Polynomials, symmetry, and dynamics: An undertaking in aesthetics
Scott Crass 
AbstractTo solve a polynomial you need a means of breaking its symmetry. In the case of a generic equation of degree n, this is the symmetric group S_{n}. If we extract the square root of the polynomial's discriminant, the group reduces to the alternating group A_{n}. An iterative algorithm for solving the equation has two ingredients:
This paper discusses these two aspects in the cases of the fifth and sixth degree equations. Motivating the project is a desire to develop an algorithm with especially elegant qualities. 
One of the basic objects of mathematical study is the polynomial in one variable: an expression made up of arithmetic combinations of numbers (coefficients) and an unknown quantity (the variable). For instance, the expression

Mathematicians have produced a long history of developing methods for solving polynomial equations: finding numbers that make the polynomial take on the value zero when they replace the variable. In the example above, the numbers 1 and 2 solve the equation

A polynomial has as many solutionsalso called rootsas its degree, provided that you use complex numbers and count them properly.
There is a correspondence between a polynomial and its rootsthe roots
determine the polynomial. If you know the roots, then, essentially, you
know the polynomial. So, we can think of a polynomial in a geometric way.
In our example, the two points (1,2) and (2,1) in a 2dimensional space
of ordered pairs of numbers correspond to the same polynomial. We
can switch the coordinates of either of these points and get a different
point but the same polynomial. In this way, every polynomial has symmetry;
if you increase the degree, the dimension of the space of roots and the
amount of symmetry also increase.
The set of points described by an ordered collection of two complex numbers (x,y) is called C^{2} or 2dimensional complex space. By treating the lines through the point (0,0) as points, this space projects to a 1dimensional space CP^{1} (called complex projective 1space or the complex projective line). The same sort of structure appears in a complex space of any dimension. Thus, the 3dimensional C^{3} projects to the 2dimensional space CP^{2}, etc.
If we restrict our attention to points whose coordinates are
real
numbers, we get real projective spaces RP^{1},
RP^{2},
etc. Since it takes two real numbers to specify a complex number (for example,
3 + 2 i where i is a square root of 1), a complex space has two
times the number of "real dimensions" as the corresponding real space.
So, the complex space CP^{3} has six real dimensions (three
complex dimensions) whereas, the real space RP^{3} has three
real dimensions.
Given a number n of things, there are n! = n ·(n1) ·¼·3 ·2 ways of arranging them. A means of turning one arrangement into another is a permutation of the objects. The set of all permutations of n things forms an object with algebraic structure called a groupspecifically, the symmetric group S_{n}. A polynomial of degree n typically has S_{n} symmetrythe basic idea is that you can permute the roots in n! different ways without changing the polynomial.
The simplest permutation is to exchange two things and leave the other
things unchanged. We can express every permutation as a succession of such
transpositions.
The permutations that decompose into an even number of transpositions also
form a groupthe alternating group A_{n}. The number of
permutations in A_{n} is half the number in
S_{n}.
When you move a set of objects S according to the structure of a group, you use a group action. In the case of solutions to polynomials of degree five, you can move the points in C^{5} (the set S in this case) that correspond to the roots by permuting their coordinates. For instance, transform the point (1,2,3,4,5) into (2,1,5,3,4) by exchanging the first two coordinates and "cycling" the third, fourth, and fifth coordinates. We say that you are "acting on" C^{5} with the symmetric group S_{5}. If you use only the even permutations, you are acting on C^{5} with the alternating group A_{5}.
The orbit under a group action of an element in S is the set of objects in S to which the element moves when you transform it according to all of the group elements. For example, under the symmetric group S_{3},


An operation that takes each point in a space A and associates it with a point in another space B is called a mapping (or map) from A to B. In this discussion, our interest is in maps from a space to itself (when B = A). To illustrate, take a point (x,y) in a 2dimensional space and "send it to" the point each of whose coordinates are the squares of the original:

(2,3) ® (4,9)
(1,i) ® (1,1)
(1+i,2i) ® (2i,34i)
This sort of map is a dynamical system, meaning that you can iterate its behaviorapply it repeatedly. For instance,

