Dynamics of
Polygons 

email: mail@dynamicsofpolygons.org 

The web of the regular pentagon (N = 5) 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 22gons known as S1 and S2.
Compass
and straightedge construction for the regular heptadecagon
(N= 17)
A P5 projection for N= 17
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 nonregular 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.
Step1
orbit of S1 center. It is period 7. The remaining points in S1 take two
circuits around M, so they are period 14. A
region above S1 for N = 7 showing the breakdown of selfsimilar structure.
The
First Family for N = 17 showing M on the right and D on the left. There are 17 such D’s 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 (17771855)
Neils Able
(1802−1829)
Evariste Galois
1811−1832) A period 12 orbit for N = 6 A P5 projection for N = 16 
This is a nonprofit 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 tau. (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.) If the edges of a regular Ngon are extended until they meet, the result is called a 'star polygon'. For the regular pentagon (N = 5) and octagon (N = 8) shown here, these extended edges have been repeatedly mapped under the inverse of tau to generate the black weblike regions. These webs W are called 'singularity' sets because by definition tau is not defined on any point in W. This means that tau is always defined on the complement of W as shown by the examples here. These white regions in the complement of W are called 'tiles'. For N = 5 and N = 8 these tiles exist on all scales  as can be seen by clicking on the pentagon image in the left panel. The web W reveals geometric and algebraic structure which is inherent in the polygon and every regular Ngon has its own unique web. The webs for N = 3 and N = 4 are not very interesting because the 'algebraic complexity' of N = 3 and N = 4 is 1  while the algebraic complexity of N = 5 and N = 8 is 2. The algebraic complexity of a regular Ngon is defined to be Phi[N]/2 where Phi[N] is the Euler totient function  which enumerates all the integers less than N which are relatively prime with N. So to find Phi[8] check which of 1,2,3,4,5,6,7 have no factors in common with 8 and the answer is just {1,3,5,7}. For N >2, this number will always be even and we reduce it by half because of symmetry. The only Ngons with 'quadratic' complexity 2 are N = {5,8,10,12} and in each case W is a fractal image which is selfsimilar on all scales. For Ngons with complexity greater than 2, it is likely that W will be multifractal with an infinite spectrum of dimensions. A recent paper summarizes what is known about the dynamics and geometry for regular polygons. It will be at the Cornell University arxiv site but it is also available here under the link to PDFs at FirstFamilies &Mutations Aside from a few simple cases, very little
is known about the dynamics or singularity set W of a regular polygon. The
regular heptagon N = 7 has 'cubic' complexity 3 so the vertices cannot be specified algebraically without a cubic equation and the
regular 11gon requires a quintic. In both cases
the resulting dynamics are very complex. At the top of this page are two ‘density’
plots showing the smallscale dynamics of the regular 11gon. (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 shown below get their name
from their ‘Step1’ 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 hexagonal ‘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 multifractal 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. N = 11 and N = 22 have the same 'quintic' complexity because doubling the vertices of a odd Ngon is a trivial operation that does not increase the algebraic complexity. See Constructions of Regular Polygons. This is no longer true for even Ngons, so N = 24 has algebraic complexity 4 which is twice that of N = 12. In the plots below N = 11 is shown embedded in the web of N = 22. The web of N = 11 can be done separately but there would be no difference. The blue and black regions are invariant under tau. Here we are interested in the region local to N = 11 at the red arrow. The First Family members of N = 11 are numbered consecutively from S1 to S5, and the most important tiles are S1 and S2 adjacent to N = 11 as shown here and at the top of the page. The edges of odd Ngons appear to have dynamics which are more complex than any region in the overall web W and the blue region here adjacent to N = 22 is better behaved. We may never know the limiting smallscale
structure for the regular 11gon because the geometric complexity tends to increase
with each new ‘generation’. The plots above resolve only the first 3 or 4
generations. N = 11 has four linearly independent scales which act across generations, so a 6^{th}
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 nontrivial scales guarantees a diversity of dynamics beyond
that of a simple fractal structure. The First Family
tiles are always part of this 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 multifractals
provides some insight, but after years of investigations, there has been only
minor progress in resolving the smallscale 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 25^{th} 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 to 11. See
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 plugin. Download the plugin at http://www.wolfram.com/cdfplayer 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 17gon is constructible. This implies that the vertices for
the regular 17gon 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 17gon.
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 17gon (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. This will be discussed below. 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 cases. In the latest paper discussed above we show that the 4k+1 conjecture appears to follow from the Edge Conjecture  which makes predictions about the evolution of the S[1] and S[2] tiles adjacent to N. In the 4k+1 case, the predicted family of S[2] includes an S[3] tile that apparently can generate 'nextgeneration' S[1] and S[2] tiles to keep the chain intact. In this case we know both the 'temporal' (dynamical) scaling and the geometric scaling of these sequences so it is possible to calculate the local fractal dimension. For N = 11 above we can make no such predications and the whole concept of 'generations' may be pointless since the longterm evolution is unknown. To peer thought these layers of complexity
we will sometimes use projections which generate ‘crosssections’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 ngon is never
higher than φ(n)/2) Note that each vertex remapping has the
form R^{17} → R^{17} 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
17gon. (This region is one of 7 invariant annuli surrounding the 17gon.
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 17gon 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 (17371794) 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. Nonregular 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 nonregular 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 the OuterBilliards Map τ :
Given a convex polygon with a clockwise (or counterclockwise) 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. 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 ngon P with
vertices {c_{1}, c_{2},..,c_{n}}
which we assume to be numbered clockwise, the corresponding edges E = {E_{1},E_{2},
..,E_{n}} have both a ‘forward’ and ‘trailing’ extension. (Just the trailing extensions are shown
below in blue.) Let E^{f} = E^{f}_{k} be the union of
the forward extensions and E^{t} = E^{t}_{k}
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 level0 web is defined to be W_{0} = E E^{t}. This is the exceptional (or
singular) set of τ. Since W_{0} is connected, the complement of
W_{0} external to P consists of n disjoint open (convex) sets which
are known as level0 tiles. These tiles are also called ‘atoms’ because all
