The geometry of self-reference (and thus the difference between the "map" and "journal") can be seen easily through the analogy of the "planet ij," a sphere occupied by a single kingdom which expands over the planet's homogeneous surface.

As the kingdom expands, it enlarges its wall by adding stones to a new construction. With every expansion, new stones are required. The wall, and the space containing it, operate according to the consistent rules of ordinary logic: a larger wall requires more stones.

At the point where the wall equals the "great circle" circumference of the planet, however, something strange happens. Every new wall past this point requires not more stones but fewer. The residents of the kingdom cannot explain this using a "map logic," but their journal experience continues to produce these "irrational" results at an accelerating rate.

Finally, the kingdom expands to enclose all but a small circular territory. It is evident that, within this new logic, 'ij', a flip has taken place. What was formally the "inside" of the kingdom now looks more like an outside. What was originally the vast expanse of territory "outside" the kingdom now looks like an inside.

diagraming the planet ij

The expansion project viewed from "above" quickly reveals the problem. The points AC are at the same "level" (we could imagine an equator connecting them) while B is at the "pole."

A traveler moving from A to B experiences curvature equally, but it is not until point B that the contradiction is encountered in an obvious way. A wall built at this point would "flip" from requiring more stones to requiring less. Similarly, a traveler starting out from C would experience the same events but in a mirror sequence. A view taking only the end points A and C into account could "ignore" the polar experience and just think of A and C as antipodes – opposite versions of the same thing.

A more accurate view would show that there are a few significant points along the way, however. At A or C, no contradiction is apparent. At B, the contradiction is realized as a slowdown in the need for new stones turns into stones left over from each expansion. Halfway between A and B, also halfway between B and C, there is an exact balance of the "ordinary" view (as if the space were flat) and the revised view of the space as curved.




We can demonstrate this metaphorically by looking down on B as if it were the "north pole" and imagining arrows drawn at the "equator," all facing south. At a position 90º away, the evidence of curvature is at its maximum because we are also 90º "vertical" to the line connecting A and C directly. Taking 'ij' as the effect of curvature and '+1, -1' as the distance between A and C without the knowledge of curvature, i and j are in "perfect balance" at a 45º position.

The view halfway between an imaginary and a transitive ("un-curved") view is the most ambiguous. It can see B in its uncurved role and also as a polar point. B oscillates with B-cross to produce the square-wave phenomenon, ij, or .

the indispensable imaginary dimension

Just as the imaginary numbers in mathematics enable operations that would otherwise be difficult or impossible, the imaginary dimension (called "curvature" here) gives the residents of the planet ij an appreciation of how their map experience is connected to their journal experience.

Mathematically, it is rather easy to demonstrate how this happens. In a second-degree equation, x-squared plus one equals zero (see left), a solution can be expressed in two ways. Either an imaginary number (i) can be invented to stand for "the square root of minus 1," or the equation can be solved by substituting first +1 and then -1 to find that the value gets "inverted." Clearly, this is an example of the "inverter switch" because x uses itself in its own definition.

Plotting the values of the possible solutions creates a diagram similar to that of the "polar" view of B. The horizontal or 'x' axis stands for +1 and -1, and the vertical or 'y' axis stands for the values of ± i. Note that the negative term -i has been changed to j in order to describe B's oscillating behavior as "ij."

One usually assumes that the mathematical use of "imaginary numbers" has nothing to do with the artistic use of the imagination in everyday life, but boundary language points to a clear and solid connection. Just as imaginary numbers facilitate transactions that are "unimaginable" in rationalistic terms, the artistic imagination plays out relationships that are not and possibly can not be explicitly visible in everyday life. That there is a connection between these two uses of the "impossible" – and that they may even be able to share a common system of notation – is an exciting prospect.

fractals and self-reference

By now, even people with little or no interest in mathematics have heard about catastrophe theory and the strange notions of improbable connectivity, inspired largely by the theoretical advances of Benoit Mandelbrot. Fractals and "self-similar forms" involve recursion and have much to offer the study of boundary language. We will limit our comments here to show how Spencer-Brown's calculus is able to link such apparently different topics as fractals, self-reference, irrational measurements (intransitivity) and imaginary constructions in art and elsewhere.

Borrowing from Louis H. Kauffman's "Some Notes on Teaching Boolean Algebra," we can look at the Golden Rectangle's Fibonacci series as a branching structure that repeats its pattern at each successive level.

Using the notion of the inverter switch, we can turn the equation, Ø=1+1/Ø into a tree structure. It's also possible to describe the same series in Spencer-Brown's form: . is a similar form, but it lacks the external cross. The calculus shows that self-similarity takes the form of a series of nested crosses. The number of crosses, odd or even, changes only the "entry point" at which one metaphorically "jumps into" the self-replicating sequence. The Fibonacci series, with its extra cross, is thus structurally equivalent to the self-similar "polar point" of the planet ij, .

the representation in front of the representation

This seemingly trivial point – that adding an extra cross does not change the fundamental structure of the form – is actually a matter of enormous importance. Particularly in works of art, the added margin, which should invert the value of the space but doesn't (one moves from inside to inside) because the movement has a self-referential function, is not an anomaly but, rather, an artistic commonplace. We shall explore it through several examples: Picasso's 1907 painting, "Les Desmoiselles d'Avignon," where a figure holds back a curtain and thus introduces a "painting within a painting"; Alfred Hitchcock's Rear Window, where the apartment of a photographer immobilized with a broken leg becomes the analog of the audience's viewing space; and The Wizard of Oz, where Dorothy returns from "away" to a home that is no longer a home, and her estranged bedroom becomes a viewing chamber into the fantastic space of the tornado.

These and other samples of self-reference (the theme in art is called "iconicity") show that self-reference is both fundamental and informative. Moreover, its formations follow regular rules that permit us to compare them if we can synchronize their uses of boundaries. The process of synchronization involves the diagram known as the "bolagram" (boundary language diagram), which attempts through an economy of terms and processes to embrace a broad range of phenomena in culture, art, and science.

© 2012, Donald Kunze, all rights reserved