When n is less than 5, the symmetric groups S_{n} act faithfully on the complex projective line CP^{1}this space has the structure of a sphere. Corresponding to each actionnot always permutations of coordinatesis a map whose dynamics provides for an algorithmic solution to a given nthdegree equation. By 'dynamics' we mean the process of applying a map repeatedly (iteratively) to points. For instance, Newton's method provides a direct iterative solution to quadratic polynomials, but, due to a lack of symmetry, not to higher degree equations. My interests here are the geometric and dynamical properties of complex projective maps rather than numerical estimates.
The search for elegant complex geometry and dynamics continues into degree five where A_{5} is the appropriate group, since S_{5} fails to act on the sphere. This reduction in a polynomial's symmetry requires the determination of the square root of a certain number associated with a given polynomial: the discriminant. Doyle and McMullen developed an algorithm for solving the quintic at the core of which is a map with icosahedral symmetry. [4] Their solution to the quintic takes place in three iterative steps each of which involves iteration in one complex dimension.
An alternative approach is to work with the threedimensional action of S_{5} that derives from the group of permutations of five variables. Section describes maps that can produce quintic solutions that run as a single iteration in three dimensions.
Moving on to the sixthdegree leads to the twodimensional A_{6} action of the Valentiner group V. In Section , the problem shifts to one of finding a Vsymmetric mapping of CP^{2} from whose attractor one calculates a given sextic's root. Providing the overall framework is the 2dimensional A_{6} analogue of the icosahedron.
For a detailed treatment of the geometry and dynamics involved here as well as how to use both in developing a solutionprocedure to the quintic and sextic see [1], [2], and [3].
The geometric and dynamical descriptions that follow might seem technically
excessive. In providing them, my intention is to indicate some of the rich
interaction between geometry and dynamics. To my mind, this coherence offers
an abstract beauty that deepens the appreciation of the visual images.
Furthermore, while graphical results can be pleasing or even striking,
they also can provide a source of mathematical information. One can scarcely
conceive of extracting such data without these visual tools.



This is a 4dimensional space which projects to 3dimensional complex projective space CP^{3} so that the action of S_{5} on H creates an action of S_{5} on CP^{3}. Let G_{120} denote the group of 120 transformations on CP^{3} corresponding to the permutations of S_{5}.
For many purposes, the most perspicuous geometric description of the
G_{120}
action employs five coordinates that sum to zero. For example, the point
(1,2,3,4,10) belongs to H.


F_{3} = x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3}+x_{5}^{3}
F_{4} = x_{1}^{4}+x_{2}^{4}+x_{3}^{4}+x_{4}^{4}+x_{5}^{4}
F_{5} = x_{1}^{5}+x_{2}^{5}+x_{3}^{5}+x_{4}^{5}+x_{5}^{5}
are S_{5}invariant. A fundamental fact is that
every polynomial that is invariant under the S_{5} action
on
Hpermutation of its variableshas a unique expression in terms
of these four polynomials.


belongs to this set. (Notice that a significant difference between real and complex spaces appears here: although there are infinitely many points in H that satisfy F_{2} = 0, the only such point with real number coordinates is (0,0,0,0,0).) This quadric surfaceQ consists of two families of complex projective lines. (Note that Q is a complex surfaceit has two complex dimensions.) Distinct lines in one family do not intersect while, at each point on Q, exactly one line in each family intersect. (See Figure 1.)
Furthermore, as a set, each family of linescalled a ruling on Qhas the geometry of the icosahedron. In addition, a transformation in G_{120} sends lines in one ruling to either another line in the same ruling or a line in the other ruling. The set of transformations of the former type form a subgroup G_{60} of G_{120} that amounts to the rotational symmetries of the icosahedron when applied to either ruling.
Figure 1: The quadric surface Q
The 3dimensional S_{5} action comes in both real and complex versions. This means that G_{120} acts on Rthe real projective 3space of points whose coordinates are real numbers that sum to zero. Table 1 in Appendix enumerates some special orbits contained in R while Table 2 describes elements of Q that are fixed by certain members of G_{120}. For ease of expression, I refer to special points (or lines, planes, etc.) in terms of the orbit size: "20points" (10lines, 5planes).
Corresponding to each special point a = (a_{1},a_{2},a_{3},a_{4},a_{5}) is the plane of points whose coordinates satisfy the equation



