Math-Art: 9 December 1995 ‐ 24 January 1996

In honour of the 1995 Canadian Math Society Winter Meeting, Simon Fraser University presents:

MA-map

To navigate this exhibit, scroll down. Those images on a yellow background have extra information about how they were created.

postcard

Three public events take place on December 9 at Simon Fraser University at Harbour Centre, 515 West Hastings Street. For more information or to reserve seats at the lecture, call 291-5100.

  1. An exhibition of art based on mathematical principles and ideas. Fractals, more so than other areas of mathematics, are visually exciting in a very real way. Fractals are a language with which to describe the geometry of nature. Images of fractals can be an important access to mathematics and surrounding philosophical questions. Noon ‐ 5 pm in the Teck Gallery (Room 1305).
  2. Interactive computer displays for visitors to experiment with making their own fractal designs; investigate E-Gems, a project to design math and science electronic games (Counting on Frank, one of their products, was chosen by PC Magazine as one of the top 100 CD-ROMs for 1995); and check out the math education and art resources available on the Internet. Noon ‐ 5 pm in the Segal Centre (Room 1430).
  3. Ingenious mathematical amateurs. 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 (Emerita) of Mathematics, Moravian College. 5 ‐ 6 pm in the Fletcher Challenge Canada Theatre (Room 1900).

We'd like to thank our sponsors, Simon Fraser University, the Canadian Mathematical Society, and the British Columbia Ministry of Employment and Investment.

The fractal on this postcard is Fiery Dragon (1990), by Daryl H. Hepting.

Introduction

Mandates of art galleries are many, but the mandate that immediately defines a university gallery is the mandate that a gallery demonstrates the aesthetic dimensions of academic disciplines. It was to this end that Simon Fraser Gallery mounted its first math art exhibition, EXPLORING THE IMAGE WORLD OF COMPUTER GRAPHICS works by Daryl Hepting ( and others from the Graphics and Multimedia Research Lab in the School of Computing Science ), in the W.A.C. Bennett Gallery, May 30 to July 26, 1994.

Daryl Hepting, a Ph.D. Candidate in the School of Computing Science and one of the four curators of this exhibition, knows the connections between computers and aesthetics as few in Simon Fraser University can. He takes us on a short course on math art in the notes on these digital graphics.

Other members of Simon Fraser University who joined me in curating this exhibition were Dr. Katherine Heinrich, Chair of Mathematics & Statistics and Dr. Malgorzata Dubiel of the same department.

These eighteen images were selected from scores of possible images so as to represent the complexities and contrasts of visual aesthetics that are possible in digital graphics. They were also selected to represent the images that are available from three sources on the WORLD WIDE WEB (internet): computer art exhibits, pioneer publications, and computer images created by Daryl Hepting of Simon Fraser University.

Simon Fraser Gallery is grateful for the assistance of Gallery staff, Daryl Hepting and the other curators for making this exhibition possible. It is also grateful to the Graphics and Mulimedia Research Lab for production assistance. The Teck and Simon Fraser Galleries are honoured to support the Department of Mathematics & Statistics in hosting the Canadian Mathematical Society 1995 Winter Meeting.

Dr. Edward Gibson, Director, December 1995.

Notes on Exhibition

The majority of works in this exhibition represent a collection which rely on the computer for their existence. And the computer involves mathematics to describe, manipulate, and translate even the simplest scenes into the forms we see here. The involvement of mathematics plays a greater role in these works ‐ it provides the inspiration and the foundation upon which they are constructed.

A prominent feature of the exhibit is its display of fractals, objects from a non-Euclidean geometry. Fractals ‐ sharing a root with fracture and fraction ‐ have been called the geometry of nature by Mandelbrot, who discovered their capability to describe the non-smooth world in which we live. Fractal geometry, which is new to us, has shed light on old things ‐ things like the historical record of water levels in the Nile river for instance. And in its turn the computer has shed light on fractals, allowing the underlying beauty of what may seem to be dry-as-dust mathematics to shine through. Works like Blessed State or Steps to Infinity provide evidence of the profound connection that seems to exist between fractals and the natural world.

It is fitting that this collection of computer-related art comes to us in large part through the internet (the World Wide Web). Although Daryl Hepting is now at SFU, his works are distributed widely across the internet. In addition to Hepting's collaborations with Musgrave (George Washington University), Prusinkiewicz (University of Calgary), Saupe (University of Freiburg), and Snider (University of Regina), the computer-based works come from:

The other pieces in this exhibit remind the viewer that the computer is responsible for only a portion of all mathematical art. The works of M. C. Escher and Marjorie Rice offer a beautiful expression of this fact.

Partial financial support for the exhibit at Harbour Centre was provided by the BC Ministry for Employment and Investment.

Fiery Dragon

by Daryl H. Hepting

Fiery Dragon

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. Click here to see a movie which 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.

Dragon Mesa

by Przemyslaw Prusinkiewicz, Dietmar Saupe, and Daryl H. Hepting

Anemones (1992)

by Kevin Suffern

Blessed State (1988)

by F. Kenton Musgrave

Carrot in Blue (1991)

by Daryl H. Hepting and Przemyslaw Prusinkiewicz

The Great Fern Dune (1994)

by Daryl H. Hepting

Fractal Emblem (1992)

by Daryl H. Hepting

Fractal World (1994)

by L. Kerry Mitchell

Gravity (1994)

by L. Kerry Mitchell

Steps to Infinity (1994)

by Daryl H. Hepting and F. Kenton Musgrave

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 1994/95 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...

Untitled

by Stan Ostoja

Captivating Rhythm (1995)

by Daryl H. Hepting

This image is based on a fractal which is similar to the one used to create Fiery Dragon and Steps to Infinity. This curve is also defined by two transformations, but with slightly different parameters:

			xform
			        scale 0.618	    /* Dragon scales by approximately .7071 */
			        rotate -137.5
			
			xform
			        scale -0.618
			        rotate 137.5
			        translate 1.0 0.0
			

This image is an interpretation of the process by which the points of the plane are captured by the transformations in the description of the curve to form the actual curve.

I've assigned an initial point to each of the two transformations and I apply the transformations to their respective points and then to the results, over and over again. As I compose the transformations into longer and longer strings, the contractivity of each composite transformation increases. These "stronger" transformations capture more and more of the plane. I've illustrated this process in a 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.

The upper black and white image above shows one possible colouring for the image used to construct Captivating Rhythm. Each band corresponds to a different level of "capture", increasing away from the curve. The reverse of "capture" (lower left above) is "escape" (lower right above). Notice how the arrangement of disks differs between them.

The image below shows another way to view the same data, ultimately selected for the final image. Here, I have broken up what were solid bands in the earlier image by using information about which transformation was applied at a particular time. In creating the final image, I applied a fractional Brownian motion texture and added an outline of the fractal.

Splash (1993)

by F. Kenton Musgrave

The Subject of All Objects (1994)

by Sophie Anapliotis

Desktop Tetrahedron (1989)

by Daryl H. Hepting and Allan Snider

Ingenious mathematical amateurs

A talk by Dr. Doris Schattschneider

Images (from left to right): Hibiscus by Marjorie Rice, Escher No. 42 by M. C. Escher, and Escher No. 20 by M. C. Escher. These images will be on display as part of the Math-Art exhibit.

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.

References: