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 Mom, so they
are period 14. A region above S1 showing smallscale
‘chaos’
The First Family for N = 17 showing
Mom on the right and Dad on the left. There are 17 such Dads surrounding Mom. P2 projection
of the S[1] center in ring 2 for N
= 17 Disquisitiones
Arithmeticae By C. F.
Gauss – 1801
C.F. Gauss (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 is inherent in the
polygon so we call it the ‘genetic structure’ of the polygon. For regular
polygons with prime number of vertices, the basic geometry can be derived
independently of the Tangent map. Regular polygons typically have structure
on all scales but it is not clear what other polygons have this property. The
pentagon and octagon above have a simple fractal structure, but most regular
polygons have a more complex structure which is often multifractal. Aside
from a few simple cases, very little is known about the genetic structure 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. The 11gon has four nontrivial
scales which define each generation. These scales are typically irrational
and 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
as each generation depends on all the previous generations. The issue not
just size, but the corresponding dynamics which determine the size. The
theory of 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. 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. The Tangent
map is not defined on the extended trailing edges of the polygon. These are
the magenta edges shown on the left for the regular (equilateral) triangle.
Since τ is not defined on these edges, τ^{2} =
τ◦τ is not defined in the inverse image of these edges. These
are the blue edges shown here. Points on these blue edges map to the magenta
edges, so they also have to be excluded from the domain of τ. Continuing
in this way, the 'web' of points which must be excluded at iteration k is
Where W_{0}
is the level 0 web, consisting of the magenta trailing edges. Taking the
limit as k→∞ yields the (inverse) web W. (The forward web is
defined in a similar fashion using forward edges and τ, and the full web
is the union of these two.) The Tangent map software will generate the
forward webs to any desired accuracy and a reflection transform will yield
the inverse webs when necessary. The complement of this web are points where
τ is always defined. For the regular triangle, this web consists of
rings of congruent triangles and hexagons as shown on the left. This is what
we call the genetic structure of the regular triangle. Any other triangle
would have a similar web because every triangle is an affine transformation
of the equilateral triangle. Run CDF Web Demo. An affine transformation T is a linear transformation together with a possible translation so it has the form T[{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 Dads and Moms survive the turmoil and
why does the ratio of consecutive Dad (& Mom) periods approach N + 1?
(vii) Which
polygons have structure on all scales ?
Are there a 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 Regular Heptagon: N = 7 We will use the regular heptagon to introduce some
of the basic concepts. For 'most' Ngons and 'most' initial points, the
orbits under τ are periodic. Above is an example of a period 7 orbit
with a regular heptagon at the origin. The larger polygons are regular
14gons. In our genetic language they are called 'Dads' and the generating heptagon
is 'Mom'. So here we have a Mom surrounded by 7 Dads. This type of symmetry
is typical for regular polygons. The positions and sizes of the smaller
progeny are also generic. These represent 'resonances' of the mapping τ.
The major resonances are clearly the Dads whose centers are period 7.
In between
these larger resonances, the orbits can be very complex, but all orbits in
this region are bounded by the ring of Dads. Outside of this ring, there are
endless concentric rings of Dads with dynamics which mimic this region. So
for regular polygons, it is sufficient to study the dynamics of this inner
'star' region. By symmetry we can also restrict our study to one of the
MomDad pairs with the corresponding progeny. Below is the canonical first
generation family for N = 7. We call it the First Family.
The parameters of this family can be derived independently of the Tangent map
because they depend only on linear combinations of the vertices of Mom. There
is much evidence that the structures seen here are actually a natural
extension of the geometry of the polygon. The regular heptagon has three
‘stepregions’ and each of these regions has a ‘canonical’ occupant. These
occupants are ‘resonances’ of the mapping τ. They are regular 14gons
and their names are S1, S2 and S3. The names come from their
‘stepsequences’. The step sequence of an orbit is the number of vertices
advanced on each iteration. The orbit of the center of S1 is period 7 and
constant 1step so the step sequence is {1,1,1,1,1,1,1} which we shorten to
just {1}. The remaining points in S1 have the same step sequence but they
need to make two revolutions around Mom, so they are period 14. The situation
is the same for S2. The center is period 7 with step sequence {2} and the
remaining points have period 14. Dad is also known as S3 because his orbit
around Mom is constant step3. There are no step4 orbits in the inner star