Another noteworthy orbit is that of the five coordinate planes consisting of points one of whose coordinates is zero. The intersection of each such coordinate plane with the quadric Q produces a 1dimensional seta spherewith the geometry of the octahedron. Some data for special twodimensional orbits appear in Table 3.
Finally, a number of special lines appear as intersections of the 5planes
and 10planes. Table 4 summarizes the situation.
Some of the geometry that has dynamical significance shows up in various collections of lines. First, the 10lines whose points have three equal coordinates form a complete graph on the 5points. Figure illustrates this in two ways. The pentagonpentagram figure displays a 5fold symmetry while the double pyramid exhibits the 6fold symmetry of a single 10linerepresented by the polar axis.
Within each of the icosahedral rulings on Q there are three special lineorbits. These correspond to the 12 vertices, 20 facecenters, and 30 edgemidpoints of the icosahedron. Intersections of lines between rulings give special point structures.
Figure 2: Configuration of 10lines and 5points
Figure 3: Special configuration of 40lines and 20points on Q. At a 20point there are two 40linesone in each ruling on the quadric. This pair of lines corresponds to a pair of antipodal vertices of the dodecahedron. Also shown are the 10lines determined by a pair of antipodal 20points.
The trajectory of a point x under a map f is the set
of points obtained by applying f iteratively to x. A point
p
is periodic if its trajectory contains p more than once.
A periodic point a in a space X is attracting when the trajectory
of every point near a gets arbitrarily close to a. The basin
of attraction of a is the set of all points attracted to a.
Also, the attractor of f is the set of all attracting points.
The primary tool to be used in solving the general quintic is a map that associates points in CP^{3} with points in CP^{3} in a way that respects the action of the group of transformations G_{120}. We want to find a G_{120}symmetric map (or simply G_{120}map) with elegant geometry and reliable dynamics; this means that its attractor
f_{1}:(x_{1},x_{2},x_{3},x_{4},x_{5})
®
(x_{1},x_{2},x_{3},x_{4},x_{5})
f_{2}:(x_{1},x_{2},x_{3},x_{4},x_{5})
®
(x_{1}^{2},x_{2}^{2},x_{3}^{2},x_{4}^{2},x_{5}^{2})
f_{3}:(x_{1},x_{2},x_{3},x_{4},x_{5})
®
(x_{1}^{3},x_{2}^{3},x_{3}^{3},x_{4}^{3},x_{5}^{3})
f_{2}:(x_{1},x_{2},x_{3},x_{4},x_{5})
®
(x_{1}^{4},x_{2}^{4},x_{3}^{4},x_{4}^{4},x_{5}^{4})
are G_{120}symmetric. Moreover, by combining these with the invariants

The G_{120}maps have an algebraic structure relative to the G_{120}invariants. This means that for an invariant F_{l} and symmetric map g_{m} of degrees l and m, the product

