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 P2 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 smallscale ‘chaos’
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 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 multifractal. 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
7gon 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
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 hexagon ‘skating
rink’ are also periodic so they have constant winding number. The period is
169·2 and ω = 41/169. The neighboring dynamics are very complex and the
skating rink is bordered by a 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. 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
nonperiodic orbit. We may never know the limiting smallscale
structure for the regular 11gon 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 nontrivial scales which define each ‘generation’.
These scales are noncommensurate and they 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 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 multifractals
provides some insight, but after years of investigations, we see little hope
of ever 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 5.0 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 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. Each extension
introduces an additional level of complexity. Very little is known about the
geometry of these field extensions, but we have found that 4k+1 primes such
as 17 have a local recursive structure which appears to be lacking in 4k + 3
primes. This 4k + 1 structure may to be a reflection (no pun intended) of
Gauss’s results on quadratic reciprocity because these results partially determine
the nature of the extension fields. To peer thought these layers of complexity
we will sometimes use projections which generate ‘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 theTangent 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. See also TangentDurer.
We will use N to designate the number of
sides (or vertices) of a convex polygon. As indicated above the case of N = 3
holds few surprises, even when irrational coordinates are allowed, but the
case of N = 4 is surprisingly complex when at least one of the vertices is
irrational. A Penrose Kite
is a quadrilateral such as the one shown above. In 2007 Richard Schwartz of Brown University showed that when q= 2, there are
orbits which diverge.We call polygons in this class
'unstable', so this irrational Penrose Kite is unstable. In 2009 Schwartz
generalized this proof to include any irrational q value and he conjectured
that ‘most’ polygons are unstable. Our numerical evidence supports this
conjecture. There is a summary table below which gives results about an
assortment of polygons.There are longer versions in
Pdf form which can be accessed via the Site Map or
the Pdf folder link above. In
his Wikipedia article on the outer billiards map,
Schwartz lists the following as the most important open questions: (i) Show that outer
billiards relative to almost every convex polygon has unbounded orbits. (ii)
Show that outer billiards relative to a
regular polygon has almost every orbit periodic. The cases of the equilateral
triangle and the square are trivial, and S.Tabachnikov answered this for the regular
pentagon. (iii)
More broadly, characterize the
structure of the set of periodic orbits relative to the typical convex
polygon. Below is a list of further questions which
we hope to answer with the help of the community (or a Deity). (iv)
For a given polygon what are the
admissible step sequences ? (This is one for the
Deity.) (v)
Which polygons have unbounded orbits ? ( These are called 'unstable' polygons.) (vi)
Why is it true for 4k+ 1 prime 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 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 W_{k} to be the (forward) exceptional set for τ^{k}
, with limiting web W^{f }= The 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. N=
7 and N = 11 share the same scales, so they have conjugate dynamics. The
local dynamics would be ‘unchanged’ if the origin is shifted from M to D.
This is illustrated on the left below where the origin is unmarked. If D is
at the origin, M is now 5step relative to D, but the progeny have step
sequences which are essentially unchanged. To illustrate this, on the right
is the orbit of DS[3] around M. The period is 14 and
the step sequence is constant {32}.(The points in Ngons such as DS3 all map together, so the center has the
same period as the remaining points.) 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. Algebraically, this conjugacy
is a reflection of the fact that the cyclotomic polynomials Φ_{7}(X) and Φ_{14}(X) are identical so φ(7) = φ(14). This
is true for all regular Ngons with N odd. Star Points
and Scaling for any 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: Definition: (i)
For a regular Ngon with Neven ,
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 Ngon 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[Half] known as GenStar[N]. The scales are strictly decreasing with scale[1] = 1 and scale[Half] known as GenScale[N]. N = 14 had 6 star points so there are 6 scales, but symmetry
reduces the ‘essential’ scales to 3, and these 3 are proportional to the 3
scales of N = 7. This is generic for N twiceodd and it explains why they
have conjugate dynamics. 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]. 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 Ms)
and regular 2Ngons 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 '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 Mom[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 ? 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. •
Small scale chaos in qstar region and Star[2]
region •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 Helen3 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 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 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. 
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
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 above is
the orbit of {4,1} which is period 6. This is a
perioddoubling orbit and the center point at {3,2}
has period 3. 
N
= 281 Regular 

•
As N increases, the star region grows while the scale shrinks. GenScale[281]≈.000062
so M[1] is microscopic and her local dynamics have little or no 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 top 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 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]]. 
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