For a nice introduction to roots and their fractal nature, see this post from 2011 on John Baez’s Azimuth blog or his 2023 UCR page here. Some of my code here is sourced from a great repository of interesting math visualizations from many others including the author of the pywonderland GitHub of Zhao Liang.
The SVG images below can be clicked to zoom in on another tab, but here is a Mathematica Notebook with Tooltip info on the selected roots. That SVG is 8000×8000, so you may have to <cntrl>- to zoom out to see any of the interesting parts (depending on your screen resolution and browser setup).
The roots are complex valued (Real x-axis and Imaginary y-axis) with normalized binVal(ue) defining the color from 0 to 1. The ColorRules are as follows:
A binVal{x,y} is incremented when the {x,y}={Real,Imaginary} root elements fall in that particular bin. The results are normalized by dividing each bin by the Log2@binVal{x,y} / Max@Log2@binVal. So the two x-axis binVal={+/-1,0}width/2 have a binVal=1 means the cumulative roots in every degree have the most roots at that 0/π location. The two y-axis binVal={0,+/-1}height/2 have a binVal of ~.903 that land in those +/-i π/2 bins.
The analysis below provides more detail on the Roots-of-Unity Circle (RUC) at the center of the light yellow area. These are interesting in that there are 4 quadrants each with 12 roots that all have 1/2< binVal <1 at the center of holes.
There are 6 roots that sit in holes surrounded by zeros (i.e. 2π/{12,10,8,6,5,4}) and 6 roots in holes with surround values <1/2 (i.e. 2π/{24, 14, 7, 11/2, 29/6 (75°), ~437/90 (77°)}). The only other binVal roots >1/2 are on the 1 pixel wide x-axis with 5936 of 8000 (~75%) from the x-y grid of 8000×8000 (64 million) bins which sum over each of the 224 (~16 million) roots-of-polynomials with -/+1 coefficients. The rest of the x-axis roots are 2020 bins equal to 0 and only 44 bins between 0 and 1/2).
The large table below shows 100 detail areas of “special” algorithmically selected locations from bin location Tally counts of 7732 unique binVals. About 50% of these occur in 2 to 4 (symmetric) locations with a binVal>1/2. The other 50% of the Tally counts have binVal<1/2 and tend to have 103 to 106 locations with all but 5980 (of 64 million) bins. This is made clear from the above MatrixPlots.
Each detail area shows a 400×400 ArrayPlot around the selected location and an 11×11 zoom-in of that binVal matrix with its ArrayPlot. They include the 48 RUC roots and 12 x-axis roots with binVal>1/2, and 40 off-axis roots with ~2/3 bins of zero and 1/3 with binVal<1/2. This off-axis list includes one set of 4 quadrant roots that is a proper hole (in the 4 yellow rings located just above/below the bottom/top apex in the full image or 10-14th from the bottom in the detail list below). The other 9×4 roots are what I call “pseudo-holes” that have no enclosed high central binVal.
The annotated upper-right quadrant of the SVG image has the RUC roots ringed in colors (shown in the zoom-in graphic above) with large colored rings assigned by their binVal. Annotated rings with binVal>1/2 are cyan, magenta, and red. The binVal<1/2 are yellow, orange, or blue rings. The smaller black ring is what is visible in the 11×11 MatrixPlots below. To the right is the text with the angle at which the root is located.
FYI – The color scheme on these detail zoom MatrixPlots are set to the default in order to maximize and highlight the numeric differences, so they don’t match the ColorRules in the large 8000×8000 MatrixPlot at the top of the page.
Click the image below to see this SVG and zoom further.