is a symmetric map of degree l+ m. When looking for a map
in a certain degree with special geometric or dynamical properties, my
approach is to express the entire family of symmetric maps for that degree
and, by selecting from a palette of parameters, locate a subfamily and
eventually a single map with interesting behavior.
The rich geometry of the quadric Q provides an intriguing setting
for dynamical exploration. Are there S_{5}symmetric maps
that send Q to itself? If so, how do they behave on and off Q?
I will describe two species of such maps: one associated with the icosahedron
and the other with the octahedron.
Were a G_{120}map to preserve the rulings on Q, its restriction to either ruling would express itself in terms of the basic maps that are symmetric with respect to the onedimensional icosahedral actionthat is, the rotational symmetries of the icosahedron. Maps of this kind occur in degrees 11, 19, and 29. [4] Consequently, the family of 11maps (maps having degree 11) comes under scrutiny; this family has a palette of 20 parameters.
Choosing parameters to satisfy the demands of the 1dimensional 11map with icosahedral symmetry, we obtain a family of rulingpreserving maps.
Restricted to a ruling, the dynamics of each g_{11} is wellunderstood.
The 20lines (under G_{120}) are exchanged in pairs. (Recall
that 20lines in Q correspond to dodecahedral vertices in the respective
rulings.) Moreover, almost every line in the ruling belongs to the basin
of one of the ten antipodal pairs of the superattracting set of 20lines.
(See Figure 6 in Appendix.) Thus, for nearly every
point x on Q, there is an "antipodal" pair of intersections
between 20lines in different rulings which the trajectory of x
approaches.
Since the orbit of the five coordinate planes has fundamental geometric significance, a map that preserves these sets might exhibit interesting dynamics. Arranging for this requires four of the twenty parameters of the family f_{11}.
The intersection of a 5plane and the quadric Q is a conic (sphere) with the S_{4} symmetry of the octahedron. One of the special maps for the octahedral action on CP^{1} is a degree5 map that attracts almost every point to the eight facecenters vertices of the dual cube. Geometrically, the map stretches each face F of the cube symmetrically over the five faces complementary to the face opposite F. As a face stretches, it makes a halfturn so that the vertices and edges land on their antipodes. (See Figure 6 in Appendix.) Under G_{120}, antipodal pairs of octahedral facecenters are pairs of 20points.
The idea is to find a reliable map that sends Q to itself and behaves like the special 5map on each of the octahedral conics. Such a map would attract points on Q to the 20points. We also want the 20points to be attracting off Q. In degree five there are too few parameters for the purpose. However, the 11maps provide enough freedom to arrange for elegant geometry.
Each of the 10lines two of whose coordinates are zero contains a pair of antipodal 20points. A map that
would act as a "superattracting pipe" to the quadric. Selecting the remaining parameters accordingly reveals a map h_{11} with these properties.
It happens that h_{11} also preserves Rthe
S_{5}symmetric
space of points whose coordinates are real numbersas well as the 2dimensional
intersections of R with 1) the five coordinate planes and 2) the
ten planes whose points have two equal coordinates. In the first case there
are four intersections of the 2dimensional space with the superattracting
10lines while in the second there is a single such intersection. Each
intersection is a real projective line as well as an "equatorial slice"
of the associated complex projective linea sphere. When restricted to
such a slice, h_{11} acts chaotically, meaning that the
trajectory of most of the circle's points gets arbitrarily close to every
point on the circle. A basin portrait for the 5plane reveals no basins
other than those of the four chaotically attracting 10lines. (See Figure
8.) The dynamics on the "real part" of the 10plane shows, in addition
to the chaotic lineattractor, three additional basins at 30points. (See
Figure 8.)
In the configuration of 10lines each 5point lies at the intersection of four lines. (See Section 2.4.) Moreover, these are the only intersections of 10lines. To take advantage of this structure, a map could have superattracting pipes along the 10lines and, thereby, have 3dimensional basins of attraction at the 5points.
The family of 6maps has six free parameters. I takes four parameters to obtain maps for which the 10lines are superattracting in the "offline" directions. For the remaining two, we get a map f_{6} whose restriction to a 10line has the simple form

when expressed so that the pair of 5points on the respective 10line are 0 and ¥. Recalling that a 10line is a sphere, such a map attracts all points in the northern hemisphere to the 5point at ¥ (the north pole) and attracts all points in the southern hemisphere to the 5point at 0 (the south pole). The equatorial circlea real projective linemaps to itself chaotically.
Of necessity, f_{6} preserves each S_{3}symmetric 10plane L whose points have two equal coordinates. Furthermore, f_{6} preserves Rthe S_{5}symmetric RP^{3}. We can get a picture of the map's restricted dynamics by plotting basins of attraction on the intersection of L and R. (See Figures 9 and 10 in Appendix.) The plot shows attraction to three 5points and one 10point. However, the 10point lies on the "equator" of a 10line. Here, f_{6} repels in the offplane direction. Thus, the basin of a 10point is 2dimensional. No other attracting sets appear.
A 15line contains one 5point, one 15point, and two 10points. When restricted to such line, these three points are attracting. (Figure 11 displays portraits.)
Finally, f_{6} preserves a certain 3dimensional real projective space that is associated with the group of transformations that fix a 5point. This RP^{3} intersects two 10planes in a space which you can think of as a sphere; this sphere has the symmetry of a double rectangular pyramid. In addition to the 5point this RP^{2} contains three 10points as well as the RP^{1} through two of the 10points. Since this line is an equatorial slice through a 10line where the map looks like

