# David Reimann

I enjoy giving visual representations to abstract mathematical concepts such as number, form, and process. I often use patterns that convey messages at multiple levels and scales using a wide variety of mathematical elements and media. Some of my work contains fine detail that allows the art to be viewed differently depending on the distance between the viewer and the art. Another prevalent theme in my work is symmetry, where the overall pattern is created by repeated rotation or translation of a smaller very similar units. My overall goal in creating art is to share the beauty and wonder I see in mathematics.

This artwork depicts a small sample of the infinite number of parabolas that pass through three points. It is based on a recent paper ("All Parabolas through Three Non-collinear Points" by Huddy and Jones; Mathematical Gazette 102, July 2018, 203-209). Three control points are rotated around the z-axis resulting in three rings. A parabola exists with an axis of symmetry at every angle in the range [0,Pi] except the three where a pair of the control points are co-linear. The parabola with an axis of symmetry at an angle theta is associated with a parabola in the sculpture in the xz-plane rotated around the z-axis by an angle twice theta.

This art is inspired by a recent article (Dresden et al., Mathematics Magazine, December 2019). Let r1, r2, and r3 be the roots of the Ramanujan simple cubic f(B,x) = x^3 - ((3+B)/2) x^2 - ((3-B)/2) x + 1. The function n(x) = 1/(1-x) permutes the roots: n(r1) = r2, n(r2) = r3, and n(r3) = r1. For a particular B value, consider the triangle with vertices that are the ordered pairs of roots {(r1,r2), (r2, r3), and (r3, r1)}; its vertices lie along the graph of n(x). The artwork is a visualization of these triangles for the values of B such that |B| is in the set {0, 4, 8, 12, 16, 20, and 24}.