“The spiral illustrated here is composed of eighteen symmetric networks, from the smallest, a triangle, to the largest, which is formed by 190 line segments. These networks are illustrations of complete graphs: A complete graph on n vertices consists of all possible connections (edges) among pairs of vertices, and is denoted Kn. Among n vertices there are n(n-1)/2 such possible edges. Thus the smallest network shown, the triangle, represents K3; the next smallest network, having 6 edges, represents K4; then comes K5 with 10 edges; etc. up through K20 with 190 edges.

In 1972-73 I was studying graph theory, and decided it could be interesting to illustrate an array of basic graphs in a spiral design. I chose to have the size of the networks increase as the number of vertices increases, in such a way that each network has the same apparent visual density. For the underlying curve I used an Archimedean spiral (http://en.wikipedia.org/wiki/Archimedean_spiral), and located the center of each network on this curve. I also imposed a slight gap between adjacent networks, rather than having them touch. Satisfying these requirements, together with some other more recondite constraints, involved a considerable amount of analysis and experimentation, almost all of it purely computational. During the design stage, there were no more than two or three trial drawings because in that era it was very time consuming to produce such pictures. In contrast, I carried out repeated numerical experimentation in order to obtain locations for the centers of the networks and radii for them which were likely to yield the image I had in mind.

This design was drawn in 1973 by fountain pen on a Calcomp drum plotter, driven by an IBM 360/75 computer. The result was then photo-reversed and reduced by fifty percent before being lithographed in black ink on white paper. (This year it was reprinted in an improved version from the 1973 negative.) The computing and drawing were carried out at the University of Waterloo Computing Centre. I am grateful to Amherst College for support during my sabbatical leave in 1972-73 and to the Department of Combinatorics & Optimization of the University of Waterloo for providing me with a visiting appointment that year. Thanks also are due to the staff of the Computing Centre at Waterloo, who taught me to use their facilities." (from Starr, 2008)

Tecumseh is in the permanent collection of the Kunsthalle Bremen.