Inside the alternating group A_{6} are twelve versions of the alternating group A_{5}. These twelve subgroups decompose into two systems of six:
We can apply the elements of the group A_{6} to these A_{5} subgroups. Doing so permutes each of the two systems individually. If we use only the members of one of the A_{5} subgroups, that subgroup remains unchanged as a whole. As for the five companion subgroups in its system, they move according to the way the icosahedron's rotational symmetries act on the five cubes circumscribed by the icosahedron. Meanwhile, the other system of six A_{5} subgroups undergo the permutations of the six pairs of antipodal vertices. As a consequence, the intersection of two A_{5} subgroups in the same system has the same structure as the group A_{4}the tetrahedral rotationswhile two in different systems specify the dihedral group D_{5}the rotational symmetries of a double pentagonal pyramid.
In the late nineteenth century, Valentiner discovered a groupcall it
Vof
360 transformations of 2dimensional complex projective space that has
the same structure as the group of permutations A_{6}. To
solve the sextic equation, we must find a map on CP^{2}
that is symmetric with respect to V.
The A_{5} subgroups of A_{6} correspond to subgroups of V. Each of the A_{5} subgroups preserves a respective 1dimensional conica spherewhich thereby has the geometry of the icosahedron. The group V permutes these two sets of six icosahedra in the same way as A_{6} permutes its two systems of A_{5} subgroups.
Some of the special icosahedral points on a conic occur at its intersections with the other 11 conics. There are two cases.
As for other special orbits, each of the 45 transpositions in A_{6} corresponds to a transformation T in V that fixes every point on a line associated with T. In addition, T fixes a point that is not on its associated line. These give Vorbits of 45 lines and 45 points. An Vmap typically preserves each of these lines and points; however, as we will see (Section 3.4), something quite different can occur. The typical points on a 45line provide Vorbits of size 180. Other special orbits occur at the intersections of the 45lines:
Figure 4: The triangle of one 36point and two 72points.
The onedimensional icosahedral group I_{60} acts on two sets of five tetrahedra each of which corresponds to a quadruple of facecenters on the icosahedron. However, no element of the group sends the tetrahedra of one set to those of the other. Such an exchange occurs by means of orientationreversing transformations. Some of these are reflections through the 15 great circles of reflective icosahedral symmetry; the remaining 45 are the compositions of odd numbers of these 15 basic reflectionsfor example, the map that sends a point to its antipode. Extending the orientationpreserving group I_{60} by such an orientationreversing transformation produces the group of all 120 symmetries of the icosahedron.
The two systems of conics are the Valentiner analogues of the two sets of tetrahedra. There are orientationreversing transformations of CP^{2} that exchange the systems of conics. By taking all combinations of such a transformation with the elements in the group V, we get a new group consisting of 720 symmetries of the Valentiner structure. In analogy to the 15 great circle reflections that produce all 120 symmetries of the icosahedron, there are 36 orientationreversing transformations that combine to make the 720 Valentiner symmetries. Each of these 36 transformations fixes every point of an associated realprojective plane. (Figure 5 illustrates a geometric construction for these planes.) These 36 planes stand in analogy to the 15 great circlesreal projective linesof icosahedral reflections. A map that respects all 720 of the Valentiner symmetries must send each of these planes to itself. This circumstance allows us to make pictures of such a map's dynamical behavior.
Figure 5: A geometric interpretation of transformations that exchange conics in different systemsindicated by the pair of points.
Given a polynomial that is unchanged by the Valentiner group, we express it as a combination of four basic invariants. We can obtain any Vmap
from combining these invariants and their derivatives (in the sense of calculus). Again, the idea is to employ a palette of parameters in designing a geometrically elegant map.
In degree 16 we find the map of least degree with Valentiner symmetry. This map has the property that it smashes a 45line down to its associated 45point. Furthermore, it "blowsup" a 45point to its companion 45line. This means that the map spreads a region near the 45point over the 45line. The basin portrait (Figure 13) fails to reveal the geometric elegance that we seek.
Associated with the icosahedron is a degree19 map that takes each of the 20 faces and stretches it around the icosahedron omitting the opposite face. When iterated, this map attracts almost any point of the icosahedron to one of the six pairs of antipodal vertices. (See Figure 14.)
In the higher dimensional case of the Valentiner group, there is a 19map h_{19} that sends each of the 12 icosahedral conics to itself. This means that when we restrict our attention to only one conic, h_{19} is the map described above. So, we understand much about its dynamics there. Recall that the vertices of the icosahedral conics make up the 72 point Vorbit. It happens that in the direction away from the conic, these points are also attracting.
Moreover, h_{19} has the additional symmetries of the transformations that exchange the systems of conics. Therefore, it preserves each of the 36 real projective planes associated with these transformations. These spaces look very much like familiar twodimensional planes. A portrait of the map's dynamical behavior on such a plane appears in Figures 19 through 17.
For each system of conics, but not for both, there is an orientationreversing map that preserves the six conics individually. When restricted to one of the conics, the map's geometry is to stretch each dodecahedral face onto its complementthe sphere minus the facewhile fixing the vertices and edges and sending the facecenter to its antipode. You can imagine pushing the face inside the dodecahedron spreading it out symmetrically onto the other 11 faces. This defines an 11map expressed in complex conjugated coordinates. When iterated, the 12 fixedpoints at the vertices attract almost all points on the sphere. Since two conics in one system intersect transversely at the vertices (60points), these points are attracting in all directions. I plan to study this map and, in a future paper, give a detailed description of its behavior.


















