Polynomiography
Polynomiography: A Computational and Mathematical Perspective
By
Rana Sohail Saleem
MSCS (Networking), MIT, Virtual University of Pakistan
Abstract— Polynomiography is basically the study of art and science of visualization to make approximations for the complex polynomial’s zeros. The study revolves around the pictorial view of polynomials where the image is created and further be used in educational and scientific world. The applications created by polynomiography are very useful in the fields of art, education and science. This paper will make an endeavour to establish the relationship between the computational and mathematical aspects.
Keywords— Polynomiography, polynomials, art, science, mathematics,
I. INTRODUCTION
Before describing the polynomiography let’s discuss the polynomial as it can be defined as a finite collection of points in Euclidean Plane where these points are basically are the roots or zeros. These roots bifurcates the Euclidean Plane into independent territories which are determined by an iteration function. Now polynomiography can be described as the painting of these points in an ordered fashion thus making it an artwork. So polynomiography is the study of art and science of mental picture to make estimates for the complex polynomial’s zeros. The study encompasses the graphic view of polynomials in which the created images are used in the world of education and science. World of polynomiography is made up of two words ‘polynomial’ and ‘graphy’, the meaning is very prominent and that is polynomials are presented with the help of graphs. Actually we require specific algorithms to do the job. Polynomial is an object which is based on the mathematical basic and fundamental class. The multiplication of algebraic expression is much faster through algorithm than before. Here the pixels are involved which may vary from hundred to millions. Today the computer makes it very easy to get the results. Millions of pixels are manipulated on the screen of a computer as in [1].
Polynomiography is the result of infinite number of iteration functions which are established for the purpose of approximation of the polynomial’s root-finding. Here the iteration function means the mapping of the plane into itself.
The paper is organized in sections. In section II, Polynomiography base and its fields will be defined. In section III, relationship of mathematical concepts and polynomiography will be discussed. In Section IV, the discussion will be concluded.
II. POLYNOMIOGRAPHY – THE BASE AND ITs FIELDS
A. The Base
Once we take the polynomial
p(z)= anzn+an−1zn−1 + ….+ a1z + a0;
where n ≥ 2, and the coefficients ai are complex numbers. This problem presents the approximating roots of p(z). Here iteration function is operative which belongs to “Basic Family” as in [2]. Among others, Basic Family has two famous members namely Newton’s iteration function;
and Halley’s iteration function;
In both cases the upper and lower bounds are declared as Um and Lm where m is a natural number and ≥ 2. The numbers are computable in terms of coefficient of p i.e. each root θ can have Lm≤θ ≤ Um.
Polynomiography gives out the visualizations of such situations of Basic Family and its properties.
B. Fields
Polynomiography is useful in three important fields of life which are very famous, details are as under:-
(1) Visual Art: Polynomiography software is very useful; it works like a camera or musical instrument. Through it can draw simple as well as complex patterns of designs which can be comparable with any classic work of human.
Polynomiography is very useful in teaching the basics and professional artistic and mathematical concepts in the classroom at any level. There is no requirement of basic teaching of such software; even a beginner can handle it comfortably. An analogous image to the camera can be easily created digitally by it. It is all possible by using iteration functions by the software.
Polynomiography makes it possible to solve the issue of “reverse root-finding”. Here the roots of any polynomial can be found through iteration function and combination of desired colour scheme can create wonderful designs. Likewise, user can create images by putting the coefficients of the polynomial or zeros locations in the software. In the field of art, polynomiography can be specific in terms of creating images in following way as in [3]:-
Polynomiographer uses one polynomial and produces a variety of images. This is possible because of the use of the variety of iteration functions.
Polynomiographer can turn an ordinary image to a very attractive and beautiful image by using colours and imaginative creativity.
Polynomiographer can combine the art and mathematics by using either polynomials or iteration functions.
Polynomiographer can use the already created polynomiographs and mixing of two or more can be done.
Figure 1 and 2 are examples of polynomiographic software creation.
(2) Education: Polynomiography can be very useful in the field of teaching. It is used to deploy and solve difficult theorems which are totally dealing with the polynomials. It is also helpful in the understanding of algebra and geometry. Figure 3 gives out Fundamental Theorem of Algebra visually as in [1]. It show the polynomiography of nine digit number.
Once the level is raised to higher education, polynomiography software still has the command over calculus, numerical analysis, notion of convergence, limits, iteration functions, fractals, root-finding algorithms etc.
