In mathematics, fractals do not only occur as the product of special curves, but can also represent the result space of typical arithmetic operations. Thus all polynomials with the coefficients -1 and 1 form the following fractal up to a certain degree:

Sample Image

In the generated image the complex roots were calculated for all polynomials with the coefficients -1 and 1 up to degree 40. The real part is used as x-coordinate and the imaginary part as y-coordinate.This results, for example, in various dragon fractals.

Additionally, the frequency of a zero at a coordinate can be coded by the color value, resulting in the following images:

Sample Image Sample Image

More background information about these fractals can be found at Link.

In the first approach to calculate the roots of all polynomials up to a certain degree, Laguerre's Method (https://mathworld.wolfram.com/LaguerresMethod.html) was used to approximate all complex roots. All roots with no imaginary part were discarded. Laguerre's method works reliably with polynomials with a degree lower than 30. With even larger polynomials the computational time of this method increases very strongly. Therefore, the roots were determined by calculating the eigenvalues of the corresponding matrix (https://en.wikipedia.org/wiki/Characteristic_polynomial). Due to the huge amount of possible polynomials, the calculation was divided into several batches and took place on an HPC cluster. More than 2.8 million CPU hours were spent to calculate the fractal.

To shrink down the amount of necessary storage, the calculated roots were binned into a 10.000 x 6.250 matrix and only the frequency of each root was exported. The exported images have therefore a resolution of 62.5 megapixels.

The data sets are provided as CSV files with x and y coordinates and the frequency. '40-fractal-binned.zip' contains the binned roots normalized to the range 0,10000 on the x-axis and 0,6250 on the y-axis. '40-fractal-coordinates.zip' contains the denormalized roots in the range -4,4 on the x-axis and -2.5,2.5 on the y-axis. As far as known there is no data set with a higher degree.

The data sets are licensed under the Creative Commons Attribution 4.0 International License.

40-fractal-binned.zip

40-fractal-coordinates.zip