the subsequent dynamics are determine by repeated action of τ on this
level0 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 levelk tile will have a corner sequence of length k+1. The union of the level0 tiles is the domain of τ
which is abbreviated Dom(τ). Dom(τ^{2})
is Dom(τ) – τ^{1} (W_{0}). The union of W_{0} and τ^{1} (W_{0}) is called the
level1 web, W_{1}. In general the level k (forward) web is defined
to be: (i) W_{k} = _{} where W_{0 }= E E^{t} The
inverse web is defined in a similar fashion using τ and the extended
forward edges: (ii) W^{i}_{k} =_{}where W^{i}_{0} = E E^{f} At each iteration, W_{k} and W^{i}_{k}
partition the plane into disjoint open convex regions which are the levelk
tiles. Below is W_{1 }for the polygon
P, showing the level1 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 ‘stepsequences’ 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[{x_{1},x_{2}}] = + = 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 (energypreserving) 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 Ngon. 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 first transformation in the graphic above is an example
of an 'elliptical' affine transformation. The corresponding web is shown in
the sidebar 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 nonaffine 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.
As indicated above the case of N = 3
holds few surprises, even when irrational coordinates are allowed, but the
nonregular 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 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 Ngons that the D’s and M’s 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 welldefined class of nonregular convex polygons with this property ? (viii)
For regular Ngons,
does the smallscale 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 ngon
has a corresponding number field which is an algebraic extension of the rationals Q. This number field is the cyclotomic
field K_{n} =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 2Ngon (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 (19291999) called "Is The Solar System Stable
?"
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 (6times continuously
differentiable). This was proven by Douady in 1982. A
convex polygon is not even 1times 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 MoserNumann 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
OuterBilliards
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 W_{k} to be the (forward) exceptional set for τ^{k}
, with limiting web W^{f}
and limiting inverse web W^{i} 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 W^{c} 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
largescale 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
D’s 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' Ngons 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 nonredundant 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 ‘stepsequence’ 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 subtiles 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 welldefined fractal dimension. For N = 7 and beyond, the web
structure is probably multifractal.
M
and D are called an MD pair because their dynamics are linked (see below). The
D tile shares the same side length as M, but it has 14sides – so it contains
two copies of M. DS[2] and DS[1] above
are also an MD pair – and these pairings continue at welldefined 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 1^{st} Generation.
Likewise M[1] and D[1] will be matriarch and
patriarch of the 2^{nd} 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:
We assume that the Ngon is in ‘standard position’ centered at the origin with ‘bottom’ edge horizontal. 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 ‘level1’ singularity set for this
‘star’ region. The third image shows the level3 web which is based on the
{14,6} star polygon, so it can be regarded as a
‘level3’ {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 Φ_{N}(X) and Φ_{2N}(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 5step 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
step3 relative to D. This orbit ‘unfolds’ from a {3,2}
to {3} as we pass from M’s world to D’s. In this process the 7 D’s act as
one. 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 nontrivial scales. The situation
is worse for other regular Ngons 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
Ngon 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 Ngons M[j] (the M’s) and regular 2Ngons
D[j] (the D’s) 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 'supersymmetric'. It is easy to show that N
= 5 (shown above) satisfies the conjecture, and it may be the only case where
the selfsimilarity begins with the first generation. The heptagon is not
supersymmetric but it appears to share some of the properties of the 4k+1
primes  namely the sequences of D’s and M’s 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 D’s and M’s, but the rest of the families show little evidence of
selfsimilarity 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 AdrienMarie
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 step8 relative to D, it is a 2Ngon. If this was a 4k+3 prime, Floor[N/2] would be odd and this last occupant would be an
Ngon. 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.
Moving
outwards, the ring of D’s is constant step8 and outside this ring, the steps
are only 8 and 9. There are endless rings of D’s 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 coordinate). These rings of D’s 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 sidebar 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 nontrivial
vertices can be grouped into conjugate pairs: s_{1}= z + z^{4}
and s_{2}= z^{2} + z^{3} ,
where z = cos(2π/5) + isin(2π/5). Note that s_{1}
+ s_{2} = − 1 and s_{1}s_{2} = − 1 so
they are roots of x^{2} + x −1 = 0. The solutions of this
equation are ± 1)/2. Since s_{1} = 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 JosephLouis 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 20^{th}
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 ngon are the (complex)
solutions to X^{n} = 1. This is called the nth cyclotomic
equation. The corresponding polynomial X^{n}^{
}1 is always divisible by X1, and for prime ngons,
this is the only divisor, so the minimal cyclotomic
polynomial, Φ_{n}(X), is degree n1.
In general the degree of Φ_{n}(X) is the Euler totient function
φ(n). (This explains why it is common to use
the symbol Φ_{n}(X) for this
class of polynomials.) For
the pentagon above there are 5 solutions to the cyclotomic
equation. These solutions can be written as z_{k}
= cos(2πk/n)
+ isin(2πk/n)
for k = 0,1,2,3,4. The trivial solution z = 1 is written as z^{5} 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 Φ_{11}(X) is degree 10 and the minimal polynomial for
cos(2π/11) is a quintic: MinimalPolynomial[Cos[2Pi/11]] yields 1+ 6x 12x^{2}32x^{3}+16x^{4}+32x^{5}
= 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 7gon and it would be necessasry to quinsect an angle –
that is, divide an arbitraty angle into 5 equal parts  to construct a
regular 11gon. 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 nonprime 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 nonregular 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 nonregular 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 Ngons
with large N, the dynamics approach that of a circle or ellipse. For
odd N, there is a infinite
family of regular 2^{k}Ngons, 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 nontrivial 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 ‘powersof2’ 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
Nonregular 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 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 M on a scale from 0 to 1, with 1 being a
full rotation, so a canonical step3 periodic orbit for a regular Ngon 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 polygon) 

