Hyperbolic reframing of the Demonique and Angelique of tradition

Engaging with Hyperreality through Demonique and Angelique? (Part #12)

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Sustainability as Heaven? The argument can be further developed by recognizing the extent to which "sustainability" has now been framed as a secular understanding of the condition traditionally associated with "heaven". This is matched by the "hell" with which global civilization is confronted if it fails to get its act together appropriately -- and the condition with which many are already confronted in their daily lives.

Through the pattern of angels and demons, the order governing these conditions has been traditionally intimated -- elusively and mysteriously. However the requisite order would seem to be equally mysterious to global governance of the present, notably given the variety of contradictory advice by experts regarding its nature. Achieving sustainability appears to be as essentially elusive as access to heaven.

The merit of the pattern of angels and demons lies in part in its complexity which would seem to exceed, possibly by an order of magnitude, the somewhat simplistic pattern through which the condition of sustainability is articulated in academic and political discourse. Of particular interest however (as noted above) is the extent to which the angels and demons continue to evoke popular attention worldwide, whereas the articulation of sustainability is far from achieving such credibility. Especially relevant is the manner in which their roles and functions are personalized as a feature of individual experiential reality. Promotion of sustainability struggles to achieve this. That belief in angels and demons is deprecated as ridiculous by secular authorities, needs to be set against the surreal absurdity of global governance which those authorities have engendered and continue vigorously to sustain (as noted above).

Use of "pattern" implies the possibility of depiction in 2D, whereas the sets of angels and demons might well call for articulation as a complex in 3D, 4D, or more -- justifying introduction of the terms Angelique and Demonique (as an elaboration of associations with the Club of Rome's resolutique and problematique). These can be understood as respectively characterizing the dynamics of "heaven" and "hell" -- whether recognized in secular terms or otherwise.

Requisite complexity: Especially as being common to the Abrahamic religions in long-standing conflictual relationship, there is therefore a case for exploring the pattern of 72 angels (and the matching pattern of 72 demons) as an alternative lens, engendered by past struggles for coherence. In terms of credibility, it could be said to have stood the test of time to a degree which the quest of sustainability has yet to match. However there is also a case for questioning the nature and adequacy of that pattern in the light of recent systemic insights and the richer articulations suggested by mathematics. Essentially there would appear to be a cognitive challenge to the meaningful comprehension of complexity in experiential terms -- and not merely in the cognitively dissociated terms promoted as scientific objectivity. Integrating design and ethical dimensions, comprehension of sustainability might then be understood through the challenges of appropriateness (Comprehension of Appropriateness, 1986).

The focus here on 72 constitutes an interesting challenge to the conventional articulations of sustainability through sets of factors of a much more limited size and a far lesser degree of systemic organization. Mathematically 72 has particular properties, notably as 2³ x 3², which suggest that the set as a whole might then be especially conducive to comprehension. Clearly smaller sets may be even more conducive, but without necessarily offering the requisite variety for understanding the dynamic nature of "heaven" -- or sustainability. This suggests exploration of optimum set size as a balance between comprehensibility and distinguishable variety. The size of some circlets of prayer beads as mnemonic aids extends to 108, for example (as mentioned above).

The traditional set of 72 is described in the Wikipedia entry on Shemhamphorasch with reference to a "hidden name of God" in Kabbalah (including Christian and Hermetic variants), and in some more mainstream Jewish discourses. It is composed of either 4, 12, 22, 42, or 72 letters (or triads of letters), the last version being the most common as the 72-fold angels of the Shemhamphorash. It is derived from Exodus 14:19-21 read boustrophedonically to produce 72 names of three letters, as explained in some detail by Lon Milo DuQuette (Angels, Demons and Gods of the New Millennium: musings on modern magick, 1997). Every other line of writing is flipped or reversed, with reversed letters -- a common way of writing in stone in Ancient Greece, with a number of modern examples (including web application). It is related to the mirror writing used by Leonardo da Vinci. The key to sustainability could itself be fruitfully explored as cognitively "hidden", justifying use of such devices.

