MATH-ART: 9 December 1995 — 24 January 1996

Martin Gardener tells the story of how this curve, first discovered by John E. Heighway, used by William G. Harter as a decoration symbolic of cryptic structure, and later analysed by them and another colleague, Bruce A. Banks. Though its final structure is complex, its description is not. The following two transformations are required to generate this shape:

        xform
        	scale { sqrt(2)/2 }
        	rotate -45

        xform
        	scale { sqrt(2)/2 }
        	rotate 135
        	translate 1.0 0.0
      

Beginning with a single point, the whole curve can be generated by repeated application of these two transformations. This specification of the shape in terms of contractive, affine transformations is called an iterated function system (IFS).

This image of a dragon curve was created by covering with disks (spheres) the parts of the plane which do not include the fractal. These parts are determined by computing the minimum distance from a point P to the fractal. If this distance is d, then a disk of radius d can be drawn about the point P and all points inside that disk are known not to belong to the fractal.

The following movie illustrates this process at work.

Like many of my other works, I've used rayshade to ray-trace this image. This process creates the final image (which you see above) by simulating a camera inside the computer.

The steps in this image come from a dragon curve, like the one seen on the poster for this event (and described in the notes for Fiery Dragon. Below, I show two different colourings for a dragon curve in the plane. The curve is made up of smaller and smaller copies of itself, as the colourings indicate.

There are two transformations which define the dragon curve each small dragon seen above can be described by a sequence of these two transformations. To create the steps, I've assigned a unique height to each small dragon. Below, I show the steps inside a white sphere, which represents the sky. It doesn't look much like a very believable sky here, but I'll fix that. The lighting I've used here is much simpler than in the final image and it shows.

If you look carefully at Steps to Infinity, you will notice (part of) a Mandelbrot set in the sky, amongst the clouds. I began with a portion of the Mandelbrot set, shown below. It may be familiar to you from Ken Musgrave's image on the cover of the 1994 Fractals Calendar. In this case, I've changed to colours more appropriate for a blue sky.

For the next step, I've mapped the square image onto the inside of a white sphere, which will represent the dome of the sky in the model of my scene. Notice that this image is considerably lighter than the first, and this is due only to the white colour of the original sphere.

I create some clouds for the sky by using a fractal texture function which selects blues and whites for each pixel based on something called fractional Brownian motion.

When the clouds are combined with the Mandelbrot set image, the final Mandelbrot sky image comes to life.

And the final image once again...

xform
	scale 0.618	    /* Dragon scales by approximately .7071 */
	rotate -137.5

xform
	scale -0.618
	rotate 137.5
	translate 1.0 0.0

I've illustrated this process in the following movie, using cones to show the effect of the transformations. As the strength of a transformation increases, a cone under that transformation gets wider and shorter.

It is possible for someone without formal credentials to make contributions to mathematics. Well-known artist M.C. Escher and unknown homemaker Marjorie Rice both tackled mathematical problems, developing unorthodox notation that was essential to their methodical investigations. Each worked alone, essentially in secret, rewarded by the exhilaration of finding answers to a large puzzle.

Lecture by Dr. Doris Schattschneider, Professor of Mathematics, Moravian College.
5 - 6 pm, 9 December 1995, Fletcher Challenge Canada Theatre (Room 1900), SFU Harbour Centre Campus.

These images will be on display as part of the MATH-ART exhibit.

References:

• Schattschneider, Doris. M.C. Escher: Visions of Symmetry, W. H. Freeman Company, 1990.
• Klarner, David A. (editor). The Mathematical Gardener, Wadsworth International, 1981.