(3) Science: Polynomiography do have importance in the field of science because almost all the science theories are based on polynomials and if we know the roots of a polynomial then we can say that we know the polynomial as well. In science special polynomials like Legendre polynomials as in [4], Chebyshev polynomials as in [5], orthogonal polynomials as in [6] etc were difficult to be understood but polynomiography made it very easy and simple to understand. Figure 4 as in [1] gives some polynomiography for a polynomial arising in physics.
The use of polynomiography software application for numerical equations is very interesting. Here numbers can be encrypted with it. We can take the examples of ID or credit card numbers that can be divided into two dimensional images which can be resembled as a fingerprint. Now different fingerprints can be presented by different numbers. We can take the example of number a 8 a 7 …a 0 which can be identified by the polynomial P(z) = a 8 z 8 +….+ a 1 z + a 0 as in [1].
Finding square root of numbers is another way of handling through polynomiography which is a very interesting and simple task. Here polynomial equations are approximated by square rooting through images as in [7]. Figure 5 shows the graphical view of computing square root of two.
Likewise irrational numbers, complex numbers and iterative methods can be more elaborative and understandable with the help of polynomiography as never before as explained in [7].
B. Basins of Attractions and Voronoi Region of Polynomial Roots
The basins of attraction of a root in relation with iteration function are the regions in the complex plane as shown in figure 6
It is basically the set of initial conditions which leads to long time behaviour approaching the attractor. So the better quality of long time motion of a system could be different. It depends upon the initial conditions as the attractors may be corresponding to periodic or quasiperiodic or chaotic behaviours of different types. In a state plane a region presenting the basin of attraction will be different because it varies from system to system. These attractions can be drawn graphically but not with ease and if at any stage of drawing a minute mistake occurs then whole process would be a waste. Polynomiography made it so easy and simple that software draws all of it once the initial conditions are set in the prescribed regions.
Julia set as defined in [8] of the polynomial roots has the nature of fractal. Images of basins of attractions of Newton's method are very similar to that of some special polynomials. Mathematical analysis of complex iterations can be dealt with polynomiography as in [9].
Voronoi Region of polynomial roots can be defined as a diagram which divides the space into a number of regions. Points in each region are set at initial stage and all points are set in such a way that they are placed closer to each other. These regions are also known as voronoi cells. These diagrams are very helpful in the fields of science, technology and artwork.
There are number algorithms which find out the Voronoi regions by computation like Divide and Conquer, Brute Force, Fortune’s Line Sweep etc. Once computing intersection of half planes the time taken will be O (n 2 log n) and while proving lower bounds require Ω (n log n) and so on.
Figure 7 shows the space divided into twenty regions with twenty points or cells placed in them in a way to be closed to each other. This sort of display will be very easy to depict with the help of polynomiograhy software.
C. Root Sensitivity
The n roots of a polynomial of degree n depend continuously on its coefficients. It can be explained as a polynomial root is a zero of the polynomial function and any non – zero polynomial and its degree has equal roots. The equation polynomial f of degree two will be:-
Polynomial roots are very sensitive to even small changes which are if carried out in their coefficients. We may take the example of
suppose n = 7 then
If we change coefficient of z 6 from −28 to −28:002 then this small change can have a large change in the roots. Some real roots will become complex. This aspect can’t be visualized in advance once solving the equation but polynomiography made it possible to have advance information on this important aspect as in [11]. Figure 8 shows the changes in the roots as the coefficient of z 6 is decreased.
D. Complex Multiplication
Multiplication of complex numbers as in [12] is an important subject and polynomiography made it very easy to understand, we can take an example
Suppose two complex numbers z 1 = (a+bi) and z 2 = (c+di). They can be added as (a+b) + i(c+d), and their product is said to be (ac−bd)+i(ad +bc).
We take another example to explain it further in detail. Once the complex plane is multiplied by i then the result will be (3+4i) x i = 3i + 4i 2 and i 2 = -1, so: 3i + 4i 2 = -4 + 3i, figure 9 shows the equation in complex plane.
One important thing can be observed that rotation of angle is right angle (90° or π/2) and same will be the results once same multiplications are carried out.
In this paper polynomiography is discussed in detail. The concept of polynomial study in the fields of art, science and education has been highlighted. The mathematical concepts have specially been discussed. The mathematical concepts which are ambiguous and very difficult to grasp by the students are given a new dimensional world where these conceptual thinking has a platform of graphical view. The calculations for the polynomial roots and their coefficients are if carried out with the help of computer or calculator will be accurate but there are lots of chances where their execution on a graph paper would be a disaster with a minute mistake. These fields were very difficult to understand but polynomiography made it very simple and easy to understand. Polynomials may be simple or complex both can be dealt very intelligently by the polynomiography. This paper has the intentions to clear the basic concepts of the beginners in the field of Polynomiography in a befitting manner.
