This is an introduction to a step-by-step tutorial that will introduce several diagrams that lead to general 'boundary language diagram" (BoLaGRAM). This diagram is useful for analyzing the potential relationships between the main "functional" building program and other programs that can be the source of depth, cross-programming, poetic-artistic value and alternative readings. A boundary language diagram is not a prescription for a design, but it is a means of determining routes of access to values lying concealed within a design.

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Lesson 1
Lesson 2 (under construction)
Lesson 3 (under construction)
Lesson 4 (under construction)

Texts used in the lessons can be found in the index of text links.

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first, a few assumptions

Boundary language begins as a notation system that uses one sign with multiple uses and interpretations. The sign is similar to a boundary drawn around a space or set of objects, but it does not always designate the common interpretation of such boundaries. The mark () can indicate an action of crossing a boundary. In the ordinary sense, crossing over and back "cancels out" the effect

But, this is the hallmark of what we call "transitive" (rationalized) space, where a distance measured from A to B and, then, B to A, is always the same. In "intransitive space," such as a return trip that we experience as shorter or longer than the original trip out, measured distances are potentially different, and the above axiom is reversed:

In other words, going someplace and coming back leaves a residual, a surplus, that cannot be integrated within the usual symbolic systems that make records of travel, crossings, etc.

This condition is illustrated, in the tutorial, by the anecdote about an imaginary kingdom on a planet ("the planet of the idiots") with a uniform surface. As the kingdom grows, the wall surrounding it takes more stones with each expansion — until the kingdom's territory covers exactly half of the planet. At this point, the next and every following new wall will take fewer stones to build a "bigger" wall. The intransitivity of the spherical surface has to do with the experience of the builders. The surface "never changes" (it is always curved in the same degree) but at some point one effect changes in an unforeseen way.

The 'value' of the wall on the planet oscillates between stone removal and stone addition. This is called the "i-j" function, a "square wave":

A more commonplace example of this "intransitive effect" occurs in cases involving representation to an audience. During a theatrical performance, for example, the audience mentally subtracts its space from the "fictional presentation." When an actor looks out into that space, the audience doesn't think it's looking at them literally, but into a fictional space that is a part of the stage. The literal presence of something that is mentally subtracted for the sake of illusion/representation also occurs when we look at paintings, films, and photographs; when we read a book or listen to a story; when we represent some story, joke, or situation to others in casual conversation; or when we read a book. Note: this situation can be comically explored, as in Woody Allen's The Purple Rose of Cairo (1985), where the actors of a film reach an impass and implore the audience to offer suggestions. One actor "leaves" the film to interact directly with the audience.

In such situations, we use two kinds of logic at the same time. We need both versions of this "axiom" to describe the full range of experiences we have in space and time. There are many approaches ("protocols") to this situation. All approaches end by condensing the situation of apparent contradiction to a simpler form, called "recursion," " suture," "feedback," or "self-reference." Basically, this means that there is an "extra cross" present, silently, in any situation involving some act of framing, describing, quoting, etc.:

This present-but-denied "extra" boundary creates a portable "internal fold" that applies to many situations in everyday life — but to every condition in the world of art and representation. For art, this present-but-denied boundary provides the opportunity to conceal things from the audience without pulling them out of view. As in the case of the close-up magician, the audience "looks at everything" but "sees" only what the magician conditions them to see.

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the central role of 'anamorphosis'

"Anamorphosis" is (conventionally) a visual phenomenon where one image can contain other images hidden inside. To find the concealed image, typically, a special viewpoint is required. We encounter examples frequently but rarely notice them — traffic signs painted on the pavement to be visible from cars but "stretched out" when viewed by a pedestrian. Some cases just involve special points of view (lookouts, scenic look-outs, etc.) but no literal distortion, just a "privileged point of view."

