Challenges to Comprehension Implied by the Logo
of Laetus in Praesens
Laetus in Praesens

28 September 2020 | Draft

From Zoom Organization to Zome Configuration and Dynamics

Integrating the doughnut, helix and pineapple models towards global strategic coherence

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Contrasting global models from plan to pineapple
Phyllotaxis and implication of "spiral integration" for governance
Requisite complementarity of governance models comprehensible through metaphor
Alternation between complementary models of global governance
Interweaving spirals as embodied in zome configuration
Approximating a pineapple model by modification of a zome configuration
Enhancing communication patterns in virtual conferences with zome architecture
Potential indication of governance communication pathways in a zome configuration
Use of a zome configuration in practice and as a focus of symbolic significance


The United Nations is celebrating a historic anniversary with its 75th General Assembly (September 2020). As a consequence of uncritically adopted policies of lockdown in response to the coronavirus pandemic, the gathering has taken highly unusual form with the use of virtual conferencing technology unimaginable in the very recent past (Julian Borger, Why the UN’s 75th general assembly could be worse than the world’s worst Zoom meeting, The Guardian, 22 September 2020). As the latter notes: The worst parts of UN events will be on display, the endless speechifying first among them, but none of what normally makes the general assembly indispensable.

The occasion has been recognized by the Secretary-General as "Our 1945 moment", referring to the call to action inspired by the generation who had survived the second world war and sought to build a new world (UN faces fears of a 'great fracture' at general assembly, The Guardian, 23 September 2020). Commentators have indicated other challenges ( Marcus Holmes, et al, UN general assembly: why virtual meetings make it hard for diplomats to trust each other, The Conversation, 22 September 2020; Nicholas Westcott, UN general assembly goes virtual: a former ambassador on what that means for diplomacy, The Conversation, 21, September 2020).

Such use of virtual conferencing by the United Nations has been preceded by its adoption by many organizations and groups in response to travel restrictions and lockdowns during 2020. Whether for them or for the United Nations such rapid uptake raises the question as to how such adaptation is to be characterized. Is it a case of "putting old wine into new bottles", or of "putting new wine into old bottles", or of "putting new wine into new bottles"?

As the early report notes, does such technology simply enhance the tendency to "endless speechifying"? More problematic, as indicated, is the new tendency in the case of the General Assembly for leaders to pre-record the contributions, framing any consideration of global issues primarily for home audiences, and avoiding any need to listen to the contributions of others. Given the possibility of avoiding the costs of attendance, there is a degree of irony to the possibility that more leaders will "participate" virtually -- with the implication of more speeches of that form.

Missing from any consideration of the new arrangements is how the pattern of contributions is to be orchestrated -- and the possibilities for new thinking in this regard. What audience will be attracted by video recordings from the 193 member states of the United Nations? How will diplomatic order of precedence be recognized -- however biased this may be held to be?

Little is said on such matters with respect to the contributions presented in many international meetings now using this format -- many immediately forgettable. This suggests that it is very much a case of "putting old wine into new bottles", replicating the inadequacies of past communication processes -- even to the point of exacerbating them, irrespective of the level of crisis. To the extent that the United Nations can be upheld as a form of "global brain", are its problematic communication processes an indication of irreversible ageing -- or worse -- despite long-standing efforts at "UN reform" (Are the UN and the International Community both Brain Dead -- given criteria recognizing that NATO is brain dead? 2019)?

Also potentially problematic in a period of heightened concern with electronic security, how can virtual gatherings be "hacked" (FBI warns of hackers hijacking online Zoom meetings, New York Post, 31 March  2020; A Must For Millions, Zoom Has A Dark Side: an FBI Warning, NPR, 3 April 2020). Clearly it is possible for some form of "hacking" to be used by organizers to control the gathering by means of which participants are not aware (12 Game-Changing Zoom Hacks For Work Meetings, HuffPost, 4 April 2020)

The questions are of particular relevance to the future operation of any World Peoples Assembly or a World Parliamentary Assembly, as variously proposed. In seeking to enhance popular participation, many of the issues became evident in the "Great National Debate" organized in France in 2019 in response to the country-wide protests by the Yellow Vests movement, as discussed separately (Claire Mufson, What will France do with 'National Debate' data? France24, 3 March 2019; France's 'great debate' is over -- so what comes next? DW, 15 March 2029). How indeed can such process be facilitated otherwise, as might be envisaged (Multi-option Technical Facilitation of Public Debate: eliciting consensus nationally and internationally, 2019)?

The concern in what follows is the quest for indications as to how the communication potential of virtual conferences might be more fruitfully organized -- beyond past patterns of enabling the many to listen to the few. Those patterns can be seen as strangely echoing concern about the global resources controlled by "the 1%", in contrast with those to which the remainder have access. This can be recognized as especially inappropriate -- if not dangerously so -- in a period in which the expertise and insights of the few have been demonstrably inadequate in many instances (Radical Disaffection Engendered by Elitist Groupthink? 2016).

The quantitative concern can be framed otherwise by reference to seating arrangements in conventional plenary assemblies of many hundreds. What facilities enable those seated to communicate with each other -- without that communication being centrally controlled? How does any sense of coherence emerge without suppression of the perspective of any minority through a voting process upheld as democratic and unquestionable?

Will the future perceive as bizarre the geographic requirement to position distinctively those groups favouring a particular strategic perspective -- even caricatured as "left" or "right"? How do people get "positioned" in a virtual gathering and, more generally, how does this then meaningfully reflect the "global" diversity of those represented? When there are many participants and many perspectives, how is memorable global sense-making enabled given the challenge of numbers? Increasingly, for many, in metaphorical terms their "mailbox" is full (Comprehension of Numbers Challenging Global Civilization, 2014).

The following exercise explores the possibility of interrelating the widely cited doughnut model of global economics with a pineapple model (Pineapple model of global governance? 2020). The former, although strategically concerned with the threatened relationships between nine planetary boundaries, offers little insight into the requisite strategic patterns of communication. The latter focuses on imagining the dynamics of communication potentially consistent with the knowledge architecture of virtual meetings -- inspired by the elegant integrity of zome configurations and by insights from nature (Fibonacci Spiral in 3D Framing Psychosocial Phyllotaxis: articulation of global governance through the language of flowers? 2020).

Despite the little-known form of zomes, they can be recognized as having structural properties in common with the pine cone. As a symbol of human enlightenment, this has been a feature of architecture and decoration in civilizations around the world over centuries. The Vatican Court of the Pine Cone offers one example; another is the award-winning addition to the London skyline -- nicknamed the Gherkin.

Contrasting global models from plan to pineapple

It is unnecessary to comment further on the facilities offered by Zoom-style meetings. As indicated above, the concern here is with the possibility of orchestrating virtual meetings more fruitfully. Rather than conventional references to their "organization", the concern is with the process of "organizing" them -- understood metaphorically in the light of the musical origin of the term with respect to the pipe organ, as described separately (Engaging Playfully with Coronavirus through "Organizing" Global Governance? 2020; Envisaging a Comprehensible Global Brain - as a Playful Organ, 2019).