The triadic naming of the angels is curiously reminiscent of the trigrams of the 64 I Ching hexagrams and the subdivision of each into 3 pairs.

Forms of presentation: lists, tables geometry: The concern here is with how a pattern of 72 might be distinctively represented as indicative of a unique form of coherence -- essential to secular comprehension of sustainability. The many traditional lists of 72 angels and demons are not helpful in this respect. More interesting is their representation in 8x9 table form -- notably with respect to the sense of 9 choirs of angels (discussed below). However tables of those proportions are themselves not especially conducive to credible integrative comprehension -- whatever that may be held to be. The experiential cognitive challenge could be more fruitfully framed metaphorically as one of juggling 72 balls.

A further possibility is to switch to visual presentations in geometrical or topological form, preferably in 3D. This would be consistent with insights into globality -- historically subsequent to the tabular presentations of tradition. It may be readily assumed that there is a quality to the spherical form of the planet which is at least indicative of the integrative connectivity of the dynamics of sustainability -- and of heaven. The challenge to comprehension of globality is however highlighted by the extent to which many are obliged to think in terms of the flatness of the Earth which they most commonly experience -- despite representations of it and images from the Moon (Irresponsible Dependence on a Flat Earth Mentality -- in response to global governance challenges, 2008; Future Generation through Global Conversation, 1997).

In this respect it is intriguing to note the variety of approaches to depicting the global nature of the Earth (in 3D) as a map on a flat surface (in 2D) -- namely the process of map projection. Wikipedia offers a List of map projections, including the following:

• Cylindrical: In standard presentation, these map regularly-spaced meridians to equally spaced vertical lines, and parallels to horizontal lines.
• Pseudocylindrical: In standard presentation, these map the central meridian and parallels as straight lines. Other meridians are curves (or possibly straight from pole to equator), regularly spaced along parallels.
• Pseudoazimuthal: In standard presentation, pseudoazimuthal projections map the equator and central meridian to perpendicular, intersecting straight lines. They map parallels to complex curves bowing away from the equator, and meridians to complex curves bowing in toward the central meridian. Listed here after pseudocylindrical as generally similar to them in shape and purpose.
• Conic: In standard presentation, conic (or conical) projections map meridians as straight lines, and parallels as arcs of circles.
• Pseudoconical: In standard presentation, pseudoconical projections represent the central meridian as a straight line, other meridians as complex curves, and parallels as circular arcs.
• Azimuthal: In standard presentation, azimuthal projections map meridians as straight lines and parallels as complete, concentric circles. They are radially symmetrical. In any presentation (or aspect), they preserve directions from the center point. This means great circles through the central point are represented by straight lines on the map.
• Retroazimuthal: Direction to a fixed location B (by the shortest route) corresponds to the direction on the map from A to B.
• Polyhedral maps: Polyhedral maps can be folded up into a polyhedral approximation to the sphere, using particular projection to map each face with low distortion.

The notable limitation of these projections is that they are necessarily designed to represent a globe in static terms -- with the necessary distortion of each constituting a further challenge to comprehension. Sustainability (and heaven) are however necessarily, and essentially, dynamic in nature, as separately argued (Dynamic Transformation of Static Reporting of Global Processes: suggestions for process-oriented titles of global issue reports, 2013).

It might be asked what form temporal mapping projections of the globe could take -- and whether a range equivalent to the spatial mapping projections is explored, for analogous reasons (see Wikipedia discussion of animated mapping). The design of some prayer wheels may reflect a quest for such representation. [It is appropriate to note that a number of the spatial projections are of a form which recalls the depictions of winged angels -- a resemblance potentially even more probable in the case of the dynamics of any temporal projection.]

Comprehension through alternation?: The argument above endeavours to emphasize that a pattern of 72 factors -- potentially vital to governance and sustainability -- poses a major challenge in terms of human cognitive constraints. Representations of the traditional angels and demons can be used to highlight the difficulty, as in the following.

