The Parallax theorem, as an investigation originated through a physical experience, I found myself within the collapsed frame of the city of São Paulo, 22 million inhabitants, no grid, de-centralised planning, a collision of ethnic immigration, a crash of diverse architectural constructions and personal interest. At that moment my senses were disorientated and my architectural historical reference toppled. São Paulo is a place where the individual has the possibility to strive and move un-noticed through the metropolis as an anarchist. This city is a great example of the parallax as a dimentional metropolitan construction constantly defying the central established order of the continuum historical map of city building. There is no past and no future, it is the present.

This metropolitan condition where the depth of field had collapsed is extremely disorientating, at the same time stimulating and exhilarating. I suddenly discovered my perceptual observation and baggage not having the distance or a solo position to view something as a whole. If there was once an archetype of metropolis township it actually alluded, this is now unrecognizable and inarticulate: there is no cosmology or ideology that unifies and interprets such effects. Under such conditions, the body as we traditionally conceptualise, perceive and represent it, its way of moving and orienting, is in this metropolis, contingent, provisional and epiphenomenon. This experience and circumstance initiated a series of investigations in how to think, articulate and represent another ordering system which is not based from a central position of view or a perspectual chart projection.

Part 1 :

“How We See Straight Lines”. by John R. Platt Scientific American, June 1960

On first thought it would seem quite impossible for human beings to see whether a line is straight or not. Our visual mechanism is apparently unsuited to the task. Consider what happens when we look at a straight line: Its image falls on the curved surface of the retina at the back of the eye. Here the light pattern stimulates the tiny receptor elements-rods and cones- that lie underneath it, and they fire off a volley of electrical signals to vaguely defined regions on both sides of the cortex of the brain. Surely the cerebral pattern is not “straight”. It is often argued, however, that the brain somehow knows the location of the rods and cones whose stimulation gave rise to the sensation. If these particular rods and cones lie along an appropriate curve, the object they record is a straight line.

Let us examine this reasoning a little more closely. The retina is a layer of tissue about one inch square, containing something like 10 million receptor cells arranged in a closely packed mosaic. How can the brain know where each cell is? Hermann von Helmholtz, the great pioneer in the theory of vision, thought the knowledge might be provided by specific “local signs”-possibly chemical in nature-from cells at different positions. His idea has recently been confirmed in some elegant experiments performed by Jerome Y. Lettvin and his colleagues at M.I.T. When they cut a frog’s optic nerve and allowed it to regenerate, they found that the neuron from each point of the retina grew back to its proper point in the brain. But this specificity can hardly be indefinitely fine. Seen under a microscope, the mosaic of retinal cells looks random, and, as living tissue, it has been subject to all the accidents and irregularities of biological growth. Surely the cells must be subject to some microscopic uncertainty of location. Thus a line that appears straight to one man should appear full of little wiggles to his twin brother. The amplitude of the wiggles would indicate the limits of accuracy of the genetic or local sign-specification.

Yet the fact is that we can tell when a line is straight, and none of us ever sees any such wiggles. Our actual precision in certain visual observations is fantastic. Our vernier acuity, or ability to detect a lateral break in a straight line, is about two seconds of arc. This corresponds to a distance of a little more than a hundred thousandth of a centimetre on the retina, about a 30th of the diameter of a cone cell! Even in mechanical construction this precision is almost impossible; a hundred thousandth of a centimetre cannot be measured in the finest machine shops except by optical methods. In a biological system such as the eye the location of every tissue cell to such an accuracy, 30 times finer than the size of the cell, is quite unbelievable. I puzzled over this paradox for a long time, until I finally began to wonder if we were not looking at the problem in the wrong way in emphasizing the precise location of the individual cells. we were unconsciously assuming that the brain can somehow examine its associated retina, as if through an external microscope, and locate each of the rods or cones in space.