Doughnut model: The response to the challenges of global governance can be understood as framed by contrasting metaphors, of which a "plan" is the most obvious -- with the "doughnut model" offering a distinctive contrast (Kate Raworth, A Safe and Just Space for Humanity: can we live within the doughnut?, Oxfam, February 2012). An illustration of the model is presented below left, although the extent to which boundaries are transgressed and social foundations are met are not visible on this diagram. The boundaries, represented below centre, derive from the study of "planetary boundaries" (Planetary Boundaries: exploring the safe operating space for humanity, 2009). This doughnut-like area is defined by combining the much-debated set of 9 "planetary boundaries" to which the doughnut model adds a new set of 11 social boundaries, based on the 11 dimensions of human deprivation that emerged from the issues raised by governments in their Rio+20 submissions.

That pattern of boundaries may be used to represent the boundaries to remedial capacity as shown below right, and discussed separately (Exploring the Hidden Mysteries of Oxfam's Doughnut; recognizing the systemic negligence of an Earth Summit, 2012; Recognizing the Psychosocial Boundaries of Remedial Action: constraints on ensuring a safe operating space for humanity, 2009; Indicators of Political Will, Remedial and Coping Capacity? 2019).

Circular governance models interrelating distinctive boundaries
Doughnut economics model
(classic image)
Nine planetary boundaries
[click image for enlargement]
Nine remedial capacity boundaries
[click image for enlargement]
Dougnut economic model Earth-System planetary  boundaries Earth-System remedial capacity   boundaries
DoughnutEconomics / CC BY-SA Reproduced from Exploring the Hidden Mysteries of Oxfam's Doughnut; (2012)

The doughnut can be recognized as a memorable representation of a torus which offers other insights meriting reflection (Imagining Toroidal Life as a Sustainable Alternative: from Globalization to Toroidization or back to Flatland? 2019).

Pineapple model: Such frameworks can be contrasted with the possibilities of the "pineapple" in the light of insights into the construction of a "zome" and its relation to patterns evident in nature (Coronavirus -- Global Plan, Doughnut, Torus, Helix and/or Pineapple? 2019). Arguments in the latter, noting a relationship between the various models, took the following form:

Global strategic plan for coronavirus pandemic?
Relevance of doughnut model to a pandemic?
Torus as a key to strategic knowledge organization?
Strategic implications of the helical core of the coronavirus?
Pineapple model of global governance?
Formalizing a pineapple model of governance through zonagons and zonohedrification?
Zomes as a key to appropriate organizational and knowledge architecture?
Perception and uncertainty in global governance -- "one model fits all"?
Nine-fold configuration of planetary boundaries?
Playful transformation between forms of global governance -- "organizing"?

The argument there stressed the inadequacy of any "plan" in response to global challenges -- if only in the light of the problematic geometrical relationship between the two-dimensionality of a plan and the three-dimensionality of the globe, with the consequent implications for comprehension and appropriate strategic articulation. The "doughnut", whether or not it is understood as a torus in 3D, highlights other concerns. As the epitome of fast-food consumer malnutrition, the choice of a doughnut as a model could be held to be seriously incompatible with its use in interrelating economic and environmental challenges.

Challenging a model framed as a "doughnut" with that of a "pineapple" should not obscure the properties which the latter shares with a "pine cone". As mentioned above, this has long featured in symbolic architecture and decoration worldwide, with new applications in architecture (if not in cognitive terms) in process of development:

Helical model: As a geometrical complexification of the torus, any helical model -- such as the Triple Helix model of innovation (and its extension to quadruple and quintuple variants) -- offers richer possibilities. However a "pineapple" model has the further advantage of combining the geometrical attributes of both helical models and the torus -- as well as being both inherently comprehensible and valued for its nutrional qualities (to a far higher degree than the doughnut). Ironically for a conceptual "model", both the root and fruit may be eaten or applied topically as an anti-inflammatory or as a proteolytic agent. The pineapple is an excellent source of the trace mineral manganese -- an essential cofactor in a number of enzymes important in energy production and antioxidant defences.

Geodesic model: The question with respect to any model, of which the pineapple is an indication, is how to translate its geometry into a form of relevance to global governance -- in the light of the challenges so significantly faced by the United Nations in having recourse to virtual assemblies. The form which could best serve as an intermediary is the zome. This relatively little-known form and its architectural manifestations, follows in the somewhat improbable tradition of the geodesic dome developed by Buckminster Fuller  (Synergetics: explorations in the geometry of thinking, 1975) as separately questioned (Geometry of Thinking for Sustainable Global Governance: cognitive implication of synergetics, 2009).

The geodesic dome is now used world wide, and is especially valued for its capacity to encompass large areas at relatively low cost -- as well as being elegantly reminiscent of the globality sought in global governance, if only to a limited degree. The geometric principles of tensegrity on which it is based have been notably recognized in cybernetic terms as of significance to new forms of dialogue (Stafford Beer, Beyond Dispute: the invention of team syntegrity, 1994).

Golden ratio? Frank M. J. van der Linden explores patterns and morphology of building elements for close-packing of dome surfaces after the example of phyllotaxis and morphogenesis of primordia on the apex of composites (Phyllotactic Patterns for Domes International Journal of Space Structures, 9, 1994, 1).

That reference to phyllotaxis in the architecture of buildings follows from an earlier argument highlighting the potential insights to be drawn from the organization of plants in terms of the perspectives of general systems and biomimetics  (Fibonacci Spiral in 3D Framing Psychosocial Phyllotaxis: articulation of global governance through the language of flowers? 2020). That argument necessarily addressed the criticism by mathematicians of the misconceptions -- deemed nonsensical -- which have been cultivated from an aesthetic perspective regarding the golden ratio and Fibonacci patterns of organization (Self-similarity design patterns as exponential bullshit?; Global governance bullshit combining exponential growth and suboptimal distribution?).

Specifically it framed the question as to whether any conventional deprecation of the golden ratio in design could be a case of "throwing the baby out with the bathwater"? Given the significant failure of science to alleviate, rather than exacerbate, the patterns of disagreement which now characterize global society and the discord between the "sciences", it is appropriate to ask whether conventional science has the capacity to appreciate the nature of the aesthetic integration of which the golden ratio may be indicative (Knowledge Processes Neglected by Science: insights from the crisis of science and belief, 2012).

In the binary mode, so systematically reinforced by science, it would then be appropriate to distinguish science of pseudo-relevance to the human condition, as variously exemplified (Challenges More Difficult for Science than Going to Mars, 2014). Rather than as "scientism". perhaps there is a case for naming it as "pseudo-relevant science".