One of the most famous examples of anamorphosis was employed by Hans Hölbein in the portrait known as "The Ambassadors" (1533). The blur diagonally stretching between the feet of the two figures can be seen clearly by looking along the line of the blur, near the surface of the canvas. The blur is really a skull, all the more significant because it completes a geometric design, an isosceles triangle those vertex intersects the horizon at a 27º angle. The date of the painting (1533) is 3 x 500 + 3 x 11, and the inscription on the back of the canvas gives the specific date of Good Friday. This date was particularly significant because many people of Hölbein's day thought that this would be the Apocalypse, the end of the world, and the involvement of the number 3 was important, because three (French "tres" = "very") is a "number of completion." The crucifix barely visible at the upper left of the canvas confirms this hidden code, making Golgotha, "the Place of the Skull," universally significant as the end of history.

John North's book (The Ambassadors' Secret, 2004) details this covert design and its possible meanings. For us, it is important to note that anamorphosis works at a "hinge-point" between two kinds of meaning, one based on the transitive order of representational-projective space, another based on topological transformations, such as folds, twists, and ruptures.

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topologies as maps

The notation system would make limited sense if space were free of twists, rips, folds, and anamorphic devices. But, it's not, and the primary reason it's not is the element of time. A plan or map is usually drawn as a "view from above" that creates an imaginary surface upon which values, elements, etc. are projected and symbolized. The cost of this projection is the dimension of inspection used by the map/plan reader, who gains access to this abstraction by giving up the dimension connecting his/her point of view with the representation's two-dimensional surface.

The dimension line connecting the map with the map-reader (the 'sagittal' dimension) is already a temporal dimension, because if anything it is proof that the map was made in the past and stands in an uncertain relation to the 'present reality' it pretends to represent. The temporality of the sagittal is complex. The viewer is in much the same position as the audience of a play or film. The time of the 'event' is both a kind of present, a kind of past, and a kind of future.

A boundary language map does not give up this dimension, because it is just this dimension (and others like it) that create within any scene, representation, or complex of designed or natural elements a multiform topology constructed by (at least) three distinctively different kinds of gaze.

The first gaze is the one employed when we look at a building, artwork, map, or plan without taking into account our point of view ("naive looking"). This gaze automatically obeys the rule implicit in the representation "not to notice" certain things, which can be dismissed as accidents, mistakes, noise, etc.

The second gaze ("complicit looking") is aware that the viewpoint has been conditioned by the representation and a part of it. This gaze recognizes the "devices" of invisibility but misses the full potential of its role. This is a "cynical" position because it accepts a role, just as the audience of a magician accepts that "the whole thing is a trick" but enjoys the act nonetheless.

The third type of gaze ("topological looking") sees that the first two gazes leave what "should be hidden" completely exposed and explores that fact. This gaze not only comprehends the structure of the situation; it realizes its potential and exploits it.

From "naive looking" to "topological looking" is a quick way of summarizing boundary language. This is not to "make the invisible visible" but, rather, a way of discovering the role of constructed invisibility.

This use of three gazes is related to suture theory and Jacques Lacan's "Seminar on the Purloined Letter"

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the three gazes and the 'fourth wall'

The "fourth wall" is a metaphor for the physical sets of film and theater, the wall that, by being missing, allows for the production equipment (camera, director, sound crew, etc.) which in turn place the audience "inside" the illusion of the artwork. A theatrical stage set is similar, where three sides of a room are constructed to allow the audience the illusory experience of being inside the room, although the auditorium exists at a considerably different scale.

The three kinds of gaze are constituents of all works of art, architecture included. The easiest way to formalize their effects is as a "theatrical geometry," where the relationship between the audience, the stage, and the illusions on the stage streamline the three gazes as a sequence that, in theater and film, transforms the "invisible/passive" spectator into an active participant. This final effect is called "suture" in film theory. It relates more generally to the inside-out process that converts small scale objects and events into "universes" of meaning.

