Consistency and Completeness in Visual Images

John Heuser

     In Chapter 4 of Hoffstadter’s Godel, Escher, Bach: The Eternal Golden Braid, he continues his examination of how we think and perceive things in terms of systems, and their relationship with the "real world". He goes into the history of Euclidian vs. non-Euclidian geometries, and points out that their differences begin in a departure within the use of the straight line in the parallel postulate . Basically, this postulate is defined by straight lines in terms of pairs of never-intersecting lines. If we reject a restricted idea of "straightness", we can jump into spaces which are defined by points on parabolas, or spaces whose points define hyperbolas.

     These geometric systems, while having fundamental differences in the shapes of their graphic representations, are still grounded in the commonality of four other postulates, which remain unchanged. Therefore it's not too surprising that we have little trouble with seeing consistency in the value of thinking of lines as being perfectly straight when laying out a building foundation, while using hyperbolic functions in determining optimum load vs. span relationships in designing its trusses. What happens to buildings when we depart from laws which we have internalized as part of "normal" experience?

     The Escher print, “Relativity” ( fig. 22 in the book), is given as an illustration of a hypothetical world which violates our sense of consistency in terms of the laws of gravity, our spatial sense of "up and down", and our sense which comes from living in a world full of normal buildings. Hofstadter describes the progression of our interpreting the picture, by pointing out that we first seek familiar frames of reference, before realizing that all the staircases are juxtaposed in ways that intertwine impossibly; if one of the stairs was "correct" in the real-world sense, all the others would be extremely dangerous for whoever walked on them.

     I am intrigued by the emotional impact caused by this picture, which is so disturbing to my "everything-should-be-straight-up-and-down" body sense. A lot of the Escher prints are disturbing in this way: there is a sense of danger, or vertigo. He wants to shake us up, wake us out of our Euclidian dreams. If I surrender to the danger, and throw myself off of one of Escher's precipices, perhaps I enter into a more plastic world of creativity.

     I am reminded of the art of the Surrealists, who described affective states as being as important as the artifact being created. They described a dreamlike state during painting as being essential for the surreal experience, and for the painting of surreal works. And many of their works reveal startling juxtapositions of objects, or dreamlike distortions of things, such as in Dali's somewhat spooky, well-known liquid clocks. These works evoke similar emotional impacts to Escher's.

     I propose that the experience of these artists, as well as our experience as their audience, is one of increased creativity that results from challenging our habitual sensory systems. As an experiment, I have tried a visual image which I feel contains elements which are somehow inconsistent, disturbing, or contradictory, in hopes of illustrating the process of creating new, disturbing understandings out of an hierarchy of familiar structures (See Figure: Mr. Snake Punctures the Real World ).

     In the picture, Mr Snake's tail escapes from the frame, and if you hold him up and look through the hole, you can see that his fangs are puncturing the scene. If you wish, you can hold him up so he is biting the head of a friend (or enemy).

     The tail disturbs the familiar hierarchy of "pictures inside frames". And the real-world view through the hole is disturbing to "pictures inside frames inside the real world". Not only that, but the hole can be held up so that the snake bites any visible object, including paradoxical Escher prints, or differential equations. In any case, it is evident that one path to creativity is to disturb habitual perceptive hierarchies, and I think this is an important theme for Hoffstadter.

Figure & Ground

John Heuser

     Hofstadter begins Chapter 3, "Figure and Ground" , with a discussion of prime vs. composite whole numbers. He wants us to see similarities among Escher's birds in the 1955 lithograph Liberation , figure and ground as understood by artists, and capturing compositeness and primeness in a typographical system.

     Hofstadter clarifies what he means by typographical systems by listing the kinds of operations permitted within them. The list describes ways that we manipulated the MIU- and pq- systems in previous chapters. The six permitted operations are:

1. reading and recognizing the symbols used

2. writing them

3. copying them

4. erasing them

5. recognizing whether any of the symbols are the same

6. listing and reference to previously generated theorems

     This list of operations would be rudimentary for a machine like my personal computer. These rudiments could be compared to components of the bottom row of triangles in Liberation . The print depicts a roll of paper being unrolled from the bottom up. As the eye moves up along this unrolling scroll, it first finds rows of black and white unilateral triangles which waver and distort until, by the fourth row, they assume the shapes of black and white birds.

     The lithograph asks us to consider basic elements of black and white areas, and same-length lines defined by the edges of equilateral triangles.  This could be analogous to the elements t, q, and - in the tq- system. The sense of upward direction in the print is determined by rules that govern our sense of space, and our sense that birds themselves fly upwardly. Forming theorems in the tq- system is also directional toward increase by its  rule of inference which Hofstadter defines to make a collection of theorems isomorphic to multiplication tables.

     In Liberation, the fifth row up forms birds whose negative spaces, or ground, are also birds. The fourth row are figures and grounds which are proto-birds, while in the sixth, the black and white birds begin to separate so that there are areas which are "true" ground (presumably sky). Subsequent rows of birds are shown in increasingly variable flight patterns. A similar movement is seen in the progression from the rudimentary typographical operations within a simple system such as tq-, to the intelligence-mode leap to identifying the tq- system as a code for multiplying integers.

     We want to leave the system when we intuit answers, and propose that we could define primes simply by first defining composites, and calling everything else primes. Unfortunately, this violates the purpose of formal systems by violating the Requirement of Formality, which only enables us to generate strings of symbols by defined rules. Unlike artists, whose ground is automatically defined by their subject-figures, we cannot generate formal  system theorems from something just by observing that they are "not-theorems" of a known system.

     The birds flying around in the top of Liberation , with unrecognizable grounds or negative spaces around them, are like theorems of formal systems for which there are no typographical decision procedures, which Hofstadter concludes exist on page 72. His reasoning is that for some systems, nontheorems cannot be inferred by simply erasing all theorems, so it follows that, for these, typographical decision procedures do not exist. This is comparable to artistic figures, whose grounds are not always recognizable shapes, but are often just left-over negative space.

     In actuality, artists are formally trained in art schools to attend to negative spaces as being meaningful design elements. In fact, the treatment of negative spaces can be a measure identifying the difference between naive and experienced artists. One training technique in drawing courses is to instruct the students to ignore the figure, and fill in only the negative spaces with solid color. Often, the result is more proportionally accurate than if only the figure is concentrated upon. "Ground" for artists has design meanings which may escape the attention of the untrained eye. I'm bringing this up as an aside, in response to Hoffstadter’s labeling negative spaces as meaningless when they do not represent recognizable objects. When he concludes, "There exist formal systems for which there is no typographical decision procedure", he is asking us to visualize hypothetical figures with meaningless grounds. My response would be that if they would seem meaningless, perhaps it would be due rather to our inability to recognize the meaning, rather than an objective fact. At any rate, I feel as though I have stretched the comparison between artistic ground and formal system- generated figure and ground as thin as I can.

     Both primes and composites, thankfully, can be generated by formal systems, as is demonstrated by the last section in the chapter, "Primes as Figure Rather than Ground" (p.73, 74). In this section, we can imagine a picture of an array of composites and primes totally filled up with figures and with grounds which are recognizably meaningful.