Phyllotaxis and implication of "spiral integration" for governance

Transcending the narrow perspective of science, the literature on the Triple Helix model of innovation (and its quadruple and quintuple extensions) has developed the case for a multi-spiral form in integrating the strategic roles of university-industry-government-public-environment interactions within a knowledge economy.

Separately the challenge of visualizing the relationship in 3D between multiple spiral forms has been discussed (Psychosocial Learnings from the Spiral Form of Hurricanes: implications of the triple helix and the 3-fold triskelion as "cognitive cyclones"? 2017; Framing Cyclic Revolutionary Emergence of Opposing Symbols of Identity: biomimetic embedding of N-tuple helices in spherical polyhedra, 2017; Visualization in 3D of Dynamics of Toroidal Helical Coils, 2016)

In its triple form, some consideration has been given to the relevance of the helical model to international institutions. However its potential relevance to the challenges of the United Nations is seemingly yet to be explored.

Phyllotaxis: Whilst the geometry of multi-spiral forms is a real challenge to comprehension, the form of the pineapple is well-known -- as with the sunflower (discussed below). Less evident is its integration of multiple spirals and the pattern they form. It is that pattern which leads the pineapple to be widely cited with respect to the vital importance of spiralling forms in nature -- described as phyllotaxis.

A valuable clarification through imagery is offered by Edmund Harriss (Prime Phyllotaxis Spirals, Maxwell's Demon: vain attempts to construct order, 18 March 2012). The author has subsequently acquired renown through discovery of a curve, the Harriss spiral, derivative of that by which an approximation to the Fibonacci spiral is constructed (John Dolva, Harriss Spiral, The Education Forum. 4 December 2015). The new spiral is valuable in its own right as indicative of more fruitful forms of distributed governance.

As argued separately, there is a case for exploring the value of an analogous role in psychosocial organization, notably in knowledge architecture, as well as conference and project organization (Fibonacci Spiral in 3D Framing Psychosocial Phyllotaxis: articulation of global governance through the language of flowers? 2020). The adaptation of phyllotaxis to the physical architecture of domes was noted above. In the quest for analogous adaptations, in the light of a general systems perspective and that of biomimicry, especially relevant is the possibility of drawing upon the insights in the remarkable magnum opus of Keith Critchlow (The Hidden Geometry of Flowers: living rhythms, form and number, 2011).

François Rothen Phyllotaxis or Self-Similarity in Plant Morphogenesis. In: Fractals in Biology and Medicine, Birkhauser, 1994

Phyllotaxis refers to the geometry governing the arrangement of inner florets of a sunflower..., of the scales of a pineapple ..., of the leaves around a stem and so on. The florets align with spiral whorls in the case of spiral phyllotaxis (daisy) or with helices in the case of cylindrical phyllotaxis (pineapple or fir-cone).

As queried in the earlier argument, is there an analogous geometry to be discovered (or employed) in the arrangement of the phenomena of psychosocial organization -- of institutions and their departments, of conferences and their participants, and of collective projects?

Spiral parastichy numbers: As noted by P. Prusinkiewicz, et al (Phyllotaxis: The Algorithmic Beauty of Plants, Springer, 1990), phyllotaxis is dominated by intriguing mathematical relationships. One of them is the remarkable fact that the numbers of spirals which can be traced through a phyllotactic pattern are predominantly integers of the Fibonacci sequence (R. O. Erickson, The Geometry of Phyllotaxis, 1983).

Parastichies are those criss-crossing spiral lines that appear on sunflower heads, pinecones and pineapples as a result of how these plants grow, for which interactive exploration has been provided by Andy Giger (Parastichies Explorer). As noted in the comment by the author:

The numbers of these lines curving in different directions vary, but they tend to be Fibonacci numbers. The phyllotactic spirals of sunflowers and other plants all have a divergence angle of 137.5° (the golden angle), but what happens with spirals based on other angles? As it turns out, you can produce parastichies with many different angles, but only the golden angle makes spirals for which all the parastichy numbers are Fibonacci numbers...

The pineapple is an example of a pattern where a given scale on the surface has six neighbours, belong to to 5-, 8- and 13-parastichies, having a corresponding divergence angle related to the golden ratio. In the case of the sunflower, a parastichy is not visible where the florets are most closely spaced towards the centre of the flower head. For parabolic curves, opposed parastichy pairs intersect perpendicularly near their points of minimum spacing (F. R.Yeatts, Another look at parastichies, Mathematical Biosciences, 144, 1997, 21).

Sunflower example: Despite the number of illustrations of the phenomenon in relation to the pineapple, there is a degree of confusion in the literature as summarized in an inset below. Attention can initially be more appropriately focused on a study enabled by The Royal Society in collaboration with the Turing Sunflower Consortium of the Museum of Science and Industry -- especially given the early attention to phyllotaxis by Alan Turing (Morphogen Theory of Phyllotaxis, 1952). Subsequent to his seminal role in information science, Turing has become known for his theory regarding how patterns appear in nature (Jonathan Swinton, Turing, Morphogenesis, and Fibonacci Phyllotaxis, In: Alan Turing: His Work and Impact, 2013).

The difficulty confronted by the project, even in the case of sunflowers, is that plant biologists have not worked out a mechanistic model that fully explains how the sunflower seed patterns arise. The problem is that plants don't always show perfect Fibonacci numbers -- real life being recognizably messy -- and data on real sunflower diversity is scarce.

The crowdsourced project published by the Royal Society Open Science notes that nearly one in five of the flowers had either non-Fibonacci spiraling patterns or patterns more complicated than has ever been reported, including near-Fibonacci sequences and other mathematical patterns that compete and clash across the flower's face. Counting the clockwise and counterclockwise spirals that reach the outer edge of the flower, typically results in a pair of numbers from the sequence: 34 and 55, or 55 and 89, or -- with very large sunflowers -- 89 and 144. The possibility of capturing sunflower development has thus become more realistic -- and more complicated (Jonathan Swinton, Erinma Ochu and The MSI Turing's Sunflower Consortium, Novel Fibonacci and non-Fibonacci structure in the sunflower: results of a citizen science experiment, Royal Society Open Science, 1 May 2016).

Unusually, as an open access report, it is possible to reproduce below some of the many indicative images -- typically a major constraint with those in comparable literature on the phenomenon in pineapples. The diagram on the left distinguishes the occurrence of Fibonacci numbers (+1, -1 or x2), Lucas numbers, or F4 (Fermat numbers).