In the first stage, "gaze 1," the dividing line of the proscenium imposes a rule of silence and passivity on the audience, who receives the work of art as a "naive" illusion, enjoying it without giving thought to its structure. The first stage accepts the rules of invisibility imposed by the conventions of art. Objects and even characters do not have to be literally present in order to play a part in the fiction.

At the second stage, a second kind of gaze becomes aware of the conventional tricks and structures but "cynically" goes along with the illusion. This could be compared to a parent who takes his/her child to watch a "Punch and Judy" puppet show. The parent "knows all about" the silly farce and laughs along but the child, experiencing the show for the first time, enjoys it fully, "naively" unfamiliar with the show and its traditions.

The third gaze sees what the first gaze conventionally ignores but, unlike the second gaze, realizes the opportunity to become actively involved in the structure of the artwork, which can be seen to extend to the audience in an unexpected way. This is equivalent to acknowledging the dimension (between the artwork and the audience) as a variable rather than a constant. The dimension itself can take up a more sophisticated role, as a "space of concealment" (poché). In this case, the dimension of representation can serve as a place where the audience "has already experienced" (but not noticed) things that refer to it at a point near the end of the artistic experience.

The three gazes further point to the contrasting use of metaphor and metonymy in the work of art and its delivery to the audience. Metaphor operates as a screen (the dotted line) that is the surface of the representation — what unifies the events, characters, and ideas that occur along a parallel "metonymical" line that organizes the "parts" of the artwork. The line perpendicular to these two parallel lines is called the "sagittal dimension," and is the line where the audience's interest and role is constructed metonymically, through conventions and the tricks of illusion and stage management.

The first gaze sees the screen as an image that is readily digestible. The second gaze recognizes the tricks that managed visibility and invisibiltiy to create the image, but nonetheless "goes along with" the metaphorical picture. The third gaze sees these tricks as not just conventions but as ways of actively entering into the "logic" of the work of art at different levels. This creates the "extra boundary" that, formerly discounted, now creates a paradoxical condition of self-reference. The viewer must acknowledge that, although he/she "does not exist," this negative role is nonetheless crucial.

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a twist: the 'fourth wall' has a 'rear window'

If you've gotten this far, you should be ready to make the leap into the room where the Big Secrets are kept. This has to do with a symmetrical situation set up by the third gaze. The back-and-forth metonymic exchange "on stage" that is the construction of the representation is not symmetrical. It represents the first two gazes, in fact, played out between characters and other elements. As a part of the representational illusion, we "look on" in different ways. We see others employing the first gaze as naive, unaware. We see those employing the second gaze as manipulative, contriving. The first gaze maintains visibility, the second gaze is like the servant who knows the master is forgetting and supplies the needed words to complete his sentences. The viewer/audience identifies with the second gaze as they develop their own awareness of structure, but with this awareness come the realization that the second gaze has its own exteriority or point of reference.

We locate this point of reference in a spot identical but opposite to our own. To accommodate it, we create a "rear window" (from Hitchcock's inspiring film of the same name). This is not a point we will ever reach, and we cannot symbolize it. It is like a fulcrum or geodesic point there for construction purposes.

Like our own "rotated view" when we take up an active role within this structure, the "rear window" exterior point is intransitive. It "adds" a boundary transition, making the process of extending the third gaze an "intransitive" one.

The symmetry between the two points constructs a line of influence/power that regulates the operation of anamorphosis employed the representational logic at the heart of the work of art/architecture. The question of the exterior of the work has always been extremely problematic. It can be both the "past and future tense" of the work as well as an inversion of the values created internally. Classical architects treated this seriously, with strategies to insure harmonious placement of openings that linked inside and outside. The transition between inside was also a transformation from one existential state to another, and as such it required a ceremonial entry, guarded by spells, amulets, and rituals that would filter out evil influence. All ancient Greek ceremonial sites, for example, used entrances for both theatrical and magic purposes. They were "Janusian knots" tied to foil the danger of symmetry that would contaminate the closed logic of the interior with the contingent logic of the exterior.

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© 2012, Donald Kunze, all rights reserved