[1] Bahman Kalantari, “Polynomiography - A New Intersection between Mathematics and Art”, Department of Computer Science, Rutger University, USA, 2000, pp. 1.
[2] Bahman Kalantari, “The Fundamental Theorem of Algebra and Iteration Functions”, Department of Computer Science, Rutger University, USA, 2003, sec. III and sec. VII.
[3] Bahman Kalantari, “Polynomiography and Applications in Art, Education, and Science”, Department of Computer Science, Rutger University, USA, 2003, para 3.
[4] J.C. Mason and D.C. Handscomb, “Chebyshev Polynomials”, New York: Washington D.C, CRC Press LLC, 2003, Ch. 1.
[5] Harry Bateman, “Higher Transcendental Functions”, vol. II, New York,
Toronto, London, McGral-Hill Book Company inc, 1953, Ch. 10, pp.178-182
[6] H.L. Krall and Orrin Frink, “A New Class of Orthogonal Polynomials: The Bessel Polynomials”, Transactions of the American Mathematical Society, 1949.
[7] Bahman Kalantari, “A New Visual Art Medium: Polynomiography” Rutgers University, Computer Graphics, Vol. 38 No. 3 Aug. 2004, ACM SIGGRAPH, Los Angeles, California, USA, Art. 21, pp. 21-23
[8] Wikipedia, The Free Encyclopaedia website, “Julia Set”. [Online]. Available: http://en.wikipedia.org/wiki/Julia_set
[9] Bahman Kalantari, “The Art in Polynomiography of Special Polynomials”, Department of Computer Science, Rutger University, USA, 2003.
[10] Wikipedia, The Free Encyclopaedia website, “Zero of a function”. [Online]. Available: http://en.wikipedia.org/wiki/Polynomial_roots
[11] Bahman Kalantari et al., “Animation of Mathematical Concepts using Polynomiography”, Department of Computer Science, Rutger University, USA, 2004.
[12] Math is Fun Advanced website, “Complex Number Multiplication”. [Online]. Available: http://www.mathsisfun.com/algebra/complex-number-multiply.html
(3) Science: Polynomiography do have importance in the field of science because almost all the science theories are based on polynomials and if we know the roots of a polynomial then we can say that we know the polynomial as well. In science special polynomials like Legendre polynomials as in [4], Chebyshev polynomials as in [5], orthogonal polynomials as in [6] etc were difficult to be understood but polynomiography made it very easy and simple to understand. Figure 4 as in [1] gives some polynomiography for a polynomial arising in physics.
III. RELATIONSHIP OF MATHEMATICAL CONCEPTS AND
POLYNOMIOGRAPHY
A. Numeric PolynomiographyThe use of polynomiography software application for numerical equations is very interesting. Here numbers can be encrypted with it. We can take the examples of ID or credit card numbers that can be divided into two dimensional images which can be resembled as a fingerprint. Now different fingerprints can be presented by different numbers. We can take the example of number a 8 a 7 …a 0 which can be identified by the polynomial P(z) = a 8 z 8 +….+ a 1 z + a 0 as in [1].
Finding square root of numbers is another way of handling through polynomiography which is a very interesting and simple task. Here polynomial equations are approximated by square rooting through images as in [7]. Figure 5 shows the graphical view of computing square root of two.
B. Basins of Attractions and Voronoi Region of Polynomial Roots
The basins of attraction of a root in relation with iteration function are the regions in the complex plane as shown in figure 6
Julia set as defined in [8] of the polynomial roots has the nature of fractal. Images of basins of attractions of Newton's method are very similar to that of some special polynomials. Mathematical analysis of complex iterations can be dealt with polynomiography as in [9].
Voronoi Region of polynomial roots can be defined as a diagram which divides the space into a number of regions. Points in each region are set at initial stage and all points are set in such a way that they are placed closer to each other. These regions are also known as voronoi cells. These diagrams are very helpful in the fields of science, technology and artwork.
There are number algorithms which find out the Voronoi regions by computation like Divide and Conquer, Brute Force, Fortune’s Line Sweep etc. Once computing intersection of half planes the time taken will be O (n 2 log n) and while proving lower bounds require Ω (n log n) and so on.