region, but they do exist outside the ring of Dads. Dad has 6 step regions but just the
first 5 are shown above. The 6^{th} region corresponds to another
symmetric Dad off the screen on the right. The S2 region is shared by Dad and
Mom. It should make sense that a step2 orbit of Mom would correspond to a
step4 orbit around Dad, because Dad is composed of two copies of Mom. All of
Dad’s oddstep regions have canonical occupants which are 7gons (female) and
his even regions have 14gons (male). We will sometimes use indices for the
step sequences and refer to S1 and S2 as S[1] and S[2]. There is a
‘conjugacy’ for prime regular polygons which says that the local dynamics are
‘unchanged’ if the origin is shifted from Mom to Dad. This is illustrated on
the left below where the origin is unmarked. If we put Dad at the origin, Mom
is now 5step relative to Dad, but the progeny have step sequences which are
essentially unchanged. To illustrate this, on the right is the orbit of DS3
around Mom. Her period is 14 and the step sequence is constant {32}.(The
points in Ngons such as DS3 all map together, so the center has the same
period as the remaining points.) Note that she visits Dad on every third
iteration so she is step3 relative to Dad. Her orbit ‘unfolds’ from a {3,2}
to {3} as we pass from Mom’s world to Dad’s. In this process the 7 Dad’s act
as one. Algebraically, this conjugacy is a reflection of the fact that the cyclotomic
polynomials Φ_{7}(X) and Φ_{14}(X)
are identical. This is true for all
regular Ngons with N odd. For a regular
polygon with N sides there are Floor[N/2] scales which determine the position
and size of the progeny. See FirstFamilies.
Each
generation provides the seeds for a new generation. M[1] fits symmetrically
on an edge of D[0] and her relationship with D[1] is an exact scaled copy of
the M[0] and D[0] relationship. This determines Scale[3], which is also
called GenScale[7]. This provides the potential for selfsimilarity  but the
regular heptagon does not support perfect selfsimilarity. The only known
regular polygons with perfect generation selfsimilarity are the pentagon and
octagon. The heptagon appears to support a 'mod2' selfsimilarity where the even
and odd generations are selfsimilar to each other. We examine the heptagon
in much more detail in N7Summary,
but note that 7 is a 4k+3 prime while 5 is a 4k+1 prime. Our numerical
evidence supports the following: 4k+1
Conjecture:
Supppose M is a regular Ngon with N odd centered at the origin with a vertex
at {0,1}. Define: (i) GenScale[N] = (1Cos[Pi/N])/Cos[Pi/N] (*this is how generations scale under
τ *) (ii) GenStar[N] ={Cot[Pi/N]*(1+Cos[Pi/N]),
Cos[Pi/N]} (*the point of convergence
*) Suppose N is prime of the form N =
4k+1 for k a positive integer. Then there will be infinite sequences of
regular Ngons M[j] (the Mom's) and regular 2Ngons D[j] (the Dads) converging
to GenStar. M[j] will have radius r[M[j]] = GenScale ^{j} and D[j]
will have height h[D[j]] = (1 + GenScale)GenScale ^{j} for j a non
negative integer. The center of M[k]
is (1
r[M[j])GenStar and center of D[j] = (1+ h[Dad[j]])GenStar. The periods of
these centers have ratios which approach N+1.
We will call
polygons of this form '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 Dads and Moms along with their families. The
even generations appear to satisfy the N + 1 rule. See the table on the side
bar. Starting with N = 11, the 4k+3 prime polygons appear to have no
canonical family structure past Mom[2] there is not even a Dad[2]. This
makes for very complex dynamics. N = 13
and N = 17 are 'twin' 4k+1 prime polygons and both appear to have the
expected strings of Dads and Moms. The corresponding families show signs
of even and odd alternation, starting
with generation 2. This is similar to N = 7, but the first generation does
not reappear. This generation selfsimilarity does not imply that the
dynamics elsewhere are 'wellbehaved'. 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 Dad, 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 Dads is constant step8 and outside this ring, the
steps are only 8 and 9. There are endless rings of Dads at equal intervals.
Three of these rings can be seen on the left below. The center to center
spacing is twice the distance from the origin to Dad's center, so it is
2*cS[8][[1]] ≈ 21.58 (where
[[1]] signifies the first coordinate). These rings of Dads have step
sequences which increment by {89}, so the first ring is {8}, then {889}, then
{88989}, yielding a limiting sequence of {89} and limiting winding number of
1/2. In between the
rings are invariant regions which have dynamics ‘identical’ to the star
region (after rotations are filtered out). That means that the two digits
{8,9} code the same information as the 8 digits of the inner star. Each
invariant region contains two copies of each annulus from the inner star.
Shown below in magenta is the second inner star annulus and its two 'clones'
in the region between Ring1 and Ring 2. This region is known collectively as
the Ring 1 region. There is one canonical ‘Mom’ and hence one S[1] center in
each such region and the P2 projection in the 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 ‘Dads’ 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 posthumous message written on
Richard Feynman’s blackboard at Cal Tech was “What I
cannot create I do not understand”. 

