Session 3
Introduction to Polynomiography
3. 1 What is POLYNOMIOGRAPHY?
The importance of complex numbers is that everything that was said earlier about
quadratics, roots, fixed points, fixed point iterations, Newton's Method, and convergence,
extends to the domain of the complex numbers and we suddenly get a much more
beautiful and colorful world with many visual possibilities.
The term Polynomiography is a simple combination of polynomial and the sufffix graphy. Throughout the history of mankind solving a polynomial equation has resulted
in numerous significant discoveries, dating back to the Babylonians and the Summerians.
The greatest mathematicians of each generation has investigated the problem for one
mathematical reason or another. It has even inspired the discovery and study of the queen
of computational algorithms, Newton’s Method, which in turn is the foundation behind
iterative methods and their visualization leading to fractal behavior, a term coined by
Mandelbrot. In fact according to historical experts the very ideas of abstract thinking and
the use of mathematical notation are largely due to the study of polynomial equations.
Solving for the unknown is a problem faced by the layman on a daily basis, for instance
in figuring out percentages which essentially amount to the solution of a linear equation.
The computer technology and sophisticated mathematical algorithm has now resulted in a
new appreciation for solving polynomial equations, going far beyond the routine fractal
images.
POLYNOMIOGRAPHY can produce spectacular and diverse images, called
polynomiographs, that not only can be appreciated as art, but induce striking appreciation
of the connections between creativity in art and the intrinsic beauty of mathematics.
Sample images can be found at www.polynomiography.com
A polynomial can be entered into the software either through its coefficients which is
normally the way we think of a polynomial, or through its roots. For instance, if we want
a polynomial to have two roots  and  , we consider
p ( x)  ( x   )  ( x   )
The POLYNOMIOGRAPHY software also allows you to select the roots by the click of
the computer mouse on the screen, by selecting locations as the roots of the polynomial
equation. We can select these locations as arbitrary points or selected to form a desired
pattern. In the latter case the software subsequently turns the points into a polynomial
equation, then an initial polynomiograph is rendered. In this sense,
POLYNOMIOGRAPHY turns the problem of root-finding upside down and into a
problem where the user can select the polynomial which will then undergo visualization
through POLYNOMIOGRAPHY.
Furthermore, the role of algorithms become more pronounced in that it is one of the main
sources of unveiling or creating beauty from such a simple looking task as solving a
polynomial equation. While a dictionary definition of a polynomial equation describes it
as a linear combination of an internal power of a variable such as x , a modern view of
solving a polynomial equation, as well as an informal description of Polynomiography
that brings it to the layman is the following from the book, ``Polynomial Root-Finding
and Polynomiography,” 2008;
Solving a polynomial equation could be considered as a game of hide-and-seek with a
bunch of tiny dots on a painting canvas. We hide the dots behind a polynomial equation,
we then seek them using a formula or an algorithm. Polynomiography is the algorithmic
visualization of the process of searching for the dots, and painting the canvas along the
way.
The polynomiograph is produced via iteration functions which can serve as algorithms
for approximation of polynomial roots, using a computer. Polynomiography is somewhat
analogous to photography in which a photographer makes use of camera and its lenses,
together with his/her personal creativity to produce an initial image, a photograph. A
photographer may subsequently alter an initial photograph in order to create or enhance
desired effects. Many parameters come to play to create the final image: the particular
subject, the composition, the angle, the lighting, the settings, the lens, the filters, and
other parameters the photographer must take into account. In Polynomiography the
polynomiographer makes use of the corresponding software and iteration functions which
act as lenses, or painting brushes together with his/her personal creativity to produce an
initial polynomiograph. The polynomiographer then may go through the same kind of
decision making as the photographer: changing scale, isolating parts of the image,
enlarging or reducing, adjusting values and color until the polynomiograph is resolved
into a visually satisfying entity. Like a photographer, a polynomiographer can learn to
create images that are aesthetically beautiful and individual, with or without the
knowledge of mathematics or art. Like an artist and a painter, a polynomiographer can be
creative in coloration and composition of images. Like a camera, or a painting brush,
POLYNOMIOGRAPHY software can be made simple enough that even a child could
learn to operate.
