In the streets of the Algerian mock casba of the French Colonial Section of the Exposition Universelle of 1900, Paris
Architectural details of the palaces on the Esplanade des Invalides for the Exposition Universelle of 1900, Paris
Mathematics and Physics Intersect : John Baez
Dan Christensen drew a picture of all the roots of all the polynomials of degree at most 5 with integer coefficients ranging from -4 to 4:
Image 1 and 2: Dan Christensen, Plots of roots of polynomials with integer coefficients, http://jdc.math.uwo.ca/roots/
Roots of quadratic polynomials are in grey; roots of cubics are in cyan; roots of quartics are in red and roots of quintics are in black. The horizontal axis of symmetry is the real axis; the vertical axis of symmetry is the imaginary axis. The big hole in the middle is centered at 0; the next biggest holes are at ±1, and there are also holes at ±i and all the cube roots of 1.
You can see lots of fascinating patterns here, like how the roots of polynomials with integer coefficients tend to avoid integers and roots of unity - except when they land right on these points!
In the second image you see beautiful feathers surrounding the blank area around the point 1 on the real axis, a hexagonal star around exp(i π / 6), a strange red curve from this point to 1, smaller stars around other points.
Let’s define the set C(d,n) to be the set of all roots of all polynomials of degree d with integer coefficients ranging from -n to n. Clearly C(d,n) gets bigger as we make either d or n bigger. It becomes dense in the complex plane as n → ∞, as long as d ≥ 1. We get all the rational complex numbers if we fix d ≥ 1 and let n → ∞, and all the algebraic complex numbers if let both d,n → ∞.
But based on the above picture, there seem to be a lot of interesting conjectures to make about this set as d → ∞ for fixed n.
Inspired by the pictures above, Sam Derbyshire decided to to make a high resolution plot of some roots of polynomials (images 3-6). After some experimentation, he decided that his favorite were polynomials whose coefficients were all 1 or -1 (not 0). He made a high-resolution plot by computing all the roots of all polynomials of this sort having degree 24. That’s 224 polynomials, and about 24 × 224 roots — or about 400 million roots! It took Mathematica 4 days to generate the coordinates of the roots, producing about 5 gigabytes of data.
Jože Plečnik: Church of St. Michael, 1940, Ljubljana Marshes. (Photo: Miran Kambič)