Figure 7 shows the space divided into twenty regions with twenty points or cells placed in them in a way to be closed to each other. This sort of display will be very easy to depict with the help of polynomiograhy software.
The n roots of a polynomial of degree n depend continuously on its coefficients. It can be explained as a polynomial root is a zero of the polynomial function and any non – zero polynomial and its degree has equal roots. The equation polynomial f of degree two will be:-
f (x) = x 2 – 5x + 6
have two roots 2 and 3, sincef (2) = 2 2 – 5.2 + 6 = 0 and
f (3) = 3 2 – 5.3 + 6 = 0
in case the function has to map real numbers to another real numbers then their zeros will be the x-coordinates of the points where their graph will be meeting the x-axis as explained in [10].Polynomial roots are very sensitive to even small changes which are if carried out in their coefficients. We may take the example of
suppose n = 7 then
p(z)=z 7 − 28z 6 + 322z 5 − 1960z 4 + 6769z 3 − 13132z 2 + 13068z – 5040
If we change coefficient of z 6 from −28 to −28:002 then this small change can have a large change in the roots. Some real roots will become complex. This aspect can’t be visualized in advance once solving the equation but polynomiography made it possible to have advance information on this important aspect as in [11]. Figure 8 shows the changes in the roots as the coefficient of z 6 is decreased.
Multiplication of complex numbers as in [12] is an important subject and polynomiography made it very easy to understand, we can take an example
Suppose two complex numbers z 1 = (a+bi) and z 2 = (c+di). They can be added as (a+b) + i(c+d), and their product is said to be (ac−bd)+i(ad +bc).
We take another example to explain it further in detail. Once the complex plane is multiplied by i then the result will be (3+4i) x i = 3i + 4i 2 and i 2 = -1, so: 3i + 4i 2 = -4 + 3i, figure 9 shows the equation in complex plane.
(-4 + 3i) x i = -4i + 3i 2 = -3 - 4i,
(-3 - 4i) x i = -3i - 4i 2 = 4 - 3i and
(4 - 3i) x i = 4i - 3i 2 = 3 + 4i
So the results are clear that rotation has completed a circle and it finished where it has started. Figure 10 shows a very clear right angle and rotation. Such complex plane calculations can confuse the readers but polynomiography made very simple, anyone with little knowledge can easily understand it.
IV. CONCLUSION
[1] Bahman Kalantari, “Polynomiography - A New Intersection between Mathematics and Art”, Department of Computer Science, Rutger University, USA, 2000, pp. 1.
[2] Bahman Kalantari, “The Fundamental Theorem of Algebra and Iteration Functions”, Department of Computer Science, Rutger University, USA, 2003, sec. III and sec. VII.
[3] Bahman Kalantari, “Polynomiography and Applications in Art, Education, and Science”, Department of Computer Science, Rutger University, USA, 2003, para 3.
[4] J.C. Mason and D.C. Handscomb, “Chebyshev Polynomials”, New York: Washington D.C, CRC Press LLC, 2003, Ch. 1.
[5] Harry Bateman, “Higher Transcendental Functions”, vol. II, New York,
Toronto, London, McGral-Hill Book Company inc, 1953, Ch. 10, pp.178-182
[6] H.L. Krall and Orrin Frink, “A New Class of Orthogonal Polynomials: The Bessel Polynomials”, Transactions of the American Mathematical Society, 1949.
[7] Bahman Kalantari, “A New Visual Art Medium: Polynomiography” Rutgers University, Computer Graphics, Vol. 38 No. 3 Aug. 2004, ACM SIGGRAPH, Los Angeles, California, USA, Art. 21, pp. 21-23
[8] Wikipedia, The Free Encyclopaedia website, “Julia Set”. [Online]. Available: http://en.wikipedia.org/wiki/Julia_set
[9] Bahman Kalantari, “The Art in Polynomiography of Special Polynomials”, Department of Computer Science, Rutger University, USA, 2003.
[10] Wikipedia, The Free Encyclopaedia website, “Zero of a function”. [Online]. Available: http://en.wikipedia.org/wiki/Polynomial_roots
[11] Bahman Kalantari et al., “Animation of Mathematical Concepts using Polynomiography”, Department of Computer Science, Rutger University, USA, 2004.
[12] Math is Fun Advanced website, “Complex Number Multiplication”. [Online]. Available: http://www.mathsisfun.com/algebra/complex-number-multiply.html
