User blog:Koinotely/Theodore Kaczynski's Boundary Functions (1967) and Self-Reflections in a Frustrated Virtual Community

"We think of personal space as something that belongs entirely to ourselves. However, Boundary Functions shows us that personal space exists only in relation to others and changes without our control.

Boundary Functions is a set of lines projected from overhead onto the floor, dividing people in the gallery from one another. When there is one person on its floor, there is no response. When two are present, a single line cuts between them bisecting the floor and dynamically changing as they move. With more than two people, the floor divides into cellular regions, each with the quality that all space within it is closer to the person inside than any one else.

The regions surrounding each person are referred to as Voronoi diagrams. These diagrams are widely used in diverse fields and spontaneously occur at all scales of nature. In anthropology and geography they describe patterns of human settlement; in biology, the patterns of animal dominance and plant competition; in chemistry the packing of atoms into crystals; in astronomy the influence of gravity on stars; in marketing the strategic placement of chain stores; in robotics path planning; and in computer science the solution to closest-point problems. The diagrams represent as strong a connection between mathematics and nature as the constants e or pi.

By projecting the diagram, the invisible relationships between individuals and the space between them become visible and dynamic. The intangible notion of personal space and the line that always exists between you and another becomes concrete. The installation doesn’t function at all with one person, as it requires a physical relationship to someone else. In this way Boundary Functions is a reversal of the lonely self-reflection of virtual reality, or the frustration of virtual communities: here is a virtual space that can only exist with more than one person, in physical space.

The title, Boundary Functions, refers to Theodore Kaczynski's 1967 University of Michigan PhD thesis. Better known as the Unabomber, Kaczynski is a pathological example of the conflict between the individual and society: engaging with an imperfect world versus an individual solitude uncompromised by the presence of others. The thesis itself is an example of the implicit antisocial quality of some scientific discourse, mired in language and symbols that are impenetrable to the vast majority of society. In this installation, a mathematical abstraction is made instantly knowable by dynamic visual representation."

https://www.snibbe.com/projects/interactive/boundaryfunctions/

= Boundary functions = https://archive.org/details/KaczynskiBoundaryFunctions/mode/2up

"The simplest way to understand a small piece of Kaczynski’s mathematical career is to picture a situation similar to one he described in a paper titled, “On a Boundary Property of Continuous Functions.” It appeared in a 1966 issue of the Michigan Math Journal and was based on part of his dissertation work.

The paper considered a circular region. Think of it as a massive Frisbee, one that might be deformed into a landscape of mountains and valleys by twisting it, pushing through it to make bulges--anything, as long as it remains circular.

For some landscapes, there will be places on the disk’s edge that Kaczynski called points of curvilinear convergence. At those points, a person walking though the Frisbee landscape could find a path that gets flatter as he nears the edge.

In a completely flat landscape, that set would include any path. So every point on the circle would be a point of curvilinear convergence.

But in other landscapes there could be no such points. For example, the circle could contain a landscape of concentric hills, all the same height, separated by level valleys. That kind of rugged landscape might not offer any flattening approaches to the Frisbee’s edge.

“You’d always be going up the hill and across the valley and back up again,” said Pennsylvania State University mathematics professor Donald Rung.

What Kaczynski did, greatly simplified, was determine the general rules for the properties of sets of points of curvilinear convergence. Some of those rules were not the sort of thing even a mathematician would expect.

“The surprising thing is you can say anything at all,” said Peter Duren, a mathematics professor who was on the committee that considered Kaczynski’s dissertation at Michigan. “He was really an unusual student.”

Others are less impressed. Rung, who reviewed the 1966 paper before it was published but has never met Kaczynski, said that the work “proved some nice theorems,” but the paper by itself didn’t constitute a brilliant career."

https://www.latimes.com/archives/la-xpm-1996-07-21-mn-26363-story.html