. For all generations M[k] and D[k] are scaled by GenScale[7] relative to the First Family, but the remaining family members of each new generation are not always scaled copies of the First Generation. The odd generations appear to be perfect scaled copies, but the even generations are slightly different. This may be due to the fact that N = 7 has two non-trivial scales. The situation is worse for other regular N-gons like N = 13 which support chains of families. The subsequent families (aside from M[k] and D[k]) appear to have little resemblance to the First Family.


Our numerical evidence supports the following:


4k+1 Conjecture: Suppose a regular N-gon is in standard position with GenStar and GenScale as defined above. If N = 4k+1 for k a positive integer, then there will be infinite sequences of regular N-gons M[j] (the Ms) and regular 2N-gons D[j] (the Ds) 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 Ds and Ms 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 M[2]- there is not even a D[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 Ds and Ms, but the rest of the families show little evidence of self-similarity such as that found with the quadratic polygons and N = 7.


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 D, 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 Ds is constant step-8 and outside this ring, the steps are only 8 and 9. There are endless rings of Ds 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 D's center, so it is |2*cS[8][[1]]| ≈ 21.58 (where [[1]] signifies the first co-ordinate). These rings of Ds 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, Gausss 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 Ds 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 ?


Note: In the recently published First Families of Regular Polygons, there is a Scaling Lemma which shows some hope of relating these diverse families. The Scaling Lemma says that two regular polygons will have related scales when one of a factor of the other. This follows because any factor polygon will have star points which are a subset of the star points of the parent. (Star points are of course relative to the dimensions of the polygon, but they can be defined using a standard height (apothem) of 1 and a lower edge horizontal, so any external factor polygon can be assumed to also have a height of 1 and lower edge horizontal, and hence the star points will be comparable a shown below for N = 24.)

For example, using N = 8 as the external factor, every third star point of N = 24 is a star point of N = 8, so the corresponding scales are related by scale[j] (of N= 8) = scale[3j]/scale[3] (of N = 24). Here scale[3] of N = 24 is what we call ScaleSwap[24,8] = Tan[π/24]/Tan[π/8] = ratio of sides (assuming same height).


In the outer billiards map and other piecewise affine maps, the scaling tends to shadow the dynamics, so the Scaling Lemma might provide a link between the dynamics of these families of regular polygons.


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 recent notebook called FirstFamily.nb encompasses all the regular cases.


The posthumous message written on Richard Feynmans 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 M 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: 2arrows.jpg

All orbits diverge



N = 3



Endless rings of Ms and hexagon Ds.

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

Step sequences of Ds are (1),(211),(21211),etc

with limiting sequence (21) so ω→.5

Non-regular triangles have similar structure.



N = 4




Description: Description: 4rectangle2nd ring.jpg

'web' structure is similar to the triangle - but with no distinction between Ms and Ds.

Ring k has 4k Ds 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: kite2.jpg


Below is the level 600 web

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.


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 s = {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 Ds have step sequences (2), (322), (32322),..,→(32) with ω→.5

N = 6


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


kth ring of Ds has 6k hexagons and odd rings have decomposition and period doubling. The second ring shown here is period 12 and has no decomposition so the centers are period 12.


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

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


N = 7


Generation 1


Generation 2 - Portal Generation


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 M[k+2]/M[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 1s and 2s and outer with 2s and 3s. Step 4 occurs only outside star region. All prime N-gons have similar decomposition.

The interaction between scales becomes more complex with each generation, and beyond the 5th generation there are regions (at star[2] and star[3] of D[1]) where the dynamics are no longer predictable.

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


N = 8



Only octagons no period doubling

Periods of D[k]/D[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] = D[k] orbits have same ω.


N = 9


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 region around S2[3] bud is enlarged on the right.


First generation canonical except that S3 has 12 sides composed of two interwoven hexagons at different radii, and DS[3] has extended edges to form a non-regular hexagon.

Ms and Ds exist on all scales and ratio of periods M[k]/M[k-1]→10 (but not dense).

Second generation is dominated by 'Portal Ms' similar to those of N = 7. In between these Portal Ms 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: 10inner&outerstar.jpg

Domain structure is identical to N = 5 but the 10 Ds form two groups of 5 and the 10 S2s form two groups of 5. This is typical for twice-odds.

The decomposition of the Ds persists for odd rings a has no effect on the outer star region, but the decomposition of the S2s creates two invariant inner star regions one of which is shown here. Together they define the inner star. The 10 pentagon Ms patrol the border between the inner and outer stars.

N = 11





Description: Description: 11Gen1all.jpg


The second 4k+ 3 prime N-gon

Normal first generation but no evidence of Ds past D[1] or Ms past M[2].

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

D[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 only one non-trivial scale which guarantees a fractal structure.


Ratio of periods of D[k]/D[k-1] →27 so the fractal dimension appears to be Ln[27]/Ln[1/GenScale[12]] ≈1.251


The six-sided S2s are determined by triplets of virtual D[1]s, as shown here.


S4 is canonical with buds of size S1.


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

N = 13


Second Generation

Description: Description: 13Gen2b.jpg





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

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

There is no obvious self-similarity in the new generations at GenStar it is possible that each new

Generation is distinct.

N= 14







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

Show here is index .95 for N = 7

Normal ring of 14 Ds is now subdivided - the orbit of the 7 large Ds 'sees' only the canonical N = 7 M tile and the 7 small Ds 'see' only the secondary M.

Star region is no longer invariant.

Rings of Ds undergo periodic oscillations where the Ds grow and then shrink.

Dynamics are very complex. Many orbits diverge rapidly at first and then fluctuate widely in distance.




A lattice polygon has vertices with integer co-ordinates. The lattice polygon M 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 here is the orbit of {4,1} which is period 6. This is a period-doubling orbit and the center point of the tile containing {4,1} is {-3,2}. (The tile is not shown here). This is the only point in the tile that is period 3. (Odd periods can only arise in this fashion because tiles are inverted on each iteration so almost all periods are even.)



N = 281








This is a 4k+1 prime so there should be chains of Ms and Ds converging to GenStar ≈ {-178.888,-1}

(the convention here for M is height 1, not radius 1)

As N increases, the star region grows while the scale shrinks. GenScale[281]≈.000062 so M[1] is microscopic and the local dynamics at GenStar appear to have little effect on the global dynamics- which are dominated by simple rotations about M like a twist map on the unit circle with minimal perturbations. The amount of twist increases smoothly with the distance from the origin. D is S[140] with maximal twist (for the inner star ) at ω = 140/281.


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


D 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 D (at 3:00) bisects DS[140]. If N was 4k+3, the shepherd would be an odd-step tile. The general formula for the shepherd is DS[Floor[N/2]].



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