•
All orbits diverge 
N
= 3 Regular 

• Endless rings
of M’s and hexagon Ds. •
Ring k has 3k D’s and 12k Ms and these are also the
periods if we plot centers of the D’s. Shown here are the 12 M's in ring 1.
They map to each other in the order given. •
Step sequences of D’s are (1),(211),(21211),etc with
limiting sequence (21) so ω→.5 •
Nonregular triangles have similar structure. 
N = 4 Regular


•
'web' structure is similar to the triangle  but with no distinction between
M’s and D’s. •
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 (Penrose Kite) 
Below is the level 600 web 
•
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 roundoff : Τ^{500000000}(p)
={−5730+ (1−q)/2 −6688q , −4417} ). 
N
= 5 Regular 
The star region below shows the location of a nonperiodic point p with orbit dense in the star region. 
•First
nontrivial star region inside ring of 5 Dads. •Decagon
and pentagon periods satisfy: d_{n} = 3d_{n−1}+
2p_{n−1 } &
p_{n}_{ }= 6d_{n−1}
+ 2p_{n−1} with d_{1}=5 and p_{1}=10 •
d_{n}/d_{n−1} →6 and
decagons are dense so fractal dimension is Ln[6]/Ln[1/GenScale[5]]
≈ 1.241 •The
point s = {c_{5}[[1]], c_{4}[[2]]}
has a dense nonperiodic orbit with ω→ .25. The plot on the left
is 50,000 points in this orbit.Note perfect
selfsimilarity. •Bounding
D’s have step sequences (2), (322), (32322),..,→(32) with
ω→.5 
N
= 6 Regular


