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There are four fundamental "protocols" or procedures for investigating boundary phenomenon. They are similar in structure and terminology to enable comparisons of a broad variety of examples, but protocols focus on particular themes and concerns represented by the "entry points." Narrative protocol is concerned with the temporal distribution of events, characters, and elements that, typically, resolves a conflict or redresses a crime. This is sometimes called the "Murder Mystery" protocol. Visual protocol uses the metaphor of spatial "rotation" and the collapse of spatial scale. Mystery protocol focuses on the intellectual solution of a fictional crime, where the identity of the perpetrator is concealed and thought and action alternate. The civic protocol is drawn from traditions and myths about civic and domestic foundation rites, hospitality, and silent trade. Emphasis on "voice" as the medium of authority and truth guides the cosmic-theological protocol, where meaning is converted into open-ended significance. Additionally, there is an approach that involves the 'calculus' of George Spencer Brown, a logical notation system using one symbol () and two axioms.
The primary graphic nucleus of the boundary's complex behavior is based on "the inside frame," a flip of a boundary between two equivalent topological conditions.
Either the boundary around C can tightly hug a small zone, or it can "flip" to enclose all non-c territory, making /c into a "ground" on which A-cross and B are contained within a single island.
The flip of the boundary around C is related, in visual phenomenon, to the general "direction" of the gaze. Depending on whether the subject is looking at or being looked at, the perceived boundary conditions are reversed. For example, the phenomenon of the "evil eye", a generic gaze without any specific location or direction would be perceived as a containing circumferential boundary. An invisible voyeur, or the guards in the central tower of the Panopticon, would generate a similar circumferential boundary, with the "victim" imprisoned within the space of A. For a step-by-step account of the ambiguity between the map and journal aspects of the boundary, visit
The normal condition of perception, where the subject sees the world as a panorama about him, is represented by the C position. However, when C finds that he himself is being watched, A becomes the inside frame and the diagram is reversed. Where does the power of this simple flip come from? It's a basic property of the general ambiguity of boundaries.
The bolagram is a diagram that summarizes this ambiguity with a triangle whose vertices represent the possible "states," or boundary conditions, that apply to a great variety of phenomena. The "transitive" or map view is represented by the ABC line, while the complementary journal view is represented by the lower part of the triangle. C is the hinge or pivot that joins the transitive spatial view with the "intransitive" temporal, or journal, view.
The most direct way of understanding how the bolagram effectively represents complex but fundamental situations is to look at the famous depiction of art, coined by Samuel Taylor Coleridge: "the willing suspension of disbelief."
1 / to believe or disbelieve?
The "transitive" system operating for belief reflects the simple ("map view") function of boundaries: to create an either/or situation. In non-art, we use belief and disbelief to distinguish between truth and falsehood, the existent and the non-existent.
2 / putting the will on hold
The two choices represent the logic of "the excluded middle," i.e. no other option is allowed. Judgment must select one or the other.
This adds a dimension of "will/unwilled" where the category of will remains blank (it is not an option for non-art).
3 / a third way out
For art, there is a "third way" involving a use of the will; that is, art requires a "voluntary" attitude adjustment.
By negating belief not once (C) but twice, it is possible to "suspend disbelief." This is a way of avoiding the either/or ultimatum of ordinary thought and life and entering into an "intransitive" space where there is a middle term that may take alternate values.
This suspension splits the audience's consciousness in two, making it possible for an "immobilized" spectator to imaginatively enter into the artifact (B) and perceive the "reality" of the work in the virtual world "anamorphically" created by the artifact. IJ becomes the system of oppositions within the artwork that generate a set of possibilities without determining a particular outcome.
Other cases of double negation also involve the C-C' element as a hinge, or "inside frame," from which a transitive situation (either/or) can find a third alternative, an oscillating value (called "ij" values) that completes a cycle of themes and values that would remain incomplete with only a transitive "map view."
In drama, art uses this third way out as a "witty solution" to an intractable dilemma. Culturally, the third option provides a means of doing two things at once: adhering to the spirit of a law, for example, while violating the letter. In visual arts, the third way involves, typically, a way of building an "anamorphic" image within a representational view. Architecture uses options of scale dysfunction, doubling, motility, etc.
The "willful" part of the bolagram relates directly to the role of the audience (the "enthymeme" is used to indicate the audience's silent addition of the logic needed to complete the art equation).
© 2012, Donald Kunze, all rights reserved.
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