3. 2 How to use the POLYNOMIOGRAPHY software?
Step 1. Open the software and you get a screen like the one shown below.
Step 2. Enter the polynomial you want to evaluate in the polynomial window.
Polynomiograph uses z as a variable instead of x. So to solve 2x2 - 3x -10 = 0, enter
2x^2 -3x -10 in the POLYNOMIAL window.
Step 3. Press the green play key
and you get a plot of the solutions of
2
2x – 3x -10 = 0 in the complex plane.
Step 4. Go to VIEW/ axis / standard and a set of axes will be added to the plot.
Step 5. Go to VIEW / roots and the two real solutions will be highlighted.
Step 5. To change the colors used go to COLORING. In the plot below COLORING
/color pallet / rainbow has been selected
If COLORING /Color Palette II- / Red-Blue, you will get
3.3 How Polynomiography plots Basins of Attraction
When we used Newton’s Method and Excel to calculate the square root of 16 we
calculated it with an accuracy of 4 decimal places using the function as shown in Table
3.3.1
x 2 - 16 x 2 + 16
N f (x) = x =
2x
2x
Notice that the number of iterations it took would be different for different initial guesses.
Table 3.3.1. Different results using Newton’s Method to find the square root of 16.
Table 3.3.1 shows the following
- It takes 4 iterations to get an approximation accurate to 4 places with an initial
guess of -10.
- It takes 3 iterations to get an approximation accurate to 5 places with an initial
guess of -6.
- It takes 4 iterations to get an approximation accurate to 5 places with an initial
guess of 2.
- It takes 3 iterations to get an approximation accurate to 5 places with an initial
guess of 6
You can argue that the basin of attraction for 160.5 using Newton’s Method is the set of
all non-zero real numbers. An interesting question is what elements in the basin of
attraction will converge after a given number of iterations? For example, guesses of -8
and -6 both converge after 4 iterations.
POLYNOMIOGRAPHY will help answer this question. Figure 3.3.1 (a) shows what
POLYNOMIOGRAPHY produced for the polynomial x2 -16. Any guess in the pink
region will produce an orbit that converges after 6 iterations. Figure 3.3.1 (b) shows what
POLYNOMIOGRAPHY produced for the polynomial x2 +16. Any guess in the blue
region will produce an orbit that converges after 4iterations.
Figure 3.3.1 (a) x2 - 16
Figure 3.3.1 (b) x2 + 16
Figure 3.3.2. The Cartesian graph shows that the graph of f(x)= x2 +16 does not have xintercepts so f(x) must have 2 imaginary roots. The POLYNOMIOGRAPHY graph
shows that the imaginary roots are -4i and 4i.
Figure 3.3.2. The Cartesian graph shows that the graph of f(x)= x2 - 16 has x-intercepts
at –(-4,0) and (4,0). The POLYNOMIOGRAPHY graph shows that the real roots are -4
and 4.
Exercise 3.1
Figure 3.3.2 POLYNOMIOGRAPHY plot for x2 +16. A guess in the upper blue region
shows that a guess of (2.58, 4.18) converges after 4 iterations as does any other guess in
the blue region.
Exercise 3.1
Use POLYNOMIOGRAPHY to answer the following
1. Find the solutions and the corresponding Basins of Attraction for the following
quadratic equations.
a. x2 – 3x – 15 = 0
b. x2 – 8x +16 = 0
c. x2 – 3x + 15 = 0
2. A given quadratic equation can have 2 real solutions, 1 repeated real solution or 2
complex solutions. Make up examples of each type, plot their solutions using
POLYNOMIOGRAPHY and write a summary of how the plots differ.
3. Match each Cartesian Plot with the corresponding Polynomiography Plot
Cartesian Plot 1
Polynomiography Plot 1
Cartesian Plot 2
Polynomiography Plot 2
Cartesian Plot 3
Polynomiography Plot 3
Cartesian Plot 4
Polynomiography Plot 4
4. Use POLYNOMIOGRAPHY to find the roots of x4 - 4x3 -14x2 + 85x –
116 = 0.