# Hartmut F. W. Höft

I like to experiment with simple planar geometric figures. Starting with one outline I move it, rotate it, and change its proportions. This process generates a sequence of intersecting outlines. With my images I want to demonstrate that applying elementary planar transformations to one simple shape produces complexity that can mask how the images were generated.

One special planar form is the ellipse: it is convex, has mirror symmetries and a shape that is easily manipulated through its two semi-axes. Furthermore, our eyes immediately recognize its outline. The centers of the ellipses are moved along a closed, planar curve that I think of as the spine of the figures. A general (polar) form of the equations that I use for the spines is r(t) = s + sin^n(n t) + cos(k t), where s, n, and k usually are integer valued parameters. Since the spine is not drawn, this one fundamental aspect of the images remains hidden to be discovered by a viewer. All images are mathematically planar.

In this picture 12 images are arranged in a spiral. A primordial sea in the center starts the process. One parameter of the spine function increases, then stays constant, while the other increases and then decreases again. As the sequence progresses along the spiral the spine function increases in complexity so that the images billow out like clouds and then start separating into two funnels as the second parameter decreases again. Each image on the spiral is composed of 289 ellipses with a closed spine curve though it does not appear to be closed since the axes of the elipses decrease to zero at the bottom. The graphics were rendered in Mathematica 8 and printed on archival, photographic digital paper.

In this image 289 ellipses are drawn in the plane along a closed spine curve with 17 lobes. The bunching of cords into apparent ridges and spikes add dimension and the illusion of three-dimensionality to the image. The semi-axes of the ellipses are changing with the square of the polar angle along the spine. The graphics were rendered in Mathematica 8 and printed on archival, photographic digital paper.

In this image 720 ellipses are drawn in the plane along a closed spine with 11 lobes. At the middle level of the image there is a single large ellipse although a viewer may see the illusion of a spiraling opening downward on the left and a like spiraling opening up. The closed spine is traversed three times. The graphics were rendered in Mathematica 8 and printed on archival, photographic digital paper.