Thinking about the microscope fallacy, as it might be termed, led me to wonder whether there could not be some high-precision physical method that would enable a system consisting of 10 million elements to make acute discriminations without knowing exactly where its individual sensory elements were located. I finally found one, a method that I call functional geometry. As its name implies, the method generates spatial relations in the course of the normal functioning of the visual system rather than through the static, point-by-point location of images.
The essence of this functioning is motion, or scanning. Several years ago experimenters discovered that, in order to keep a static pattern steadily in view over even a short period of time, a person must continuously shift his eyes in tiny scanning motions. If he does not, the image fades away. I suggest that the same scanning can provide the sense of straightness.
The basic idea is a follows. Imagine the image of a scene-any scene-projected on the retina. The arrangement of light and dark areas stimulates a particular set of rod and cone cells, which then transmit a specific array of signals to the brain. If the eye scans the scene, moving so as to shift the image slightly, a somewhat different set of receptors is stimulated, and the signal array changes accordingly. But suppose the scene is a straight line, and the scanning is parallel to the line. Then the motion does not change the set of stimulated receptors, and the signal array remains constant. This constancy, or “self-congruence,” after displacement is what the brains recognizes as straightness.

Evidently an ability to detect the sameness of an array is about the weakest demand one could make of a communication network, however it may operate. Moreover, the perception can be made without knowledge of where the individual receptor-cells are located. All that is necessary is an external object that is congruent to itself under a displacement such as the eye can carry out. A straight line fulfils the condition. So do parallel lines. A crooked line, or a set of nonparallel lines, does not.

One of the important features of the method is that the images of these lines on the retina or on the cortex can be as crooked as you please without destroying the self-congruence; all that is required is that the image fall on the same locus after displacement, and it makes no difference what that locus is. The discrimination is therefore for straightness or parallelism in the external field. Clearly the brain does not know-and, if it uses functional geometry, does not need to know-how the image on its cortex would look when observed from the outside. The fact that we see self-congruence in the external field is what makes straightness. We do not see objects, but relationships; and the relationships are public. This is a point of considerable importance in linguistics and theories of knowledge.

Another important feature of the self-congruent method of perceiving patterns is that it is not affected by damage or loss of receptor cells, or blind spots. An array of signals can be the same after displacement as before, regardless of what cells have high or low sensitivity. This means we do not have to assume uniform sensitivity in all the cells of the eye. And it is consistent with the fact that we do indeed perceive patterns as passing straight across our blind spots.
In addition to scanning back and forth along a line, our eyes may move transversely across it. By making a series of such perpendicular passes at various points along a line, and by comparing the times at which signals come from different receptors, we can also form a judgment of straightness. here we are limited only by the time taken in observation. The longer the time, the closer the check on possible discrepancies between time-sequences at various points along the line. It is probably this mechanism by which we make visual judgments of the highest acuity.

Either type of scanning of straight lines, or of parallels, is carried out by combinations of two rotations of the eye-ball: around a horizontal axis (looking up or down) and around a vertical axis (looking left or right). However, our eyes are capable of still another motion though it is sharply limited: rotation around a longitudinal axis pointing along the line of sight (see bottom of illustration on page 128). (To observe this rotation, closely examine a marking on the iris of your eye in the mirror as you tip your head from side to side). By adding a component of this twisting motion we can scan along a curve, and, if it has constant curvature, keep the image over the same set of receptors on the retina. Thus arcs of circles exhibit the same sort of self-congruence as straight lines, and concentric arcs the same sort of self-congruence as parallels.
As a matter of fact, it is not easy to judge whether a gentle arc is curved or straight. Uniformity of curvature (including the zero curvature of a straight line) is more readily perceived than is straightness or curvedness. If the curvature is sharp enough, however, we probably become aware of it through muscular cues arising out of the twisting motion.

Both the method of self-congruence and the approach to greater accuracy through repeated trials are central concepts in high-precision optical work. Every amateur who has made his own telescope mirror knows that spherical and plane mirrors and precision screws can be brought virtually to perfection by being polished with a matching tool until they are self-congruent under lateral or rotational displacement. Within a finite time the error can be made less than any preassigned value.