• Domain
structure is identical to N = 3 with any hexagon as M and the adjacent
triangle as S1. •
kth ring of D’s 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 D’s are (2),(32),(332),..→(3) 
N
= 7 Regular 
Generation 1 Generation 2 
Portal Generation 
•
First prime Ngon with multiple scales. It is not
supersymmetric but it retains some of the properties of 4k+1 primes, so it
is a 'hybrid'. • Odd
generations are selfsimilar. •
Even (Portal) generations are selfsimilar. •
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
supersymmetric 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 Ngons have similar
decomposition. •The
interaction between scales becomes more complex with each ‘generation’, and
beyond the 5^{th} generation there are regions (at star[2]
and star[3] of D[1]) where the dynamics are no longer predictable. •Bounding
D’s have step sequence (3),
(334), (33434),..→(34) with limiting ω = .5 
N
= 8 Regular 

•Only
octagons – no period doubling •Periods
of D[k]/D[k1] →9 and they are
dense so fractal dimension is Ln[9]/Ln[1/GenScale[8]]
≈ 1.246 •Dense
nonperiodic 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 Regular 
The small rectangle above outlines a portion of the second
generation which is shown below. There are 'islands' of chaos amid perfect
selfsimilarity. 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 nonregular hexagon. •M’s
and D’s exist on all scales and ratio of periods M[k]/M[k1]→10
(but not dense). •Second
generation is dominated by 'Portal M’s' similar to those of N = 7. In between
these Portal M’s 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 Regular 
The central star region showing all of the outer ring and
half of the inner ring. 
•
Domain structure is identical to N = 5 but the 10 D’s form two groups of 5
and the 10 S2’s form two groups of 5. This is typical for ‘twiceodds’. •
The decomposition of the D’s 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 ‘M’s’ patrol the border between the inner and
outer stars. 
N
= 11 Regular 

•
The second 4k+ 3 prime Ngon •
Normal first generation but no evidence of D’s past D[1]
or M’s 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 budforming process breaks down. •No
obvious selfsimilarity but small invariant ‘islands’ exist on a scale
between generations 3 and 4. 
N
= 12 Regular 

•Complex
geometry due to the factors of 12, but only one nontrivial scale – which
guarantees a fractal structure. •Ratio
of periods of D[k]/D[k1] →27 so
the fractal dimension appears to be Ln[27]/Ln[1/GenScale[12]]
≈1.251 •The
sixsided S2’s are determined by triplets of virtual D[1]’s,
as shown here. •S4
is canonical with buds of size S1. •S3
is nonregular octagon with center at {0, } 
N
= 13 Regular 
Second Generation 
•The
second ‘supersymmetric’ prime polygon so at GenStar,
ratio of periods of D[k+1]/D[k] → 14, and same
for M’s. Ratios alternate high and low, so there is some evenodd
differentiation. •Dynamics
around M[1] are characterized by dense halo of
noncanonical buds. There are
protected pockets at GenStar and under D[1] for 3rd generation. •There
is no obvious selfsimilarity in the new generations at GenStar–
it is possible that each new Generation
is distinct. 
N=
14 Woven 

•
A woven Ngon 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 D’s is now subdivided  the orbit of the 7 large D’s 'sees' only
the canonical N = 7 M tile and the 7 small D’s 'see' only the secondary M. •
Star region is no longer invariant. •
Rings of D’s undergo periodic oscillations where the D’s 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 coordinates. The lattice polygon
M shown here has vertices
{{2,0},{1,2},{3,1},{1,1}} •
Any polygon with rational vertices can be rewritten 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 coordinates. Shown here is
the orbit of {4,1} which is period 6. This is a
perioddoubling 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 Regular 