(0,0,1,w_{3},w_{3}^{2})
(0,0,1,w_{3}^{2},w_{3}) 
antipodal pair of eight octahedral facecenters on intersections of
Q
and
a coordinate plane ({x_{1} = 0}, etc.)
w_{3} = e^{2 p i/3} 

(0,1,i,1,i)
(0,1,i,1,i) 
antipodal pair of six octahedral vertices on intersections of Q and a coordinate plane ({x_{1} = 0}, etc.) 

(0,1,1,g,g^{}) 
antipodal pair of 12octahedral edgemidpoints on intersections of Q
and a coordinate plane ({x_{1} = 0}, etc.)
g = 1 + Ö2 i 
























The basin plots that follow are productions of the program
Dynamics
and Dynamics 2 that ran respectively on a Silicon Graphics Indigo2
and a Dell Dimension XPS with a Pentium II processor. Its BA and BAS routines
produced the images. (See the manuals [5] and [6].)
Each procedure divides the screen into a grid of cells and then colors
each cell according to which attracting point its trajectory approaches.
If it finds no such attractor after 60 iterations, the cell is black. The
BA algorithm finds the attractor whereas BAS requires the user to specify
a candidate attracting set of points. Each portrait exhibits the highest
resolution availablea 720 ×720 grid.
Each of the ten pairs of antipodal dodecahedral verticesblack dotsis a period2 superattractor. Their basins fill up the sphere. (Bear in mind that points in the space of this plot correspond to lines in either ruling on the quadric surface Q.)
This plot indicates the behavior of h_{11} restricted to an S_{4}symmetric conicthe intersection of a coordinate plane and the quadric Q. The four pairs of antipodal vertices of the cube are period2 superattracting 20points whose basins fill up the conic.
These show the behavior of the octahedral map h_{11} on a 15line and a 30line respectively.
In the former case, the superattracting points at 0 and ¥ are a pair of 30points on Q that h_{11} exchanges. A pair of fixed 10points accounts for the remaining two basins. At each of these attracting points, the map repels in at least one direction away from the line.
On the 30line, the superattracting points at 0 and ¥ are a pair of 60points (antipodal edgemidpoints) on an octahedral conic; the map exchanges the two points. The remaining two basins belong to a pair of 20points on R. As before, the map repels in at least one direction away from the line at each of these attracting points.
We see the restriction of h_{11} to a plane with S_{4} symmetry. The attractor consists of the four lines formed by intersections of R, a coordinate plane, and four of the 10lines with two zero coordinates. The six intersections of these 10lines occur at 10points with three zero coordinates. (See Table 4.) In the picture, two of these intersections occur on the line at infinity. The pictured "lines" are the images of small circles centered along the edges of the inner square. This graphical technique specifically exploits the chaotic and attracting behavior of h_{11} along each line.
Here, h_{11} is restricted to a plane with S_{3} symmetry. The attracting line is the intersection of R, a 10plane with two zero coordinates and the 10line at infinitythe light gray basin. The three "attracting" 30pointsthey are blowing upare the vertices of an equilateral triangle.
We see the restriction to the plane determined by the intersection of R and a 10plane with two equal coordinates. Since this plane is S_{3}symmetric, we select the coordinates so that the three 5points are vertices of an equilateral triangle that is symmetrical about the center. Three of the superattracting pipes contain this triangle.
The map sends the indicated circle to the ``triangle" shown here. This triangle strongly aligns itself with the superattracting pipes through the 5points. The attractor at the center is a 10point. In the direction away from the plane, f_{6} repels at this site along a superattracting pipe. The three "spokes" at basin boundaries are pieces of 15lines each of which passes through a secondary basin that contains a point that maps to the central 10point.
This shows f_{6}'s critical setwhere the map folds the plane oversuperimposed on the basin portrait. The critical contour is a Mathematica plot. The curve crosses itself at the 5points. All but six critical points appear to belong to the basin of either a 5point or the central 10point. The six exceptions lie on the 15lines at basin boundaries. If this is so, then there is no other attracting site.
