I started creating mathematical art while teaching a first-year seminar based on the work of M.C. Escher. For that course students learn about tessellations and fractals, among other topics, and then create their own artwork. To help that process, I created examples to try to motivate and inspire them to create their own versions. And I was hooked. In the past I have used Geometer's Sketchpad to experiment with geometric shapes. I'm currently using R to explore patterns and symmetries. Sometimes I try to create something that I find beautiful or pleasing. Other times I allow myself to just explore and am delighted when I discover something beautiful.
This image shows a plot of the 5th vs. the 6th eigenvectors of a one-dimensional evenly spaced inter-point distance matrix for three different divisions of the one-dimensional segment. The first used 250 divisions, the second 500 and the third 1000. It is interesting to see how the image develops as the number of divisions is increased and how the result grows to resemble a knot with overlapping regions. The use of two colors further emphasizes and enhances the symmetry of the resulting pattern.
This work is an extension of an undergraduate research project on the patterns that are produced when the eigenvectors of a one-dimensional evenly spaced inter-point distance matrix are plotted against each other. The first row is the 2nd eigenvector vs. the 3rd through the 7th eigenvectors; the second row is the 3rd vs. the 4th through 8th; etc. The interesting patterns result from the symmetry/skew symmetry of the odd/even eigenvectors.Two colors were used to emphasize the alternating nature of the plotted points. Extensions to higher order eigenvectors reveal surprising repeating patterns. The undergraduate research project was suggested by and mentored by researchers at a nearby Navy base.