Design of an Experiment Using Fourier Transform

We propose a method for analyzing and correlating ordered sets of icons by which we may construct a paradigm of analogical forms.

The first stage is the construction of a data base in the form of optical transforms of iconic images that have a recognized order in a cultural context--i. e.,the set of 94 Nyingma drawings of Buddha, the cards of the Tarot deck, etc. Subsequent stages entail Fourier analysis of the transforms, and the conducting of experiments in which subjects are asked to order the transforms with or without reference to the original images. We would expect to find significant correlation between the results of the experiment and the traditional ordering of the sets.

We expect to generate a model useful for translation of cultural teachings, split-brain and artificial intelligence research, computer translation between iconic and linear languages, and understanding the mechanisms of quantum information transfer (ESP).

Data Base

We project a laser through a 35mm slide and a lens of long focal length. Just in front of the focal point of the lens, we make a new photographic record, a grainy pattern which will be the two-dimensional Fourier transform of the image on the slide.

Images suitable for study, in addition to the Nyingma icons and the Tarot, include:

• Hexagrams of the I Ching               

• Musical scores                              

• Ehrenfried Pfeiffer's plant patterns

• Yantras                                

• Tangrams

• Children's drawing sequences

• Knots

• Letters of the alphabet & Arabic/Celtic/designs

• Rohrschach blots

• Consequences of the Laws of Form

Many Eastern images are deliberately designed to affect the consciousness of the viewer; in Western terms, different patterns are processed differently by the brain*. Experiments performed with the icons and the optical transforms may shed some light on the nature of consciousness and on how iconic data is processed.

We propose a series of tests in which subjects are asked to match icons and transforms, or to order a set of transforms. Subjects for some of these experiments might be drawn from a population accustomed to right-brain processing: artists, designers, actors, children. The results can be compared with those from tests given to scientists and practical men and women; we would expect to find significant differences. The optical transforms would also be useful in ESP ability; senders could transmit transforms and receivers tested for the matching image.

Analysis

Fourier analysis of the optical transforms will be of importance in showing the deep mathematical structure  manifested in cultural forms. Thus may the temporal be related to the universal. How we may interpret this relation may be elucidated by reference to Laws of Form, the calculus of indications revealed by G. Spencer Brown. Since the mathematics of Laws of Form is about unchanging archetypes, any existential  interpretation conceals far more than it reveals. We are reduced to particularizing a general state.  

Laws of Form, however is about and reflects in its own structure the relation of form to content,and is thus valuable in illustrating this relation much as i, the imaginary number, is useful in constructing oscillating electronic circuits because i is itself an oscillation. From contemplation of formal states, whose relations are given by Laws of Form, the right brain perceives the pattern from which the left brain can construct an acausal paradigm of how object and image, figure and ground, are in fact related. Here, by crossing out of the state, we arrive at an elegant answer to the chicken-or-egg question.

We may now state the value of the project in more general terms: we present archetypal patterns in such a way that they are recognizable from within a given discipline.

"The internal relation by which a series is ordered is equivalent to the operation that produces one term from another."
--Tractatus 5.232

The equivalence applies to the image and its optical transform, as well as to the members of a set of images or transforms. The image and the transform compare as the marked state and the unmarked state of G. Spencer Brown's calculus of indications, Laws of Form. The ordered sets are arrangements in the calculus. (Call the form of a number of tokens considered with regard to one another ((that is to say, considered in the same form))an arrangement--LOF., p. 4).

“If there were a law of causality, it might be put in the following way: There are laws of nature. But of course that cannot be said; (sich zeigen) it makes itself  manifest.”
--Tractatus 6.36
”And anyway, is it possible that in logic I should have to deal with forms that I can invent? What I have to deal with must be that which makes it possible for me to invent them.”              
--Tractatus 5.555

Gestaltists talk about laws of nature as natural rhythms, such as contact or withdrawal. There is a hint of structure that is unchanging, no matter how we may perceive it. There is nothing in Gestalt theory to indicate just why contact or withdrawal is experienced.

There is a postulate that existential choice is possible, together with assurance that "the organism" (meaning that which organizes) will somehow regulate itself, meaning obey its own rules. Why questions have a bad name in Gestalt therapy. We are discouraged from asking for reasons. We can even offer a reason why this should be so: because such questions encourage thinking rather than feeling, and frequently mask an accusation that some rule, spoken or unspoken, has been broken. Nevertheless, in constructing a theory of Gestalt, we may find it permissible to ask why it is, that the theory should look as it does look, and not some other way--why, for instance, Fritz should have distinguished five "layers" of personality, and not three, or twenty-two or sixty-four. He found something containing five parts in a particular relationships (continence). He found a structure.

The internal relation of a series may be seen in the operation by which we produce one term of the series from the next, the transformation rules between terms. In mathematics, the member of the binomial expansion with five terms looks like this:

(a+b)⁴ = a⁴ + 4a³b² + 6a²b² + 4ab³ + b⁴

How do we obtain "4a³b²" from "a⁴"? These terms are, like Fritz's layers, nodal points in the process that is expressed as a continuum as (a + b)⁴. Each term is a collection:

a⁴ = aaaa
4a³b = aaab + aaba + abaa + baaa
6a²b² = aabb + abab + baab + baba + bbaa + abba
4ab³ = abbb + babb + bbab + bbba
b⁴ = bbbb

We may collect the terms differently and express the expansion this way:

(a + b)⁴ = aaaa + aaab + aaba + aabb + abaa + abab + abba + abbb + baaa + baab + baba + babb + bbaa + bbab + bbba + bbbb.

In this arrangement we can see a simple transformation rule. If the rightmost letter is a, change it to b. If the rightmost letter is b, change it to a, and repeat the operation on the next letter to the left until the process ends. The transformation process is isomorphic to binary addition, where the transformation rule is: “starting with zero, add one to  each term to get the next." By this rule, the 17th term in the series, in this modulo 16 arithmetic, would be aaaa again. By collecting the 16 terms into groups with like numbers of a's and b's, we obtain the five-term expression, which we see is not in its simplest form. There is no indication that Fritz's series of terms is in its simplest form either, and the likelihood is that it is not. We may however see something of the relations of the five groups by reference to another paradigm that will generate the coefficients of the expansion, 1, 4, 6, 4, 1. This is the fifth line in the number arrangement known as Pascal's triangle, after the mathematician/theologian who discovered it.

Each number is generated by adding pairs of numbers in the line above. In this case the operator is a whole matrix, itself expressible as successive powers of the number 11. Why the number 11 should have these strange powers is a function of the rules of arithmetic, which define how we may combine two quantities.

We seem to be a long way from a theory of personality. And yet we can see in the mathematics the formal way in which we can combine two quantities, such as figure and ground, so as to produce a series of five terms that represents a continuum. The four degrees of the equation may be mapped into the formal categories of existence, archetype, indication, and form. When we depart from form into Void, and move to a fifth-degree equation, we are no longer able, due to the curious aspect of the number five mentioned above, to obtain a value for any of given variable. So in this sense the five terms are a kind of natural limit between a discrete element and a continuum, beyond which we can no longer recognize where we are in the form.

The question of fifth-degree equations was long of great interest in mathematics, it being assumed that eventually someone would learn how to solve them until Galois proved that it was impossible. The branch of mathematics discovered by Galois, in which solutions may be approximated, is called the Theory of Groups.

Kurt von Meier
Walter (Clifford) Barney
Circa 1976