Selected images of sunflower heads and contrasting spirals
Occurrence of parastichy numbers in sunflower heads Sunflower head with contrasting spirals Sunflower head with contrasting spirals Sunflower head with contrasting spirals
Sunflower head with contrasting spirals Sunflower head with contrasting spirals Sunflower head with contrasting spirals
Images reproduced from Jonathan Swinton, Erinma Ochu and The MSI Turing's Sunflower Consortium (Novel Fibonacci and non-Fibonacci structure in the sunflower: results of a citizen science experiment, Royal Society Open Science, 1 May 2016)

"Light" and "photosynthesis"? Of potential relevance in the quest for psychosocial analogues is the fact that the sunflower is especially recognized for turning during the day to follow the Sun in order to absorb the maximum amount of light. There is the curious possibility that a psychosocial analogue can be recognized in the the manner in which there is a reorientation to absorb the maximum amount of what would be recognized metaphorically as "light" -- potentially described superficially with terms relating to fashion, popularity, and the like.

The question meriting clarification is whether the geometry of phyllotaxis suggests the possibility of optimization with respect to the processing of insight and innovation by analogy to photosynthesis, as previously argued (Possibility of "psychosocial photosynthesis"?; Optimization of psychosocial "light capture", 2020). Is this similarly suggestive with respect to structural morphogenesis, as might be inferred from the work of René Thom (Structural Stability and Morphogenesis, 1972)?

The question could be epitomized by the primary process of a Zoom lecture or talk -- a webinar -- with an attentive set of spectators. The eminent focus of the lecture -- inviting a solar characterization as a source of "light" -- can then be understood as a source of "insight" whose presentation is tracked by an audience, potentially comparable with a set of sunflowers. The problem for the "sunflowers" is how best to organize their uptake of insight, anchoring it with an analogue to photosynthesis in order to render it memorable for the future. Given the charisma often recognized in the source of insight and those framed as "gurus", this also invites comparison with exposure to a so-called reality distortion field.

Dynamics of the "doughnut" implied by the sunflower? The argument above indicated the merit of switching the modelling focus from the form of a doughnut (or a torus) to that of pineapple. It is however remarkably curious the extent to which the sunflower images could be recognized as doughnuts enhanced by geometry of spiral form. The curves highlighted in colour suggest a 2D projection of a 3D toroidal form.

More remarkable is the manner in which the many available technical images of the sunflower fail to distinguish by name the obvious central core from the portion of the flower on which the curves are highlighted -- a core so clearly recalling the "hole" in a doughnut. In biological terms that core is best recognized as the portion of the flower in which florets are at an earlier stage of growth -- effectively a zone of emergence. This dynamic would be significant in any representation of the doughnut model in 3D as a torus embodying various flows (as with any smoke ring or vortex).

Systematic comparison of phyllotaxis in sunflower and pineapple: The Royal Society study offered a valuable clarification of the confusion associated with the spiral organization of the sunflower. A study, remarkable in other respects, provides a comparison of spiral organization in the sunflower and the pineapple (Riichirou Negishi, et al. Determining Parastichy Numbers Using Discrete Fourier Transforms, Forma (Society for Science on Form), 32, 2017). The study provides a valuable indication of earlier endeavours.  As in the case of The Royal Society study, the early work of Alan Turing (1952) with respect to seed patterns and Fibonacci phyllotaxis is noted.

The method used distinguishes between the alignment of florets in spiral phyllotactic whorls in 2D (in the case of the sunflower) and with 3D helical whorls in cylindrical phyllotaxis (in the case of the pineapple). As summarized:

We report a practical method to assign parastichy numbers to spiral patterns formed by sunflower seeds and pineapple ramenta using a discrete Fourier transform. We designed various simulation models of sunflower seeds and pineapple ramenta and simulated their point patterns. The parastichy numbers can be directly and accurately assigned using the discrete Fourier transform method to analyze point patterns even when the parastichy numbers contain a divergence angle that results in two or more generalized Fibonacci numbers. The presented method can be applied to extract the structural features of any spiral pattern...

A pineapple model is expressed by points on the surface of a cylinder, as shown in Fig. 5 [reproduced below]. The height l and the argument angle θ are determined by n, p, and φ....Large Fourier peaks at 13, 21, 34, and 55 were observed, which corresponded to the Fibonacci sequence (see also Fig. 6 [reproduced below]). Therefore, parastichy numbers on curved surfaces such as the pineapple model could be found using the presented Fourier method. In the case of φ = 137.8°, the Fourier peaks of 13, 34, and 47 were observed... The number 47 was a Lucas number, so the parastichy numbers were a mix of Fibonacci and Lucas numbers.

Modelling parastichy numbers in pineapple
Modelling parastichy numbers in pineapple
Reproduced from Riichirou Negishi, et al. Determining Parastichy Numbers Using Discrete Fourier Transforms, Forma (Society for Science on Form), 32, 2017

Further valuable clarification between the "flat" phyllotaxis of the sunflower and the "cylindrical" form in the pineapple is offered with illustrations by David Bachman (From the Golden Ratio to Fibonacci Phyllotaxis Spirals, Math Horizons, 26, 2018, 3). Unfortunately, despite efforts to do so, it proved problematic to illustrate the spiral patterning on the pineapple here, due to a combination of copyright constraints and the confusion in the literature between the occurrence of the 5, 8, 13 and 21 spiral types -- essentially in pineapples of two types. There are however many images available on the web. Preference was therefore given to modelling the interlocking spirals of the pineapple as explored below.

Challenging confusion in explaining interlocking spirals in phyllotaxis without illustrations
The argument with respect to a potential pineapple model calls for clarification of comments in the literature. which can be bypassed by the casual reader

The earliest extensive study was made by M. B. Linford (Fruit quality studies II: Eye number and eye weight, Pineapple Quarterly, 3, 1933):

Confusion may arise with extraordinarily small or large fruits from the fact that two other series of spirals may sometimes be recognized. These include steep, nearly vertical spirals, of which there are 21, and very flat spirals of which there are only 5. Thus the numbers of spirals of the several types are 5, 8, 13 and 21. The number of eyes counted in any spiral must be multiplied by the number of similar spirals. In other plant parts where spiral patterns occur, chance alone usually determines whether the spiral winds to the right or to the left. This was tested for pineapples by recording direction of spirals along with eye number for a large number of fruits during season 1932.... If chance determines we should expect to find equal numbers of right and left patterns. Actually among 9,008 fruits, right spirals were found 49.4 percent of the time, left spirals 50.6. Only one of the lots..., and the 24 separate plots in that group showed no agreement among themselves; some had more left spirals and some more right. Chance alone seems to determine the direction of these spirals in the pineapple,

Paul C. Ekern Phyllotaxy of Pineapple Plant and Fruit Botanical Gazette 129, 1968,  1):

Re-examination of the phyllotaxy of Cayenne plant-crop fruit from old photographs as well as from current fruit confirms a 21/55 phyllotaxy, with n + 8, 13, and 21 contact parastichies. Earlier reports of an 8/21 phyllotaxy are not sustained, even for very small fruits from ratoon plants....