Summary of Results Below is a
summary of dynamics for various polygons. There are more detailed summaries
in Pdf format which can be accessed via the Site Map or the PDF folder.The
table below sometimes refers to the 'winding
number', of an orbit. This is also called the
rotation number or the 'twist'. For a given orbit, it measures the mean
rotation around Mom on a scale from 0 to 1, with 1 being a full rotation, so
a canonical 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 Moms and hexagon Dads. • Ring k has
3k Dads and 12k Moms and these are also the periods if we plot centers of the
Dads. Shown here are the 12 Mom's in ring 1. They map to each other in the
order given. • Step
sequences of Dads are (1),(211),(21211),etc with limiting
sequence (21) so ω→.5 • Nonregular
triangles have similar structure. 
N = 4 Regular 

• 'web'
structure is similar to the triangle  but with no distinction between Moms
and Dads. • Ring k has
4k Dads and these are also the periods. There is no period doubling. Shown
here is the second ring with period 8 and step sequence (2,1) • Step sequences
are (1), (21),(221), etc with limiting sequence (2) and ω→.5 • All
trapezoids have similar structure. 
N = 4 (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 p =
{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 Dads
have step sequences (2), (322), (32322),..,→(32) with ω→.5 
N = 6 Regular 

• Domain
structure is identical to N = 3 with any hexagon as Mom and the adjacent
triangle as S1. • As with
every 'twiceodd' regular Ngon, the canonical Dad orbits decompose into two
groups with N/2 in each group. So the first ring of Dads is 6 hexagons  but
linked together as 3 and 3. • kth ring had
6k hexagons and odd rings have decomposition and period doubling. • Dad center
periods are 3k for odd rings and 6k for even. • Step
sequences of Dads are (2),(32),(332),..→(3) 
N = 7 Regular 
Generation 1(with rightside Dad) 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 Mom[k+2]/Mom[k]→200 which factors as 8 and 25 for transitions
from odd to even and back. The value
of 8 matches the N+1 rule for 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 Dads
have step sequence (3), (334),
(33434),..→(34) with limiting ω = .5 
N = 8 Regular 

•Only octagons
– no period doubling •Periods
of Dad[k]/Dad[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] =
Dad[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 tiny rectangle around the 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. •Moms and Dads
exist on all scales and ratio of periods Mom[k]/Mom[k1]→10 (but not
dense). •Second
generation is dominated by 'Portal Moms' similar to those of N = 7. In
between these Portal Moms are regions with small scale chaos. One of these regions
is shown here. •The chaotic
region surrounding the S2[3] bud is called the Small Hadron Collider. The gap
between the central S2[3] bud and the three octagons is determined by a
sequence of (virtual) buds of S2[3] so it is 2r[GenScale[9]^{0} +
GenScale[9]^{1} + ...] where r = rDad·GenScale[9]^{4}/Scale[1]
is the radius of the first bud. (r ≈.000077) 
N = 10 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 Dads form two groups of 5 and the
10 S2’s form two groups of 5. This is typical for ‘twiceodds’. • The
decomposition of the Dads persists for odd rings a has no effect on the outer
star region, but the decomposition of the S2’s creates two invariant inner
star regions – one of which is shown here. Together they define the inner
star. The 10 pentagon ‘Moms’ patrol the border between the inner and outer
stars. 
N = 11 Regular 

