Artist's Profile:

The Art of Spirolateral Series

Algorithmic generators and transformations are used to develop a series of designs that at one time elicit images of a far off future, at the same time remind me of past ritual icons. The starting linear and resulting circular forms remind me of the past while the lack of color and the sharp edges envision the future.

Artist Statement

The series of images included are part of a larger series investigating the formation of artwork using basic mathematical concepts. This series is based on a mathematical figure called a "spirolateral". It is simply a square spiral with increasing length of turns and the turns repeating themselves. The turns can be all in one direction or certain turns can go the opposite direction. The most interesting of these are ones that close on themselves, not all do. Investigating a series of possible turning angles, number of turns, number of repeats, and trying all revered turns, I identified over 10,000 spirolaterals that closed. The web site displays over 500 such spirolateral designs plus technical information.
To curve these spirolaterals an antiMercator, Circular, Inversion, and Harmonic Mean transformation was performed in a variety of parameters controlling center offset.

The following pieces are included:

1: 004, 142 and 139: linear spirolateral

2: A22, A23, and A25: antiMercator transformation

3: B31, B41, and B53: Circular transformation

4: C06, C09, and C13: Inversion transformation

5: F02, F05, and F11: Harmonic Mean transformation

The web site displays over 250 other curved pieces.
All of these pieces are created by custom computer software which enables me to control the angle of the turns, the number of turns, and the number of repeats. The line thickness is also controllable, as can be which lines can be included; center, edges and corner. The BitArt web site has a web-based version of the linear and curved software that anyone can create their own design. The results of the software are not modified.

My overall interest is to investigate methods which can develop forms which is in one sense predictable, but has the element to generate the unexpected. The unexpected in a predictable way. The software becomes the instructions for producing the work itself. My interest is in developing methods and instructions and leaving the production to the computer. This approach allows variations to be investigated in a repeatable way, this enables me to fine tune an idea by repetition. The repetition can be manually controlled, randomly controlled or exhaustive in nature. With this approach even the viewer can create the artwork, within the parameters set by the software. The interesting aspect is that variations can be discovered by the viewer that I have never seen. Even more interesting is that the programs that I have written have actually "seen" all the possible spirolaterals, the problem is that they can not tell me which ones I should look at.
It is important not to hide the technology and the mathematical basis on how this art is created. There is a natural beauty that needs to be seen, one that does not need color or context. They are powerful enough to stand on their own.


Robert J. Krawczyk is on the faculty of the College of Architecture at the Illinois Institute of Technology. During his nineteen years at IIT, he has developed and taught a series of CAD and digital design courses covering 2D and 3D CAD, image composition, animation, and form generation methods. In addition to teaching, he is an advisor in the PhD program on form generation, fractals, 3D blob modeling and other related digital design methods. His digital design and computer-generated artwork have been published at the Mathematics and Design, Bridges, MOSAIC, ISAMA, ACADIA, and ACSA conferences. Digital art and design exhibits have included the ACADIA Digital Media 2000 and 2001 exhibits, InterSculpt 2001 and 1999, and TeleSculpture 2001 and 1999. Digital artwork includes the Spirolateral Series, have been recently shown at the University of Michigan, University of London, and the SOMARTS Gallery in San Francisco. Twenty pieces of the Curved Spirolateral Series were displayed at the SIGGRAPH 2001 N-Space Art Gallery in Los Angeles and also selected for the SIGGRAPH 2001 international Traveling Art Show.
The spirolateral web site,, was covered in the Mathematical Association of America's MathTrek web column by Ivars Peterson, was selected web site of the week by the Canadian Mathematical Society, and selected as a finalist in the 2001 JAVAMuseum competition.