Imagining the nature of cognitive flight in terms of the enneagram

Representation of Creative Processes through Dynamics in Three Dimensions (Part #6)

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Further insight is also to be derived from Stafford Beer's recognition of the association between the enneagram and the icosahedron, which he describes as emerging from collaboration with Joseph Truss -- in a chapter on The Dynamics of Icosahedral Space (Beyond Dispute, 1994, pp. 196-209). For Beer:

But it is a matter of great interest that in the whole of the literature... the enneagram occurs as a plane figure. Nowhere had there been the slightest hint that a three-dimensional manifestation existed... No wonder the search took so long, given that the diagram was discovered spread across four vertical planes... The icosahedron is the actual origin of the enneagram... (p. 206)

Various efforts have been made to depict the enneagram in 3D -- readily available on You Tube (Francisco Meana, Enneagram from 3d perspective, 2007; Francisco Meana, Sufi Enneagram, 2009; Chuck Middaugh, 3D Enneagram MOD 9, 2013), The degree of relationship to the icosahedron is not especially evident.

Beer provided no depiction, but this is offered in subsequent documents (J. Truss, et al, The Coherent Architecture of Team Syntegrity: from small to mega forms, 2003; J. Baldwin, BuckyWorks: Buckminster Fuller's Ideas for Today, 1996, p. 220). The Beer/Truss argument is also discussed by Andrew Pickering (The Cybernetic Brain: sketches of another future, 2010). Since those promoting syntegrity are especially sensitive to copyright issues, a different depiction is offered in the following generated with virtual reality software..

9-fold enneagram embedded within an icosahedron with addition of an indicative central sphere (constructed by manual modification of a virtual reality model of the icosahedron generated by Stella Polyhedron Navigator software)
View of enneagram associated with only one pattern of vertices of icosahedron (view in 3D with virtual reality plugin)	View of enneagram from left image with the icosahedron framework hidden (view in 3D with virtual reality plugin)
Note the colour coding and positioning of the icosahedral vertices -- which offer guidance when rotating the above models in virtual reality in order to render visible the enneagram pattern. Green-Magenta links of the enneagram are the only links embodied within icosahedral edges (on the left), where they are invisible. Three of the 12 vertices, positioned on the vertical axis (of the image on the right), do not form part of the 9-fold enneagram pattern (Red, Cyan and Black). The various possibilities for rotating the models in three dimensions affect the proportions of the enneagram as portrayed and the relative visibility of the Cyan and Black vertices.

Rotation of the above icosahedral images offers insights into the complex relationship between the elements of the enneagram. Of particular interest is any sense in which the Green vertices are to be considered as generative polarities of relevance to the more general argument here. The added central sphere offers a sense of the locus of the controller of the cognitive vehicle in relation to the dynamic suggested by the significance commonly associated with the lines of the enneagram. (see Susan Rhodes, Ennagram Overview, Enneagram Dimensions)

Indication of the 3-dimensional structure of the enneagram within the icosahedron (rotation of the above presentations by approximately 90 degrees)
Icosahedral framework visible	Icosahedral framework hidden
Note that the image on the left (above) offers edge-on images of some planes of the enneagram, whereas the image on the right (rotated to a lesser degree) makes apparent those planes. The separate triangular form can be understood as traversed by an axis through the unconnected Cyan and Black vertices -- and passing through the added central sphere.

Screen shots of rotation of enneagram in three dimensions (relationship between vertices fixed by invisible icosahedron)

As remarked by Beer with respect to detection of the enneagram "hanging" within the icosahedron:

Consider: if it can be detected when the icosahedron stands on one vertex, it must be detectable when the model stands on any vertex. Moreover, if it is present when these two poles are aligned, it must be present when any two poles are aligned. Thus it comes about that points 4 and 5 on the enneagram refer to any side [meaning edge] of the icosahedron -- which therefore enfolds 30 three-dimensional, four planar, enneagrams... The icosahedral model conceived as a spinning sphere could be regarded as "an interpenetration of phi-ness"... an interpenetration of three-dimensional four-planar enneagrams... (p. 206-207).

The images displayed statically above are totally inadequate as depictions of a single four-planar embedding -- especially when the lines are indicative of information flow patterns and loops (vital to control processes). To the extent that the depiction of the enneagram, from a conventional 2D perspective, can be interpreted as pillars and an archway, of particular interest is the implication that the symbolism valued so highly with respect to passing between pillars (and under archways) merits reconsideration. In Freemasonry, for example, the pillars are named Jachin and Boaz and represent one of the group's most prominentaly featured symbols. The complexity of the enneagram in 3D is indicative of the counter-intuitive cognitive complexity of such a passage. This is especially highlighted by the collection of Zen koans -- known as The Gateless Gate.

An animation is required to offer a better understanding -- and the manner in which the 3D enneagram is associated with each of the 30 edges of the icosahedron. However it is appropriate to note that representations are available otherwise in that the final stellation of the icosahedron (the 17th) is an irregular star (self-intersecting) polyhedron with 20 identical self-intersecting enneagrammic faces, 90 edges, and 60 vertices. The 20 faces correspond to the 20 faces of the underlying icosahedron with each face being an irregular 9/4 star polygon, namely an enneagram.

Animation of stellations of an icosahedron (reversed from final to core icosahedron) The stellation has 473 cells of 11 types. Numbers are indicated for each stage. (generated with the stellation function of the Stella Polyhedron Navigator)

A contrasting depiction of the stellations of the icoahedron is given separately in a discussion of Systemic comprehensiveness of sets in (Harmony-Comprehension and Wholeness-Engendering: eliciting psychosocial transformational principles from design, 2010).

Beer remarks with respect to mentation, both in individuals and groups, and the role of an information set (an "infoset"):

It is at least possible that the multiple enneagrammatic structure, reverberating as it does, provides a complex of linkages to constitute such a "corporate brain" that would then give rise to its own [infosettic] consciousness. (p. 208)

Given the inspiration that Beer drew from the work of Arthur Young -- and the latter's preoccupation with "flight" (as more generally understood) -- it is appropriate to note the use of "wings" in enneagram personality type theories (Anthony Blake, The Intelligent Enneagram, 1996). The basic type associated with any one point of the enneagram is understood to be modified by the personality dynamics of the two adjacent types as indicated on the enneagram figure.

Use of "wing" in a 3D context suggests that it might be more appropriately extended to the eight paired portions of the enneagram -- recalling the distinctions of the BaGua dynamic (noted above). As wings, the 3D enneagram might also be understood in terms of a membrane stretched (and twisted) in some way over the lines between the points (as is familiar in topology). At one extreme this recalls the concern of theoretical physics with branes (and brane cosmology), as separately discussed (Global Brane Comprehension Enabling a Higher Dimensional Big Tent? 2011). The latter referred to another extreme represented by the intuitions of the polymath Omar Khayyám -- derived from sails and tent-making.

With respect to "flight", how might the psychosocial implications of "beating" wings be understood in relation to sustainable governance (Enabling a 12-fold Pattern of Systemic Dialogue for Governance, 2011). Could the wing dynamics be related to the set of archetypal morphologies of René Thom?

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