My friend Dan Christensen has created a fascinating picture of all the roots of all polynomials of degree ≤ 5 with integer coefficients ranging from -4 to 4:
Click on the picture for bigger view. 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 sixth 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! You can see more patterns if you zoom in:
Now you see beautiful feathers surrounding the blank area around the point 1 on the real axis, a hexagonal star around exp(i π / 3), a strange red curve from this point to 1, smaller stars around other points, and more....
People should study this sort of thing! Let's define the Christensen set Cd,n to be the set of all roots of all polynomials of degree d with integer coefficients ranging from -n to n. Clearly Cd,n gets bigger as we make either d or n bigger, and it becomes dense in the complex plane as n approaches ∞, 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 → ∞. Based on the above picture, there seem to be lots of interesting conjectures to make about what it does 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. 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. He then used some Java programs to create this amazing image:
The coloring shows the density of roots, from black to dark red to yellow to white. The picture above is a low-resolution version of the original image, which is available as a 90-megabyte file on Dan Christensen's website, here. We can zoom in to get more detail:
Note the holes at certain roots of unity and the feather-like patterns as we move inside the unit circle. To see these pattern, let's zoom in to certain regions, marked here:
Here's a closeup of the hole at 1:
Note the white line along the real axis. That's because lots more of these polynomials have real roots than nearly real roots.
Next, here's the hole at i:
And here's the hole at exp(iπ/4) = (1 + i)/√2
Note how the density of roots increases as we get closer to this point, but then suddenly drops off right next to it. Note also the subtle patterns in the density of roots.
But the feathery structures as move inside the unit circle are even more beautiful! Here is what they look near the real axis — this plot is centered at the point 4/5:
They have a very different character near the point (4/5)i:
But I think they're the most beautiful near the point (1/2)exp(i/5). This image is almost a metaphor of how, in our study of mathematics, patterns emerge from confusion like sharply defined figures looming from the mist:
There's a lot to explain here: each picture demands a theorem or two to explain it! For more on this sort of thing, see:
Odlyzko and Poonen proved some interesting things about the set of all roots of all polynomials with coefficients 0 or 1. If we define a fancier Christensen set Cd,p,q to be the set of roots of all polynomials of degree d with coefficients ranging from p to q, Odlyzko and Poonen are studying Cd,0,1 in the limit d → ∞. They mention some known results and prove some new ones: this set is contained in the half-plane Re(z) < 3/2 and contained in the annulus 1/Φ < |z| < Φ where Φ is the golden ratio (√5 + 1)/2. In fact they trap it, not just between these circles, but between two subtler curves. They also show that the closure of this set is path connected but not simply connected.
