The basic game is to wind a curve through a cubic grid, grouped in eights into larger cubes, grouped by eights into larger cubes, grouped by eights into larger cubes — exactly once entering and exiting each cube of each size.

But up to symmetry, how many such curves are possible?
This number is part of the real name of this sculpture.

We have left this calculation as a puzzle, and in the meantime, call the sculpture 1 of N.

We made a quick model out of sixty- four pieces of plumbing.

The task would be to count the ways of recursively generating such a path; the total number of words of a given length is not difficult to count, but the paths must close—the beginning must match the end.

Moreover, we needed to be able to rummage around the set of all such curves, to try to make some rational choices about the final form of the sculpture.

(Chaim’s formalism of “Regular Production Systems” are designed to capture just this sort of non-deterministic production system acting under regular constraints, arising in a wide variety of settings.)

Extensive hand-written notes and diagrams developing the structure were used to make models write code. This first set of wooden tiles consists of eight bricks, and eight clumps of three bricks, in the shape of an L. It is not difficult to enumerate all arrangements of these, up to symmetry, within a 4x4 cube. In turn, arrangements of these bricks were encoded on another set of blocks, so that the system could be explored by hand.

Each digit encoded a turn from one module to the next: 0 for none, 1 for a quarter turn, 2 for a half turn and 3 for a quarter turn the other way. Eugene made special jigs, one for each digit, so the seams could be smoothly welded. All together, each block of fifteen digits encoded a brick made up of sixteen modules. There were thirty-two bricks, each with a unique place in the sculpture. The individual pieces weren’t very uniform—next time we will use artist- grade plumbing.