There usually are eight gently sloping spirals composed of every eighth fruitlet (taken in the order of their formation)  and 13 steeper spirals composed of every thirteenth fruitlet. Hence the phyllotaxy of the inflorescence and fruit based on contact parastichies is 8 and 13. A very steep spiral, with 21 such spirals composed of every twenty-first fruitlet, is also present. Since this spiral is not vertical, the method of expression of phyllotaxy based on a vertical array of every twenty-first fruitlet is not valid; therefore the 8/21 fruitlet phyllotaxy given by Lindford (1933)... is not correct. The true vertical coincides with the fifty-fifth fruitlet; hence the phyllotaxy should be 21/55. The fifty-fifth is readily located by counting up two fruitlets from the fruit base, in the steepest spiral (2x21, or 42 fruitlets), and then over one in the next steepest spiral (every thirteenth), for a total of 42 plus 13, or the fifty-fifth fruitlet...

Summarizing the study by Lindford (1933), without reference to Ekern (1968), Philip B. Onderdonk Pineapples and Fibonacci numbers, The Fibonacci Quarterly, 8, 1970. 5) notes:

The vast majority of pineapples checked had 8-13-21 rows of fruitlets (eyes). A few runts were 5-8-13. Since a pineapple with more fruitlets for a given size would likely have a finer texture and would be better for eating the author was interested in finding a pineapple with 13-21-34 rows. No such pineapples were ever found however.

 Commenting on Onderdonk's study, Michael Wirth: noted that he

... found that the majority of pineapples had 8-13-21 rows of fruitlets, with a few smaller ones at 5-8-13. It was suggested that a pineapple with more fruitlets for a given size would have a finer texture, and it was hoped that a pineapple with 13-21-34 rows could be found, however Onderdonk never found any pineapples exhibiting such a pattern. In general, pineapples have three series of spirals, derived from the roughly hexagonal pattern of its fruitlets, or scales. (Fibonacci and Pineapples, The Craft of Coding, 28 May 2020)

John Catlan The Fibonacci Sequence and Pineapples, Bromeliad Encyclopedia)

We find the leaf phyllotaxy is 5/13. The five is the number of spirals around the stem and thirteen the number of leaves removed in the five spirals until another bud, the fourteenth is found directly above bud number one. When a pineapple is formed hormones change the phyllotaxy from 5/13 of the leaves to 8/21 of the fruit. In normal fruit the number of rows of each type of spiral is constant, there being eight of the long gently sloping rows and thirteen of short steep ones.The fruit hormones cuts out and the phyllotaxy changes from 8/21 of the fruit to 5/13 of the leaves abnormalities in fruit and top development, such as double fruit, fan tops, multiple tops are the results of irregularities in these hormone driven phyllotaxis change.

Requisite complementarity of governance models comprehensible through metaphor

It can be argued that the complexity of the global condition, and the variety of perspectives it engenders, call for comprehension of the situation through a complementary set of metaphors.

The reference above to the relevance of insights from flowers and nature in general is consistent with the arguments of Jozef Keulartz (Using Metaphors in Restoring Nature, Nature and Culture, 2. 2007, 1) and of Thomas Wiben Jensen and Linda Greve (Ecological Cognition and Metaphor, Metaphor and Symbol, 34, 2019, 1).

The argument can be presented otherwise:

The question requiring clarification is how any such understanding relates to the possible models of global governance -- and especially the conditions under which each model may be especially relevant. As indicated above, approaches variously include use of:

These metaphors variously rely on increasingly complex geometry, as can be otherwise clarified (Metaphorical Geometry in Quest of Globality -- in response to global governance challenges, 2009; Engaging with Globality through Cognitive Lines, Circlets, Crowns or Holes, 2009). Curiously such geometry has been variously embodied in ceremonial objects, much-valued for their symbolic implications (Engaging with Globality through Cognitive Crowns, 2009; Imagining local-global connectivity through innovative mace and vajra design, 2019).

Comparison of governance models based on different geometries
Geometry Depiction Governance Integration Comprehensibility Memorability Symbolism
Points / Lines Checklist Declaration Low High Low Sceptre
Plane / Matrix  Plan Spreadsheet High Relatively low Low Domain
Circle / Cycles Doughnut Roundtable/Ring Relatively high? Relatively high Lower Circlet/Tiara/Ring/Sash
Torus Space habitat? ? Higher Higher? ? Crown
Helix "Spiral" stair? ? Higher Lower? Lower "Spiral" stair
Spiral "Black hole"? Financial drain Higher Lower? Lower ?
Braid Wampum Wampum Higher Relatively high Relatively high Wampum
Sphere Globe Sphere of influence Higher Relatively high Relatively high Orb

Of further relevance are the references to degrees of integration of an even higher order, as required in the organization of a hypothetical "global brain" as it might be enabled by artificial intelligence (Peter Russell, The Global Brain: speculations on the evolutionary leap to planetary consciousness, 1983; H. G. Wells, World Brain, 1938; Francis Heylighen, The Global Superorganism: an evolutionary-cybernetic model of the emerging network society, Social Evolution and History, 6, 2007).

The possibilities are especially relevant to governance given the reference to the studies of phyllotaxis by Alan Turing (cited above) and his own speculation with regard to some form of "hypercomputer" (Michael BrooksTuring's Oracle: the computer that goes beyond logicNew Scientist, 16 July 2014).

Such possibilities can be variously explored in relation to the challenges of governance:

Somewhat ironical, in the currently surreal framing of global governance, is the manner in which transcending models based on spherical geometry is framed through the popular slogans used by many leaders to make their country "great again". This could be seen as strangely associated with the use of rockets to get "off the planet" and into orbit -- whether to live in toroidal space habitats or/and to travel "elsewhere".

Alternation between complementary models of global governance

Models and their associated metaphors can indeed be recognized as complementary, each being preferred for one reason or another by different interest groups. More intriguing however is the manner in which they are interrelated, and how the focus on one may be transformed into the focus on another. What indeed are the transformation pathways between them and what are the metaphors which help to comprehend and enable such shifts in governing patterns of organization.