In biology the principle of self-congruence generates perfectly helical elbow-joints and spherical sockets. The spheres are self-centering; they know nothing about the point centers and fixed radii of Euclidian geometry. This suggests a new approach to the study of geometry. It might be more natural to start not with points, distances, lines and coordinates, but with self-congruences, which are biologically more primitive. Do we actually apply functional geometry to every judgment of straightness (and other patterns)? The experienced adult eye may not need to scan every new line afresh to determine its approximate straightness. Possibly certain receptors on the retina have been associated so often in past straight-line perceptions that when these elements are excited again and give off the same chorus of signals, we are satisfied of the straightness of the new object without further scanning. It is self-congruence to an old straight line, with a long time-delay. In a sense, the pattern has been learned.

If this is the mechanism of pattern perception, then we should expect to find that an infant or a visually naive adult (for example, a person who has had congenital cataracts removed) would require long scanning and study to determine the straightness of a line. Such, in fact, appears to be the case. The finding is consistent not only with the theory of perception developed here, but also with the doctrine of D.O. Hebb at McGill University. They hold that perceptual organization of even such apparently primitive relationships as straightness or triangularity is acquired-learned-only through visual experience.

Arthropods (such as insects and spiders) can learn almost nothing, and birds can learn only certain things. It follows that much, if not all, of their pattern-perceiving system must be pre-located and pre-connected, determined by genetic information alone. Pattern perceiving that involves learning, perhaps using methods such as functional geometry, is a way of escaping this genetic limitation. Such an escape is obviously needed for really big brain with more inputs. This suggests that pattern learning may be the faculty that grew most rapidly in the sudden evolutionary expansion of our brain and cortical capacity in the last few hundred thousand years.

Perhaps the most noteworthy feature of functional geometry with a mosaic system is that it necessarily picks out certain patterns as fundamental or primitive. A mathematician of curved spaces might say that an S-curve in one curved space is a straight line in another, and that these are equally good descriptions of the line. But a functional mosaic will accept as straight only those Euclidean lines that satisfy self-congruence under displacement. Thus straightness is a primitive and unique category of perception for all mosaic systems. So is parallelism, concentricity and so on. It is interesting to note that the various relationships belong to the “synthetic a priori” categories of Immanuel Kant -unique categories that impose themselves on all minds regardless of particular experiences and comparisons.

I suggest that there is only a small number of unique symmetry categories for a visual mosaic receptors, and that they are determined by the three possible rotations of the eyeball (see table below). When the rotations are continuous, we get straightness, parallelism and the like. When the eye moves in discrete jumps, it perceives relationships such as equidistance, congruence and the equality of angles. On this view every visual pattern-relationship that can be perceived is some combination of the primitive elements.

It would be interesting to try to construct artificial mosaic receptors, complete with scanning motions, that might be able to make discriminations similar to those our eyes make. Evidently any such system would have to be able to learn. The receiving network would somehow have to grow or to establish new connections guided by experience. If we could design such a system, it might teach us far more than we now know about the human eye and brain organize external information.

Part 2 :

"Parallax, n. Apparent displacement, or difference in the apparent position, of an object, caused by actual change (or difference) of position of the point of observation." from The Oxford English Dictionary.

"Parallax' is the difference in direction of an object caused by a change in the position of the observer. This method in astronomy is the direct way of measuring the distance of a celestial body. At the level of theory, the measurement of the distance to an inaccessible point is elementary exercise in geometry". from The Encyclopaedia Britannica.

The parallax theorem method which I describe was mostly developed by mathematicians and astronomer in the 1600 to measure and rectify distortion of the actual measured object in space. The first a documented use, for this purpose, was by an astronomer to chart the planet Erros. This instrument was immediately appropriated by navigators to chart within precision their position on the vast ocean.

On this particular parallax theorem drawing, there are two distinct observed vantage points (vantage point A and vantage point B), these are placed on the surface of the earth in relation to the star object. For the parallax theorem to actually work one needs minimally two distinct vantage points of observation. Having the angle where these two points or other points from other observation station crosses, one can then calculate an accurate measurement from the triangular formation, lines and angles, finding a measurement without any distortions, which is one of the main purpose of the parallax, to abolish any possibility of distortionand to distinguish its position from the other.

Looking at the plan section of the brain, by shifting the position from where the lines crosses to a measurement closer to the retina, one is able to isolate what is being viewed into two distinguished views. This is where the parallax can occur and inform the brain of two unconnected information or viewed position.