Animation of sequence of 72 Angel names from the Shemhamphorasch (in two contrasting representations)

Variously understood as corresponding to the 72 angel names are those of the set of 72 demons (List of demons in the Ars Goetia; Goetia Demons). Especially interesting is the sense in which the diagrams below might in each case be understood as systemic pathways or patterns within the larger set, as discussed separately (Evil Loops and Sigils as a Pattern Language, 2016). As noted there, the sigils could well be understood as corresponding to the individual patterns of a pattern language -- through which the multiple faces (facets) of evil manifestation are to be recognized. The pattern language could then be considered as corresponding to any proposed pattern with respect to wicked problems featuring within the global problematique, as indicated with respect to Patterning the Problematique.

Animation of sequence of 72 demonic sigils from the Ars Goetia (with matching reversed images)

It is noteworthy, as implied by such animations, that there is a difficulty in representing a pattern of 72 distinctions in 2D. This "difficulty" may be significant in its own right, justifying any bias to limit representations to sets and patterns of a more limited size only -- a bias constraining any capacity to encompass adequately the problematique or the resolutique as total systems.

Comprehension through mapping in 2D? One experimental layout in 2D is illustrated in the following for the case of the demons. An equivalent could be presented for the 72 angels of the Shemhamphorasch (possibly as animated alternative). These have some resemblance to the Tree of Life which is the focus of the traditional Kabbalah. The angels are held to be the inhabitants of that 10-fold pattern, with its 22 interconnecting pathways.

Alternative experimental configurations alternating between the 72 angels and demons
Animation of 8 sets of 9 (enlargements for detail: angels / demons)	Animation of 9 sets of 8 (enlargements for detail: angels / demons)
The allocation of sets to the star "tables" in the above schematics is based on the tabular form in which the 72 angels and demons are typically presented. The rows are presented "around the tables" in one schematic, and the columns are presented "around the tables" in the other. The sequence around the tables is questionable, demanding further consideration.

The patterns above can also be understood as a systematic representation of modes of failure -- the demons of governance of a viable system (as discussed in the main paper). Governance could then be understood as the art of juggling the 72 forms of negative feedback (demons) and the 72 forms of positive feedback (angels) -- recalling various traditional references to 144. The traditionally recognized functions and qualities of both could be fruitfully explored for clues to a modern articulation (possibly in management jargon) as has been done with the 64 hexagrams of the I Ching (Transformation Metaphors derived experimentally from the Chinese Book of Changes (I Ching) for sustainable dialogue, vision, conferencing, policy, network, community and lifestyle, 1997).

An equivalent mapping challenge is evident in the representation of the periodic table of chemical elements, now numbering over 100. Whilst its conventional representation as a table is a convenience offering a variety of insights, it is of limited value in enabling comprehension of the pattern as a whole -- as might be desirable in the case of the pattern of governance-significant factors. Using that table as a metaphor, aspects of the challenge are discussed separately (Tuning a Periodic Table of Religions, Epistemologies and Spirituality -- including the sciences and other belief systems, 2007).

Comprehension through mapping in 3D? As noted above, any 2D representation only faclitates "global" comprehension to a limited degree. A fruitful 3D possibility is to associate the demons or angels with polyhedra. A number of these have 72 features (notably vertices or faces). This was the approach previously taken with the 64 hexagrams of the I Ching as traditionally indicative of conditions of change (Proof of Concept: use of drilled truncated cube as a mapping framework for 64 elements. 2015; Eliciting Memorable Spheres and Polyhedra from Hyperspace: integrative connectivity of problems, strategies, themes, groups or people, 2015).

The polyhedral library of Stella Polyhedron Navigator enables display of the following polyhedra onto whose features angels or demons might be mapped:

• with 72 vertices: 18 polyhedra (animation in preparation)
• with 72 faces: 11 polyhedra (animation in preparation)
• with 72 edges: 49 polyhedra
• with 72 faces and 72 vertices: 2 polyhedra

The suitable polyhedra are themselves relatively complex as mnemonic aids. Another approach is to make use of simpler polyhedra with different constraints, enabling the display of subsets of demons on several variants, as shown below using the polyhedron best known for its pattern in football stitching (as noted above).