•
This is a 4k+1 prime so there should be chains of M’s and D’s 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’ outerstar 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 oddstep tile. The general formula
for the shepherd is DS[Floor[N/2]]. 
Bibliography 
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 119133 Alperin R.C., Trisections and totally real origami, American
Mathematical Monthly, Vol 112, No. 3 (March 2005), pp 2002011 Arnold V.I. , Proof of a theorem
by A. N .Kolmogorov on the invariance of
quasiperiodic 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 232:409416 Ashwin P,
Deane J., Fu XC., Dynamics of a bandpass
sigmadelta modulator as a piecewise isometry.
Proceedings of the IEEE International Symposium on Circuits and Systems,
Sydney Australia,2001:III811III814 Ashwin P.,
Goetz A. , Polygonal invariant curves for a planar
piecewise isometry. Transactions of the AMS,
2006 350:373390 Ashwin P., Deane J., Fu XC. ,
Properties of the invariant disk packing in a model sigmadelta bandpass modulator. International Journal of Bifurcations
and Chaos 2003 13:631641 Ashwin P.,
Goetz A,. Polygonal invariant curves for a planar
piecewise isometry. Transactions of the American
Mathematical Society 2005 358:373390 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.,
Nonsmooth 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 invariatn
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. 215226 Baez J.C., Division algebras and quantum theory, arXiv:1101,5690 Baez
J. C., The octonions, Bulletin of the American MathematicalSociety,
Vol. 39, page 1452002. Paper and additional bibliography at http://math.ucr.edu/home/baez/octonions/ 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)
14111438 Breslin S., DeAngelis, V.,The minimal polynomials of sin(2π/p) and cos(2π/p), Mathematics Magazine, April 2004 Kahng B. ,
Singularities of twodimensional 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: 35:648658 Chua L.O.,Lin T. Chaos and fractals
from third order digital filters. International Journal of Circuit Theory and
Applications, 1990 18:241255 Chua L.O., Lin T. Fractal pattern of
secondorder nonlinear 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 353:8599 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 3eme 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 sigmadelta modulation  an
analysis from the perspective of nonlinear dynamics. Proceedings of the IEEE International
Symposium on Circuits and Systems, Atlanta, USA,1996 3:146149 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 pp357371 Gleason A. , Angle trisection, the heptagon and the triskaidecagon,
American Mathematical Monthly, Vol 95, No. 3 ,1988, pg. 185194 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. 173,101133 (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: 71:0438031 04380311 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) 161178 Hughes G.H.,
Probing the strange attractors of chaos (A.K. Dewdney), Scientific
American, July 1987, pg 110111 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 Hughes G.H. , Outer Billiards on Regular Polygons,
arXiv:1311.6763 Hughes G.H. , First Families of Regular Polygons, arXiv:1311.6763 Jackson L.B,
Kaiser J.F., MacDonald H.S. , An approach to the implementation of digital
filters, IEEE transactions, Audio and Electro acoustics, AV16, No 2 (1968)
413421 Julia G. , Sur l’iteration
des fonctions rationnelles,
Journal de Math Pure et Appl (1918) 8:47245 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): 527530. 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 447448 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)
413451 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.
NorthHolland, Amsterdam, 2002. MR1928530 (2003j:37002) Mather
J., Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207, 169207(1991). Mather
J., Variational construction of orbits of twist diffeomorphisms, J. Amer. Math. Soc., 4, 207263(1991). Moore F.L.,
Robinson J.C., Bharucha C.F., Sundaram
B, Raizen M.G. Atom optics realization of the
quantum deltakicked rotor. Physical Review Letters 1995 75:45984601 Moser J.K., On invariant curves of
areapreserving 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: 6571 Moser J. K., Siegel
C.L. Lectures on Celestial Mechanics, SpringerVerlag,(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)185198 Poincare
J.H., (189299) Les methodes nouvilles de la mechanique celeste, Paris: GauthierVillars 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. www.math.brown.edu/~res/Papers/PET.pdf, 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 9780691142494 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) Schwartz R.E., The Octogonal
PETs, AMS monograph, Vol 197,
ISBN9781470415228 Scott A.J., Holmes C.A.,Milburn,G.J.
Hamiltonian mappings and circle packing phase spaces. Physica D 155 (2001) 3450. MR1837203 Siegel C. L. , Iteration of Analytic Functions, Ann of Math
(1942) 42: 607612 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 7^{th} edition, originally translated in 1966
from the German text Moderne Algebra (193031.) 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) 