In my practice I have been able to demonstrate this effect through the use of binoculars by splitting the monucular view into two distinct views. Binoculars are binary devices which speak of the tactical history of surveying terrain and the wild, and brings the two separate viewing eyes into a monocular focal plane. They also empower the observer by dislocating the subject to a closer view. The moment that these two distinct views of the same object is viewed through the binocular and the view is placed before the crossing of the lines, (see plan-section of brain), those views being distinct and without reference to each other, our brain immediately receives two distinct image signals, only similar in content, the brain’s memory and expectation wants to create one image from the new scanning actions in combination with the an already learned experience are distraught and distressed since it can’t view it monucular. By this means, an apparent displacement of reference within a field can be articulated. Unlike stereoscopic representation an object is seen by two eyes successively but so rapidly so as to form the impression of a solid. One applied use of the parallax is in the U.S.A. air force, where pilots through a series of extensive eye exercises can split their eyes and gather distinctively information during a combat air zone.

The interested in developing this particular diagram of parallax as an instrument of perception and critical record is to locate an apparent displacement of the content as it has moved off the modernist grid.

The theoretical dimension of the parallax, the measured space is the distance between what you are expected to see and what you are actually seeing.

If we were to apply the Modernist perspective model of observation as a working method, it would register only changes in proportion. Perspective is a tool that relies on aggrandizement and diminishment, with greater differences in these producing greater accuracy in locating objects in a field. Parallax, on the other hand, as an ideological and critical position can locate insignificant and inaccessible bodies by using diverse vantage points. Parallax is particularly effective as an observation method for singular objects where the depth of field has collapsed or a singular position of observation is impossible because the object has filled the frame of reference.

Our present method and structure of observation and social practice is based in the monocular system and its hierarchy. The foundation of perspective is linearity, its objective is the geometrical representation of depth on a two-dimensional medium. It assigns to the spectator of the universal theatre the place of the sovereign from which to assess the sphere of his dominion, the dimensions of his knowledge and the extent of his power, a monological concept of truth. A tool of perception and representation of ideals. Linear perspective, then, with its dependence on optical principles, symbolizes a harmonious relationship between mathematical tidiness and nothing less than God’s authority and will. The representation, is constructed according to the laws of perspective, is to set an example for moral order and human perfection.

Applying the parallax theorem as an ideological and critical position we can locate, examine and begin to understand with greater accuracy, inaccessible issues which are not apparent within monocular perspective which anchors its authority in the growth of ideal, utopic or central beliefs. Issues developed by the parallax method become democratized by diverse differences in positioning the dislocation of the central authoritarian posture characteristic of the monocular system. Thus speaking of its failure and shifting its authority.

Observed issues in question then become democratized by the requirement of various distinct positions of observation and requiring that these positions "communicate" in order to locate distortions and impossibilities. The applied metaphorical logic of the critical parallax method is a cognizant effort to reveal and promote a theoretical inquiry into the sociopolitical geographic differences particular to the a specific site.

Parallax consider the tradition of perceiving and experiencing issues through mediated ways: constructed viewpoints. Through it, one can negotiate the content of what is being considered. The constructed parallax as a perceptual apparatus, democratizes the metaphor of vision from a singular, monocular point of view (established in the Renaissance), to a multiple point of view.

The experience engages the visitor's perception, to articulate a position and posture from which site and specific social questions may arise: the contradictions inherent between beauty and the grotesque, technology and primary tool making, order and anarchy, the individual and the collective, the natural and an artificially constructed nature, single source and multiple source. At this moment one begins to examine what is in between the expected extremes.

The intent, when hierarchy is no longer the applied measured space is to reflect and expose our belief systems towards civil responsibility, the monocular system articulates its ordering codes from probabilities, what is possible, the parallax as a tool articulates the discourse of differences and impossibilities.

The monocular architectural vision is a direct representation of a singular body in space. Architectural strategies can also be made independent of the body: architecture need not always proceed as if the human body itself is being extended in potency and perception. It is evident that by applying the parallax as an instrument and ideological position that the process of reference, and the process of making architecture can respond and articulate other systems besides the body and its own history.

alexander pilis
2003.