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Polynomial Plot: Simple Math Expressions Yield IntricateVisual Patterns [Slide Show]Plotting the roots of run-of-the-mill polynomials yields dazzling results
Dec 28, 2009 | By John Matson |
Polynomials, the meat and potatoes of highschool algebra, are foundational tomany aspects of quantitative science. But it would take a particularly enthusiasticmath teacher to think of these trusty workhorses as beautiful.
As with so many phenomena, however, what is simple and straightforward in asingle serving becomes intricately detailed—beautiful, even—in the collective.
On December 5 John Baez, a mathematical physicist at the University of California,Riverside, posted a collection of images of polynomial roots by Dan Christensen, amathematician at the University of Western Ontario, and Sam Derbyshire, anundergraduate student at the University of Warwick in England.
Polynomials are mathematical expressions that in their prototypical form can bedescribed by the sum or product of one or more variables raised to various powers.
As a singlevariable example, take x2 x 2. This expression is a seconddegreepolynomial, or a quadratic, meaning that the variable (x) is raised to the second
power in the term with the largest exponent (x2).
A root of such a polynomial is a value for x such that the expression is equal to zero.In the quadratic above, the roots are 2 and –1. That is to say, plug either of thosenumbers in for x and the polynomial will be equal to zero. (These roots can befound by using the famous quadratic formula.) But some roots are more complex.
Take the quadratic polynomial x2 + 1. Such an expression is only equal to zero when
x2 is equal to –1, but on its face this seems impossible. After all, a positive numbertimes a positive number is positive, and a negative number times a negative numberis positive as well. So what number, multiplied by itself, could be negative?
Imaginary numbers were, well, imagined into existence to fit the bill. Based on thenumber i, the square root of –1, imaginary numbers are unusual in that they do notrepresent a tangible physical quantity. (You cannot have i dollars—at least, not ifyou wish to pay your bills.) Polynomial roots can be either real or imaginary—that
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is, they may or may not have an imaginary component.
What Christensen and Derbyshire did was plot the roots of entire families of singlevariable polynomials, imposing constraints on the polynomials' degrees andcoefficients. (Coefficients are the multipliers of the variable terms—in thepolynomial 4x 2, the coefficients are 4 and –2, respectively.) For example,Christensen plotted the roots of every polynomial whose degree is six or less andwhose coefficients are integers between –4 and 4.
The horizontal axis in Christensen's and Derbyshire's plots is the real numbers; thevertical axis is the imaginary numbers. So a real root, such as –1, would fall on thehorizontal axis; a purely imaginary root such as 2i would fall on the vertical axis.The rest of the imaginary numbers—those with both real and imaginarycomponents—fill out the quadrants of the graph. For instance, the imaginarynumber 3 2i would be represented by the point aligning with 3 on the horizontal(real) axis and –2 on the vertical (imaginary) axis.
What happens when these families of roots are plotted en masse? Intricate andintriguing patterns emerge that should appeal even to the most mathaverse. Take alook at Christensen's and Derbyshire's images to see for yourself.
Slide Show: Polynomial Plot
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