We see f_{6} restricted to a 15line that maps to itself. The coordinates place the single 5point at the center and the two fixed superattracting 10points at symmetrical locations on each side. At the latter points, the map repels in all directions off the line.
This approximately shows the boxed region of the previous plot.
The space here is the plane where an S_{4}invariant RP^{3} and a 10plane intersect. The intersection of the plane and the associated 10line with three equal coordinates is the central vertical axis. By plotting the trajectory of one of its generic points, this line reveals itself as a chaotic attractor; the plot shows roughly 20,000 iterates. The map also attracts at a 5point and 10point.
Figure 6: Platonic dynamics
Figure 7: Three basins of attraction for h_{11}restricted to a 15line (left) and a 30line (right)
Figure 8: Chaotic attractors for h_{11} on a plane with S_{4} symmetry (left) and a plane with S_{3} symmetry (right)
Figure 9: Four basins of attraction for f_{6} restricted to an plane
Figure 10: Detail of the left cusp of central basins in Figure 9
Figure 11: Three basins of attraction for f_{6} restricted to a 15line (left) and magnified view of the boxed region (right)
Figure 12: Chaotic attractor for f_{6} on a plane
When "restricted" to a 45line, the degree16 map "mostly" converges to one of the 45points on the line. Does this occur for almost every point on the line? Do the black specks consist of points whose trajectories fail to converge to one of the four attracting 45points that lie in the large central basins? The BAS algorithm checked 60 iterates before concluding that a trajectory did not converge.
The degree19 map with icosahedral symmetry attracts almost all points in the sphere to an antipodal pair of vertices. Each of the six colors corresponds to such a pair and the three large basins each contain a vertex. For the conicpreserving h_{19}, the basin plot on each conic looks like this one. Moreover, each basin is the 1dimensional intersection of a 2dimensional basin.
The image shows the behavior of h_{19} on one of the 36 real projective planes determined by the basic conicexchanging transformations. Each large radial basin contains one of the 72points and come in pairs as do the period2 attractors. Notice the repelling behavior along the 45lines and particularly at their intersection in the 36point at the center.
Shown here are trajectories, colored according to their destinations, of the points in the vertical strip on the left. Many of the points in the strip map inside the hazy pentagon whose vertices lie on the 45linesthe inner "star" is nearly filled. Circumscribing this pentagon is the outer starlike piece of the critical set shown in Figure 16. Furthermore, the pentagon seems to be the image of the inner pentagonal oval. Accordingly, the map folds the plane along the pentagon's edges just outside of which the 72points make their presence seen in the dense streaks. Compare this pattern of streaks to that of the 72lines shown in Figure 16. Figure 17 illustrates this local "squeezing" at a 72point.
The lines tangent to a conic at the 72points form an orbit of 72 lines. For any one of the 36 planes of reflection associated with transformations that exchange the systems of conicscall such a plane R, there are five pairs of the 72lines that intersect R in lines. The picture shows their configuration in the plane of Figure 15. Each pair receives a single color according to the scheme of the basin plot. A given pair of lines passes through the associated pair of 72points; they intersect in the corresponding fixed and repelling 36point.
Here is a Mathematica contour plot on R of the set of points in R that satisfy the equation F_{6} = 0. This curve is the critical set of h_{19}where folding over takes place. The superattracting 72points are the inflection points.
The green horizontal line corresponds to the intersection of the reflection plane R and the 36line passing through the pair of green 72points from the basin plot. The dark curve is where h_{19} sends the line. Sitting at the sharp cusps are the 72points which the map exchanges. As indicated in the caption to Figure , the line folds over at these critical points. The upper two sharp turns correspond to where the line passes through the yellow and red "streaks" that approximate 72lines.
Figure 13: Dynamics of the 16map
Figure 14: Icosahedral dynamics of the 19map
Figure 15: Dynamics of h_{19} on a plane with 10fold symmetry
Figure 16: Configuration of 72lines and critical set of h_{19}
Figure 17: Image of a 36line under h_{19}