Geometry is especially helpful in this respect, given that such transforms have long been a focus of interest associated with the generalization of 2D and 3D forms to their higher dimensional analogues. The animation on the left is indicative of the relationship between any strategic "plan", any circular organization, and a torus. The relation between models based on torus and/or sphere is suggested by the second animation. The images on the right are indicative of the role of 4D in configuring the challenges of oppositional logic in governance discourse, as discussed separately (Oppositional logic and its requisite polyhedral geometry, 2019; Oppositional Logic as Comprehensible Key to Sustainable Democracy: configuring patterns of anti-otherness, 2018)

Animation engendering
Villarceau circles
Logic Alphabet Tesseract
by Shea Zellweger
3D projection of a 4D tesseract
performing a double rotation
Animation engendering Villarceau circles Torus-Sphere transformation Logic Alphabet Tesseract by Shea Zellweger 3D projection of a 4D tesseract
By Lucas Vieira - Own work, Public Domain, Link Lucas Vieira / Public domain By Warren Tschantz
(reproduced from the Institute of Figuring)
Jason Hise / CC0

Such insights are however far from helpful for those unskilled in that discipline. Of interest is therefore the metaphors which enable comprehension of such shifts and any ability to map the pathways along which (or via which) transformation is possible. Such possibilities can be variously envisaged:

Especially valuable as metaphors, given the widespread accessibility of such devices and their comprehension, are:

Mnemonic aids to a set of contrasting strategic patterns?
As suggested by a well-known song? As suggested by a gear shift pattern in trucks Single pattern: UN SDGs as "strategic gears" ? Juggling pattern
symmetric drawing of 3-ball state graph
On the 12th day of Christmas my true love sent to me:
12 drummers drumming
11 pipers piping
10 lords a-leaping
Nine ladies dancing
Eight maids a-milking
Seven swans a-swimming
Six geese a-laying
Five golden rings
Four calling birds
Three french hens
Two turtle doves, and
A partridge in a pear tree
Example of gear shift pattern in trucks
  1. No Poverty
  2. Zero Hunger
  3. Good Health and Well-being
  4. Quality Education
  5. Gender Equality
  6. Clean Water and Sanitation
  7. Affordable and Clean Energy
  8. Decent Work and Economic Growth
  9. Industry, Innovation, and Infrastructure
  10. Reducing Inequality
  11. Sustainable Cities and Communities
  12. Responsible Consumption and Production
  13. Climate Action
  14. Life Below Water
  15. Life On Land
  16. Peace, Justice, and Strong Institutions
  17. Partnerships for the Goals
Juggling pattern: symmetric drawing of 3-ball state graph

In the light of recognition of juggling as a metaphor of governance (as noted above, the image on the right is instructive, a s reproduced from Burkard Polster (The Mathematics of Juggling, 2006, p. 62). This fruitfully integrates perspectives regarding polyhedra (and their great circles), Hamiltonian cycles and distinctive complex juggling patterns (with an annex of stereograms of more complex Hamiltonian cycles).

A particularly complex gear shift pattern (above centre) was used as an illustration in a discussion of Global Coherence by Interrelating Disparate Strategic Patterns Dynamically (2019). The latter focused on the topological interweaving of 4-fold, 8-fold, 12-fold, 16-fold and 20-fold patterns in 3D. Particular focus was given there (in the follow sections) to the UN's Sustainable Development Goals, namely a set of 16 with one additional coordinative goal:

Framing an operating context of 16 "dimensions"
Functional dynamics of a 16-fold configuration of strategic goalsModifying the form: unfolding, exploding, stellation morphing

Modifying the cyclic symmetry of the star torus
Imagining further implications: helical coil, toroidal knot or crown?

Interweaving spirals as embodied in zome configuration

Absence of detailed spiral imagery of relevance to governance: Given the current interest in the Triple helix model of innovation, and its extension to quadruple and quintuple variants, (as noted above), it is surprising to note the relative (if not complete absence) of imagery illustrative of the manner of interweaving of such helices. By contrast, this is of course a major focus of study in the case of the double helix by which those models have been inspired.

Part of DNA molecule Triple helix Triple Helix concept 3-ball juggling pattern 3-strand helical spiral
DNA animation Triple helix Triple Helix concept 3-ball juggling pattern 3-strand helical spiral
By brian0918 [Public domain], from Wikimedia Commons Source: webpage on triple helix of Loet Leydesdorff   Juggling pattern as a braid
Monash University, Melbourne

It could be said that global governance is faced with a riddle as to how to interweave its variously incommensurable preoccupations, and to associate detailed meaning as in the case of DNA  (Global Governance as a Riddle: but is a solution the answer to the question? 2018). This has been expressed in terms of the legend of Alexander the Great's Gordian Knot. as discussed separately (Mapping grossness: Gordian knot of governance as a Discordian mandala? 2016). Knot theory is associated with the theory of braiding (as mentioned above). Metaphorically this frames the challenge of governance as one of weaving (Warp and Weft of Future Governance: ninefold interweaving of incommensurable threads of discourse, 2010).

Zome configuration: The question is how this might apply to interweaving the kinds of helix highlighted, or those taking spiral form. It is in this respect that the architecture of zomes is of particular interest as featuring unusual geometries. The term derives from a combination of dome (as promoted by Buckminster Fuller) and zonohedron. Zomes have acquired a related sense through the role of Zometool as a model-construction learning device.

The inherently comprehensible architecture of a zome (as physically constructed) is of particular relevance to this argument in the light of the sophisticated elegance of the underlying mathematics of the zonohedron. This is described as a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric. Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in three-dimensional space, or as the three-dimensional projection of a higher dimensional hypercube. Many images of zomes are available on the web. The London "Gherkin" skyscraper (mentioned above) is recognized as the largest approximation to a zonohedron that has been constructed physically.

There is a degree of irony to the fact that as a "dome", a zome is typically constructed as half of a zonohedron. On the other hand, as a zonohedron, the "under half" of a zome offers a valuable pattern for mapping what is typically "back channel", "under the table" or repressed in collective discourse -- if not unconscious, as argued by John Ralston Saul (The Unconscious Civilization, 1995).

Zonohedra have the property that all edges belong to one of a few sets of parallel lines. In the case of a polar zonohedron, with polar symmetry, it follows that the edges are all equal and the faces are all rhombi. Note the sine curves formed by the edges. They feature in the indication by Steven Dutch of the Best Representations of 3-Dimensional Symmetry: Equilateral (2009), as later updated with respect to zonohedra (Symmetry, Crystals and Polyhedra: zonohedra, 2018).

The specific focus here is how zome architecture can render comprehensible the interweaving of spiral forms -- and as such to be of value with respect to the organization and knowledge architecture of governance, as previously argued (Zomes as a key to appropriate organizational and knowledge architecture? 2020). As noted above, earlier interest in the dome and its underlying principles of tensegrity, have not been significantly translated into forms of psychosocial organization, despite their potential in this respect and the cognitive implications of the title of the magnum opus of Buckminster Fuller (Synergetics: explorations in the geometry of thinking, 1975/1979), as separately emphasized (Geometry of Thinking for Sustainable Global Governance: cognitive implication of synergetics, 2009).

The argument above has emphasized the role of the golden ratio in the spiralling of phyllotaxis. Clarification in this regard with respect to zome construction is provided by Tom Davis (The Mathematics of Zome, 4 July 2007). He shows that the lengths of zome struts and the locations of the connectors in an arbitrary configuration satisfy a very simple rule that involves the golden ratio.

It should be emphasized that the concern here is not with the physical construction of zomes as currently explored, but rather with their potential role in virtual organization -- as implied by the challenge of articulating communication in virtual conferences fruitfully, as with the current UN General Assembly. The question is then how can communication pathways be "constructed" and "organized" electronically in a complex virtual conference -- as has long been a preoccupation of collaborative software ("groupware").

