Hartmut F. W. Höft

Professor of Computer Science
Computer Science Department, Eastern Michigan University
Ypsilanti, Michigan, USA

The sickle of the moon can be drawn with two ellipses in the plane and a pair of sickles with two orthogonal ellipses. I asked myself what other designs I might be able to create with such pairs as the only geometric element in the plane. With my images I want to demonstrate to the viewer the conundrum of the easy and the difficult, the simple and the complex. The image is based on two fundamental geometric elements: a curve in the plane that provides the "spine" of a figure in an image and pairs of orthogonal ellipses that are placed at regular intervals along the spine and that create the shape in the image. The spine is not drawn, only the ellipses are drawn. I render my images in Mathematica. What surprised me most, however, is that many of the resulting images appear three-dimensional. Our visual system apparently gets fooled into seeing a three-dimensional object when we draw many ever smaller slices after larger ones in a single plane. All images are mathematically planar.

Purple Tubeworm Rearing Up
Purple Tubeworm Rearing Up
8.5 x 11 (16 x 17.25 framed)
color print on archival paper

In this image 180 pairs of orthogonal ellipses are drawn in the plane along a closed curve that belongs to a family that I used in the classroom when teaching polar coordinates. The ratio of the semi-axes in the ellipses is 1.05 which is close to the ratio of the equatorial and polar radii of earth and moon. Starting colors for the ellipses are green and magenta. The lengths of the semi-axes are modified synchronously by the Gaussian Bell Curve f(t)=e^-(t^2) over the interval [-1, 1]. The underlying spine is a section from -pi/2 to pi/2 of the curve r = sin^4 (4t) + cos (3 t) rotated clockwise by 90 degrees. This is a closed curve symmetric about the x-axis with three loops and one lobe on each side of the axis of symmetry. The ellipses are rotated a total of 3*pi, but not all of that rotation is visible because of the large differences in the scale of the loops and the semi-axes of the ellipses. The graphics were rendered in Mathematica 8.