• The second
4k+ 3 prime Ngon • Normal first
generation but no evidence of Dads past Dad[1] or Moms past Mom[2]. • Second
generation shown here has some small Mom[2]’s on edges of Dad[1], but no
Dad[2]’s. Mom[1] is almost devoid of
canonical buds. •Dad[1] and
most family members are surrounded by ‘halos’ of complex dynamics as the
normal 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 perfect fractal structure with GenScale[12]
scaling. •Ratio of periods of Dad[k]/Dad[k1] →27 so the
fractal dimension is Ln[27]/Ln[1/GenScale[12]] ≈1.251 •The sixsided
S2’s are determined by triplets of virtual Dad[1]’s, as shown here. •S4 is
canonical with buds of size S1. •S3 is
nonregular octagon with center at {0, } • S1=Mom[1] is
surrounded by a ‘halo’ which is period 24 vs. period 12 for Mom[1] (no
doubling). This halo is unique among all Ngons studied. 
N = 13 Regular 
Second Generation 
•The second
‘supersymmetric’ prime polygon so at GenStar, ratio of periods of
Dad[k+1]/Dad[k] → 14, and same for Moms. Ratios alternate high and low,
so there is some evenodd differentiation. •Dynamics
around Mom[1] are characterized by dense halo of noncanonical buds. There are protected pockets at GenStar and
under Dad[1] for 3rd generation. 
N= 14 Woven 

• A woven
Ngon consists of a canonical Mom at
radius 1 and a secondary Mom at radius between h0 (height of Mom) and
1 + GenScale[N]. This covers the full range of convex proportions. • Show here is
index .91 for N = 7 •Normal ring
of 14 Dads is now subdivided  the orbit of the 7 large Dads 'sees' only the canonical
N = 7 Mom and the 7 small Dads 'see'
only the secondary Mom. (For index 1 this is the canonical N = 14
case.) • Star region
is no longer invariant. • Rings of
Dads undergo periodic oscillations. In the first cycle the secondary Dads
grow and become dominant. The first Big Crunch at ring 11 sees the canonical
Dads almost disappear. Interring spacing (and periods) are same as N = 7, so
ring 11 is at radial distance of about 91.6. (The exact parameters of the
Crunches are easy to calculate but they do not generally correspond to ring
centers.) • Second Big
Crunch at about ring 22 involves Moms.
• Third Big
Crunch at ring 32 is a close repetition of the first so complete cycle is
about 22 rings. • Dynamics are
very complex. Many orbits diverge rapidly at first and then fluctuate widely
in distance. The plot show here is the first 900 million points for an inner
star point. The obit faltered at ring 26 which has a very small dynamical
gap. Every ring is unique. Most local points are trapped by rotations, but
small tunnels do exist. These are reminiscent of the gravitational assist
channels which exist throughout the solar system. 

Level 400 web: 
• 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. • The web
shows that there are 'rings' of period
doubling orbits surrounding Mom. The
large inner ring is the period 3 ring, The next two rings have centers with
periods 7 and 9. • The region
indicated by the arrow has period 154 which means it spirals out and returns,
but unlike the Penrose Kite, these orbits must be bounded and periodic. 
N = 281 Regular 

• As N
increases, the star region grows while the scale shrinks. GenScale[281]≈.000062
so Mom[1] is microscopic and her local dynamics have little or no effect on
the global dynamics which are dominated by simple rotations about Mom – like
a twist map on the unit circle with minimal perturbations. The amount of
twist increases ‘smoothly’ with the distance from the origin. Dad is S[140]
with maximal twist (for the inner star ) at ω = 140/281. • Shown on the
top left is the inner ring which appears to be invariant. Its largest
occupant is S[93]. •Dad is
surrounded by the ‘canonical’ 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 Dad
bisects DS[140]. If N was 4k+3, the shepherd would be an oddstep Helen. The
general formula for the shepherd is DS[Floor[N/2]]. 
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