Wednesday, April 25, 2012

Hypercubes: Strange & Loopy

Abstract: Hypercubes, or tesseracts, are proposed as instances of, or equivalent to "strange loops," self-referential, paradoxical constructs discussed by Douglas Hofstedter in Gödel, Escher, Bach (GEB).

Strange loops are often characterized as "level-crossing feedback loops" that inexplicably create cycles from hierarchies.  On traversing the hierarchy in an apparently monotonic manner, one is returned to the origin, often signaled by a change in context.

 geometric hierarchy
We begin with a geometric hierarchy of points, lines, and areas; these give rise to volume. Points or vertices in space give rise to line segments or edges.  Connecting line segments end-to-end to form a closed figure yields area. Areas connected by their edges can enclose space (think "soccer ball").

Some form of subspace, however, is presumed to permit the very existence of points.  The very location of a point begs the question, "where?"  What is the position of these points?  If points are to be assigned definite locations, axes are also presumed to exist.  In the graphic at upper right, the colored arrows denote the axes of extension/projection.

 "Seed, Tree, & Fruit"
Once space, itself a "container," is enclosed, the strange loop manifests .  A tesseract may be produced by retracing our sample strange loop and substituting cubes for points.  Descriptive rhetoric lacks the force of mathematical argument, but compensates for this with  simplicity and accessibility.

The crux of GEB is Hofstedter's proposition that strange loops are the prime material of consciousness itself.