Experimental use of truncated icosahedron to display 2 sets of 32 demons linked by an octahedron displaying a set of 8 demons (thus totalling 72) (generated using Stella Polyhedron Navigator)
Version A mapping of 32	Mapping of 8	Version B mapping of 32

Clearly a similar approach can be taken to the display of any set of images of 72 angels. Of interest from a traditional perspective is the degree of resemblance to the vajra (Sanskrit) or dorje (Tibetan). This is used as a ritual object to symbolize both the properties of a diamond (indestructibility) and a thunderbolt (irresistible force). This is equivalent to the sceptre traditionally emblematic of a ruler, as with the ceremonial mace (for which an official, appointed as Black Rod, is responsible in some modern parliamentary procedures). Curiously the form also recalls both the dumbell used in weight training, as well as the use of that term to refer to the form of a 2-lobed atomic orbital.

Use of the intermediary octahedron with its 8 faces potentially reconciles the set of 64 (mapped onto the two truncated icosahedra) with the set of 72. The set of 64 has of course been long associated, through use of the I Ching (as presented above), with an understanding of governance and decision-making in China. However as a distinctive set, in addition to the 64, the pattern of 8 can be understood as a supernal or controlling pattern, perhaps to be associated with the functions and insights of "archangels" -- the ninth "choir" in the pattern of 9x8 angels (as discussed below).

With respect to holding together cognitively both the 72 patterns of the angelique and the 72 patterns of the demonique, a further possibility is indicated by the 4-fold dorje. Two vertices of the octahedron might then be linked to two more truncated icosahedra to enable mapping of both sets. However, with respect to the cognitive challenge of governance (for which any such symbol has mnemonic functions), the relation between angelic and demonic is essentially dynamic, potentially better illustrated by having the images alternate on the equivalent of a 2-fold dorje.

A suggestion of this alternation can be mnemonically enabled by rotating the 2-fold complex on an axis through the octahedron. With any such dynamic, the 2-fold mapping surface then holds the complete set of patterns for the sustainable governance of a viable system. The cognitive significance and requisite skills may well be implied by variants of the prayer wheel, baton twirling, pen spinning, and astrowheeling.

The range of possibilities for holding complexities memorably -- in terms of the set of 72 (or 144 or 288) is usefully illustrated by the following "mapping surfaces" which enable various "stories" to be explained in terms of positive and negative dynamics. Following the argument above, given its 32 surfaces, use is made of the truncated icosahedron (a "football") in various configurations. Two such polyhedra may be linked by an octahedron to give 72 surfaces. Four such polyhedra may also be linked, with the addition of two tetrahedra, to give 144 surfaces.

Exploration of possibilities of holding complexity memorably in terms of 72 angels and/or demons (visualizations constructed using X3D to enable interactive virtual reality display and/or use of WRL browser capacity)
72 surfaces (64+8) (interactive X3D / WRL)	136 surfaces (interactive X3D / WRL)	144 surfaces (interactive X3D / WRL)	Rotational pathway? (interactive X3D / WRL)

Each of the above can of course be rotated around the central axis through the octahedron (as shown in animations). Of particular interest, benefitting from design/display possibilities, are the lighting effects causing the colours to switch and flash during rotation, as partially illustrated in the screen shots. This helpfully suggests the dynamic between angelic and demonic dynamics (without the need to alternate the colours programmatically). The relevant files can be readily modified to change the proportions, colours and transparency. Due to such transparency, the image on the far right partially illustrates the manner in which the tetrahedra -- as "archangels" -- transcend and interpenetrate the other elements. Through the addition of the circle linking the truncated polyhedra in that image, a suggestion is offered of the pathway of those polyhedra during rotation.

