Polynomial Plot: Simple Math Expressions Yield Intricate Visual Patterns [Slide Show]

Plotting the roots of run-of-the-mill polynomials yields dazzling results

7 This crop shows the detail of an inner edge of the main ring in Derbyshire's image. It is a favorite of John Baez, a mathematical physicist at the University of California, Riverside, who posted these images on his Web site. "This image is almost a metaphor of how mathematical patterns emerge from confusion like sharply defined figures looming from the mist," Baez wrote in his online column. SAM DERBYSHIRE
6 Around the number 4/5, a white line again traces the real axis—Derbyshire's polynomials have a seemingly continuous collection of real roots in the vicinity. But above and below the axis, the imaginary numbers form intricate flame-like patterns. SAM DERBYSHIRE
5 This crop of Derbyshire's image shows intriguing patterns around the imaginary number i, the square root of –1. SAM DERBYSHIRE
4 Derbyshire's image of 24th-degree polynomial roots with coefficients of either 1 or –1 features a serrated hole around the root 1. The white line slicing through the image traces the horizontal (or real) axis—the line comprises the real roots of the polynomials. SAM DERBYSHIRE
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3 Sam Derbyshire, an undergraduate student at the University of Warwick in England, took a large but very specific set of polynomials to make this rich plot, comprising hundreds of millions of individual roots. Derbyshire plotted only the roots of polynomials of degree 24 whose coefficients are either –1 or 1. The color of each point represents the density of roots at that point—black being the least dense and white being the densest. Again, the points 1 and i are surrounded by interesting-looking holes, which you can see in greater detail on the following slides. SAM DERBYSHIRE
2 This crop of the previous slide reveals details of the hole near i on the vertical (imaginary) axis, which appears at the top center of the image and in greater detail in the inset. (The different colors of the points represent roots of polynomials of different degrees.) Note that i appears as a polynomial root but its closest neighbors do not. DAN CHRISTENSEN, UNIVERSITY OF WESTERN ONTARIO
1 Dan Christensen, a mathematician at the University of Western Ontario, plotted the roots of polynomials of degree six or less whose coefficients are integers between –4 and 4. The collection has large holes surrounding the points 0, 1 and –1 on the real axis—although some polynomials have those numbers as roots, the numbers nearby (especially those incorporating an imaginary component) seem to be off-limits. On the next slide, Christensen zooms in to show another hole—this one on the imaginary axis, around i, the square root of –1. DAN CHRISTENSEN, UNIVERSITY OF WESTERN ONTARIO

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