Nine-fold exercise: Given the emphasis that has been placed on the global governance strategic challenge of 9 "planetary boundaries" (as noted above), an earlier exercise focused on 9-fold symmetry, with images as reproduced below (Concordian Mandala as a Symbolic Nexus: insights from dynamics of a pentagonal configuration of nonagons in 3D, 2016). More complex examples of relevance to complex virtual conference organization are considered below.

The earlier exploration of nine-foldness presented the following model of a zonohedrified 9-gonal antiprism --  a 9-fold zome. This has 74 vertices (5 types), 72 faces (4 types), and 144 edges (9 types). This combination of 72 and 74 is fortuitous given the animations developed previously with regard to a coronavirus form assumed to have 72 or 74 protein spikes. The wireframe variants below are also helpful in giving a sense of the interwoven sine curves formed by the edges.

Although the actual structure of the zonohedrified 9-gonal antiprism does not exist in distinct right- and left-handed forms, such asymmetric chirality is evident when the zones are coloured as shown below. Of particular interest with respect to any mutual entanglement of the nine planetary boundaries is the alternation in perspective according to how the curving patterns of zones are coloured.

Views of a 9-frequency zome -- a zonohedrified 9-gonal antiprism with 9-fold symmetry (9*2m)
Side views Polar views
Rings colour coded Zones colour coded Rings colour coded Zones colour coded
9-frequency zome  with 9-fold symmetry 9-frequency zome  with 9-fold symmetry 9-frequency zome  with 9-fold symmetry 9-frequency zome  with 9-fold symmetry
Chiral contrasts as a result of colour coding
right side left side right polar left polar
9-frequency zome  with 9-fold symmetry 9-frequency zome  with 9-fold symmetry 9-frequency zome  with 9-fold symmetry 9-frequency zome  with 9-fold symmetry
Wireframe variants (of models above)
9-frequency zome  with 9-fold symmetry 9-frequency zome  with 9-fold symmetry
Animation of folding/unfolding of 9-fold zome -- a zonohedrified 9-gonal antiprism with 9-fold symmetry
Rings colour coded Zones colour coded
Animation of folding/unfolding of a 9-frequency zome Animation of folding/unfolding of a 9-frequency zome
Images and animations made using Stella Polyhedron Navigator

In endeavouring to get a sense of the coronavirus as a whole, of some value are the animations above in which the two colour-coded variants are presented in a cycle of folding and unfolding. These could be understood as of some relevance to comprehension of the set of nine planetary boundaries, namely how they might be considered interrelated and interwoven in nature -- in systemic terms.

Zome spirals: Given the preoccupations of those primarily interested in zomes and their construction -- whether mathematicians, alternative architects, or teachers -- the spiral patterns (of relevance to this argument), which are evident in the models above, are not especially evident in the literature, to whatever degree they are mentioned (George W. Hart and Henri Picciotto, Zome Geometry: hands-on learning with zome models, 2001; Zome Spirals: additions to zome geometry; additional photos). A valuable interactive demonstration is available (Sándor Kabai, Polar Zonohedron with 18 Zones, Wolfram Demonstrations Project, November 2019).. Further clarification is provided by René K. Müller (Zome, Simply Different. 2009),

It was therefore strange to discover the emphasis given specifically to a "zome spiral" (and its depiction) as belowin the construction of a memorial shrine of Islamic inspiration -- an 8-fold zome (Eric Doud, Creating the Dargah, Faith and Form, 52, 2019, 1; The Dargah Project: zome in place, WindRiverTimberFrames, 22 May 2017).

Zome spiral featured in building construction
Section of Dargah shrine Plan of Dargah shrine
Zome spiral shown in section view of Dargah shrine Zome spiral shown in plan view of Dargah shrine
Images by Eric Doud as architect for Designworks Architecture (with kind permission)

Approximating a pineapple model by modification of a zome configuration

The argument has emphasized the relative comprehensibility and memorability of a pineapple relative to various less familiar geometrical abstractions which may be oversimplified or excessively complex. It is intriguing to note that the pineapple can be understood as a form of geometrical "compromise" between:

There is therefore a case for exploring how a zome configuration can be modified to approximate to a greater degree to the form of the pineapple, especially in order to reflect the interlocking helical patterns by which it is characterized. The earlier focus on nine planetary boundaries remains a valid exercise given the preoccupation of global governance -- and hence the provisional value of the 9-fold zome. Ironically the examples offered below also suggest a degree of approximation to the form of a sunflower.

A challenge to the strategic imagination, if lessons are to be learned from combining insights from both the coronavirus and from a pineapple model, is whether the dynamics between one form and its geometric "reflection" are of particular significance to global governance. To this end the transformation through various methods of morphing are indicated in the animations below (Carl Erikson, Morphing Three Dimensional Polyhedral Objects, 1994; Robert Webb, Morphing polyhedron compounds in Stella 5.0. YouTube, 2012). Such morphing reflects an aspect of the the long-standing consideration of "variable geometry" in institutions (Alternation between Variable Geometries: a brokership style for the United Nations as a guarantee of its requisite variety, 1985).

Animations of various modes of morphing between a 9-frequency zome and its dual
(alternation between 72/74 and 74/72 configurations)
Dual Morphing by tilting triangles Morphing by titling to compound Morphing by titling to rectify
Dual of a 9-frequency zome  with 9-fold symmetry Morphing between a 9-frequency zome  and its dual Morphing between a 9-frequency zome  and its dual Morphing between a 9-frequency zome  and its dual
Animations made using Stella Polyhedron Navigator

Augmenting the polyhedra with "pine spikes" -- in a further approximation to the form of a pineapple? The previous discussion focused extensively on global configurations of spikes. The base model and dual can however be modified by spikes (in the form of prisms) on each face -- whether projecting outwards or inwards. The version on the left is slightly tilted with the top face rendered transparent.

Augmenting faces of 9-frequency zome and dual with prisms (projecting out or in)
Externally oriented spikes? Internally oriented spikes?
Base with prisms
(288 F; 146 V; 432 E)
Dual with prisms
(218 F; 126 V; 342 E)
Base with prisms
(234 F; 128 V; 360 E)
Dual with prisms
(236 F; 126 V; 360 E)
Prisms on faces of a 9-frequency zome Prisms on faces of dual of a 9-frequency zome Prisms projected into faces of a 9-frequency zome Prisms projected into faces of dual of a 9-frequency zome
Animations made using Stella Polyhedron Navigator

Use of the software to manipulate the spikes on such approximations to a pineapple, recalls the discussion of alternation between an explicate order and implicate order by the theoretical physicist David Bohm are ontological concepts for quantum theory (Wholeness and the Implicate Order, 1980). Imagining such alternation, described by Bohm as a holomovement, can be aided to some degree by animating the projection outward and inward of prisms of a given type on the pineapple models above -- and their separation from the model.