In addition to rotation around the central axis through the octahedron, the truncated polyhedra can also be individually rotated (possibly with counter-rotation). Such effects could also be achieved with colour alternation -- even to all white, or all black -- according to variants of the "story" regarding the relation between angelic and demonic dynamics. A more extensive range of colours could be used. It is also appropriate to note that, with the exception of the cube, the mapping complex uses all the Platonic solids, explicitly or implicitly (the truncated icosahedron being a conflation of the icosahedron and the dodecahedron).

"Technological theology" or "Theological technology" ? Rather than the emphasis here on mathematical theology, the degree to which such a structure holds the extremes of the symbolism associated with twirling and the form of a frequently imagined space station (image on far right above) suggests that such a dynamic meta-pattern has an intuitively attractive appeal -- possibly far more fundamental than is readily appreciated. It is even possible that enthusiasm for investment in either extreme, especially a permanent space station in orbit, constitutes a form of cognitive displacement from the possibility of engagement with hyperreality -- understood otherwise.

Currently of potentially greater significance is the invasive role that it is expected that drones -- now widely available -- are expected to perform. There is considerable irony to the possibility that the future will shortly see use of personal drones as guardian angels, hovering protectively above the wealthy (at least), with their analogues in cyberspace. More curious is that their form may readily recall depictions of the halos associated with the holy.

The necessary rotation of any outer space vehicle offers a frame for discussion of "cognitive gravity" and the possibility of its transcendence in orbit -- notably in terms of the viability of such an environment as a habitat. Of relevance in that context, and at a time of detection of "gravity waves", is potential recognition of Arrogance as an analogue to gravity -- equally fundamental and mysterious (2015). The challenge of "arrogance waves" could be considered fundamental to the future organization of a time-bound civilization in that everyone perceives themselves to be normally "right" -- if not unquestionably "upright" -- irrespective of their respective orientation to the sun and the shadow thereby cast.

The space station form readily recalls a rotating mandala, variously valued as enabling "ascent", if not "escape" from planetary constraints -- as discussed in Clues to 'Ascent' and 'Escape' in (Navigating Alternative Conceptual Realities: clues to the dynamics of enacting new paradigms through movement, 2002). With respect to "ascent" and transcendence, the last point recalls the inspiration for the designer of the Bell helicopter, Arthur Young, for the possible design of a "psychopter" -- as separately discussed (Engendering a Psychopter through Biomimicry and Technomimicry: insights from the process of helicopter development, 2011).

Given the requisite complexity (as implied above by interlocking Möbius strips), of particular interest from a mathematical perspective is the figure-eight twirl characteristic of variants of the plate trick (or belt trick). As shown below, this is used as a demonstration that a double rotation is a loop in rotation space that can be shrunk to a point.

Demonstration of how the end of a set of belts can be continuously rotated without becoming twisted or tangled. (Screen shots of an animation by Jason Hise via Wikimedia Commons)

Curiously this 8-fold pattern (as with that of the intermediary octahedron above) recalls the framing by Stephen Prothero (God Is Not One: The Eight Rival Religions That Run the World -- and Why Their Differences Matter, 2010). Failure to comprehend the integrative pattern implied by the angelic dynamic effectively transforms it into the more explicitly tangled demonic equivalent.

Comprehension through mapping in 4D? Given the constraints on integrative displays and global comprehension, it is is appropriate to note the 4D polyhedra indicated by Stella Polyhedron Navigator as being potentially useful for display of 72 demons or angels -- or the set of 144. Their four-dimensional nature is consistent with the sense that the sets of angels and demons are effectively four-dimensional entities and can only "exist" in time rather than in any static depiction. However the following example of such a "pseudo-static" structure also makes the point that these remain a challenge to comprehension.

4D Polyhedron: 68-Odho with 72 faces 72 faces of 3 types (3-gon, 4-gon, 6-gon); 96 edges; 24 vertices Cat4-60-Ico-regiment (derived from 4D Library of Stella Polyhedron Navigator)

Analogous to polygons and polyhedra, forms such as the above are 4-polytopes, or polychora. These can be tentiatively explored in terms of their potential temporal significance (Comprehending the Shapes of Time through Four-dimensional Uniform Polychora, 2015). The latter formed part of a discussion with respect to governance and sustainability (Enhancing Strategic Discourse Systematically using Climate Metaphors, 2015).

