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Polynomials, the meat and potatoes of high-school algebra, are foundational to many aspects of quantitative science. But it would take a particularly enthusiastic math teacher to think of these trusty workhorses as beautiful.
As with so many phenomena, however, what is simple and straightforward in a single 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, a mathematician at the University of Western Ontario, and Sam Derbyshire, an undergraduate student at the University of Warwick in England.
Polynomials are mathematical expressions that in their prototypical form can be described by the sum or product of one or more variables raised to various powers. As a single-variable example, take x2 - x - 2. This expression is a second-degree polynomial, 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 those numbers in for x and the polynomial will be equal to zero. (These roots can be found 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 number times a positive number is positive, and a negative number times a negative number is 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 the number i, the square root of –1, imaginary numbers are unusual in that they do not represent a tangible physical quantity. (You cannot have i dollars—at least, not if you wish to pay your bills.) Polynomial roots can be either real or imaginary—that is, they may or may not have an imaginary component.
What Christensen and Derbyshire did was plot the roots of entire families of single-variable polynomials, imposing constraints on the polynomials' degrees and coefficients. (Coefficients are the multipliers of the variable terms—in the polynomial 4x - 2, the coefficients are 4 and –2, respectively.) For example, Christensen plotted the roots of every polynomial whose degree is six or less and whose coefficients are integers between –4 and 4.
The horizontal axis in Christensen's and Derbyshire's plots is the real numbers; the vertical axis is the imaginary numbers. So a real root, such as –1, would fall on the horizontal 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 imaginary components—fill out the quadrants of the graph. For instance, the imaginary number 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 and intriguing patterns emerge that should appeal even to the most math-averse. Take a look at Christensen's and Derbyshire's images to see for yourself.
Slide Show: Polynomial Plot