Although visually ncomprehensible to humans, it is appropriate to recall their potential future relevance in the light of foressen emergence of higher orders of artificial intelligence (Is A.I. an existential threat to humanity? Quora; Kevin Drum, Why Artifical Intelligence Is Closer Than We Realize, Mother Jones, 17 November 2011).

"Hyperbolic comprehension" in the light of time inconsistency? The argument above noted the extensive literature on hyperbolic discounting in responding to the challenge of time inconsustency in anticipating the future, most notably in decision-making. As a cognitive constraint, this adds to the challenge of any dependency on alternation or 4D representation. Further insight may however be derived from the acknowledged relevance of the hyperbolic form.

Especially interesting is that both angels and demons, as indicative of problematique and resolutique, may be best understood through a hyperbolic framing of human experiential engagement with time. Understood as a complex of positive feedback loops, the set of angels then indeed merit recognition as the Angelique (as argued above) -- but one in which a hyperbolic framing is fundamental, most probably of higher dimensionality. The same may then be said of negative feedback loops with respect to a Demonique. Explicit integration of temporal inconsistency, framed in hyperbolic terms, would seem to be especially appropriate in the light of the role it plays in the global governance of the civilization described by John Ralston Saul (The Unconscious Civilization, 1995).

There is a delightful irony to the possibility that the form of both angels and demons could well be associated, if only for mnemonic purposes, with the hyperbols. As temporal entities, transcending human cognitive incosnsistency in some way, they could well be understood as "existing" at the nexus with which the origin of the hyperbola is associated. Their "wings" are then to be recognized in the two curves of that form -- potentially indicative of learning pathways in time, distinguished directionally to reflect the angelic and demonic feedback loops.

Imagining angels and demons in mathematical terms? Some consideration has been given to the "physics" of angels, as noted above (Rupert Sheldrake and Matthew Fox, The Physics of Angels: exploring the realm where science and spirit meet, 2014). Given the arguments above, how might the mathematical form of angels and demons be imagined -- when informed by cognitive psychology (George Lakoff and Rafael Núñez, Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2000)? What recent mathematical insights could fruitfully "transform" angelic and demonic depictions beyond their anthropomorphic limitations?

One interesting example is provided in an interactive applet by Jon Rogness (The Hyperbolic Paraboloid, Math Insight) as part of an Interactive Gallery of Quadric Surfaces

Hyperbolic paraboloid as suggestive of angelic form Screen shots of an interactive applet by Jon Rogness

The applet permits the form of the hyperbolic paraboloid to be variously modified. The central image is indicative of the underlying structure. That image usefully recalls both the detailed organization of antenna arrays (radiotelescopes, etc) and the wings of birds and insects.

Implications of the "shape of the universe" for the shape of the "Angelique" and "Demonique"? Astrophysics has long been preoccupied with the seemingly extraordinary question of the shape of the universe -- in terms of its curvature and topology. Hypotheses have notably explored the possibility of a universe with zero curvature, a universe with positive curvature, and a universe with negative curvature.

Use is made of "universe" as a metaphor (Clare Beghto, From the Universe of Knowledge to the Universe of Concepts: the structural revolution in classification for information retrieval, Axiomathes June 2008; Charles van den Heuvel, Elementary Structures: universe of knowledge or universe of concepts? 2011). This then raises the question of whether the "universe of meaning" merits similar exploration, as may be variously argued (Towards an Astrophysics of the Knowledge Universe: from astronautics to noonautics? 2006; Eliciting a Universe of Meaning -- within a global information society of fragmenting knowledge and relationships, 2013; Samuel Reiss, The Universe of Meaning, 1953).

The question here is whether the problematique, understood experientially as the Demonique, could be usefully explored in terms of its curvature and topology. The same applies to the Angelique as the complex of inspirations associated with the resolutique. What indeed might be the "shape" of the global problematique -- especially as it is mysteriously embodied cognitively in the Demonique?

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