The relevance of Bohm's perspective from quantum theory has since become greater with the insights in the extensive study from an international relations perspective by Alexander Wendt (Quantum Mind and Social Science: unifying physical and social ontology, 2015). Arguably the case for the pineapple model was reinforced by the interlocking helical waves aligning the spikes on the pineapple (following a Fibonacci spiral, as discussed above). Any reference to wave-like phenomena, necessarily extensively discussed by Wendt, emphasizes the degree to which any viable global strategy might fruitfully take account of a form of subtlety typically ignored by strategists. It is of course the case that the appeal of any strategy to the wider population -- as being reasonable -- may well be reinforced by a pattern of connectivity and coherence characteristic of both rhyme and wave-like rhythm, as characteristic of music song and poetry (Caspar Schwabe, The Zonohedra Music Chart).

Enhancing communication patterns in virtual conferences with zome architecture

This argument has developed the implications of a form of "psychosocial phyllotaxis" as a means of enhancing insight capture in conferences, then to be understood as a form of "psychosocial photosynthesis" (Possibility of "psychosocial photosynthesis"?; Optimization of psychosocial "light capture"). The argument was inspired by the widespread use of Zoom technology as a result of pandemic lockdown, and the unusual degree of dependence of the United Nations General Assembly on such technology.

It is remarkable to note that the uptake of virtual conferencing has not been matched by new insights into how the communication patterns might be more fruitfully organized -- especially in assemblies involving hundreds of participants. It would appear that procedures for conventional gatherings have simply been adapted to the virtual environment with no effort to derive advantage of the potential flexibility of that context and the necessarily enhanced role of computer technology, and that expected from artificial intelligence.

As noted, the argument with respect to enhancing light capture through phyllotaxis in nature has been variously explored (Takuya Okabe, Biophysical optimality of the golden angle in phyllotaxis Scientific Reports, 5, 2015, 15358; Sören Strauss, et al, Phyllotaxis: is the golden angle optimal for light capture? New Phytologist, 225, 2019, 1). The question is how some analogue might be orchestrated in virtual conferencing to enhance "insight capture" as suggested by the spiral patterns in zome configurations. As emphasized, the focus is on communication patterns among a virtual configuration of participants, whether in terms of intervention, dissemination of messages to sub-groups, or thematic organization -- all potentially evolving during the course of the conference, rather than being predefined as is otherwise the conventional practice.

The possibilities have long been envisaged and variously explored through proprietary groupware packages (available under licence), although their relevance to assemblies of international institutions and parliaments is less evident (The Challenge of Cyber-Parliaments and Statutory Virtual Assemblies, 1998). The matter could be deemed increasingly urgent given the degree to which parliamentary assemblies are simply cancelled in the face of pandemic threats, or the results of "national debates" are far from being what is claimed (Multi-option Technical Facilitation of Public Debate: eliciting consensus nationally and internationally, 2019).

Rather than articulate the possibilities verbally, the following images are presented as an indication of communication templates which merit further exploration. In interpreting their potential value in mapping conference-related dynamics, the following possibilities could be considered:

21-freqeuency Zome -- a zonohedrified 21-gonal antiprism (side and polar views)
(420 faces of 10 types; 840 edges of 21 types; 422 vertices of 11 types)
Zones coloured Wireframe variant
21-frequency zome 21-frequency zome
21-frequency zome 21-frequency zome
Images made using Stella Polyhedron Navigator
Animations of transformation of 21-fold zome to related forms by morphing
By sizing By truncation By augmentation By expansion
Morphing of zonohedron by sizing Morphing of zonohedron by truncation Morphing of zonohedron by augmentation Morphing of zonohedron by expansion
By tilting quadrilaterals By tilting triangles By tilting to compound By tilting to rectify
Morphing of zonohedron by tilting quadrilaterals Morphing of zonohedron by tilting triangles Morphing of zonohedron by tilting to compound Morphing of zonohedron by tilting to rectify
Animations made with Stella Polyhedron Navigator
Examples of modification of scaling along polar axis of 21-fold zome
Increased scaling along polar axis of 21-frequency zome Increased scaling along polar axis of 21-frequency zome Increased scaling along polar axis of 21-frequency zome Increased scaling along polar axis of 21-frequency zome
Images made using Stella Polyhedron Navigator and X3D-Edit

It is somewhat ironic to note that the zome configurations shown could be recognized as variously approximating both the pineapple model (on which emphasis has been placed) and the sunflower pattern in relation to which it has been elaborated. There is a further irony that, in developing the argument from floral phyllotaxis, the zome configurations are consistent with distinctive aspects of the work of Keith Critchlow (The Hidden Geometry of Flowers: living rhythms, form and numbers, 2011; Islamic Patterns: an analytical and cosmological approach, 1976).

Comparison of alternative zome configurations: The approach taken to the creation of zomes using the application Stella4D, derived from the advice originally given by Robert Webb who developed that application and its facility for creation of zonohedra. The advice related to the earlier focus on a zome of 9-fold symmetry in the light of the strategic issue of 9 planetary boundaries. Without the advantage of other critical insight into zome mathematics, that procedure has been followed for the creation of the 21-fold pattern above, and in the following exploratory creation of analogous patterns. Zome configurations could well be developed by zonohedrification of other polyhedra and by other methods.

The selection of examples is based on a mix of numbers of N-fold zomes -- zonohedrified N-gonal antiprisms:

This exercise in identifying polyhedra of value to mapping strategic preoccupations, follows from earlier exercises (which did not focus on zonohedra):

With respect to the following images, it is  surprising to note with respect to potential memorability (although requiring mathematical clarification), that:

Exploration of zome configurations of potential relevance to governance
with numbers of faces, edges and vertices, in each case (number of types of each in parentheses)
20f (2); 40e (5); 22v (3)
56f (4); 112e (7); 58v (5)
72f (4); 144e (9); 74v (5)

552f (12); 1104e (23); 554v (13)

156f (6); 312e (13); 158v  (7)
5-frequency zome 8-frequency zome 9-frequency zome  with 9-fold symmetry 12-frequency zome 13-frequency zome

992f (16); 1984e (31); 994v (17)

272f (8); 544e (17); 274v (9)
20 (no symmetry))
1560f (1560); 3120e (3120); 1562v (1562)
420f (10); 840e (21); 422v (11)
30 (no symmetry)
3540f (3540); 7080e (7080); 3542v (3542)
16-frequency zome 17-frequency zome 20-frequency zome 21-frequency zome 30-frequency zome
Images made using Stella Polyhedron Navigator

As noted earlier with respect to the 9-gonal pattern, the mix of 72 faces with 74 vertices is somewhat ironic given the current global challenge represented by a coronavirus with a number of spikes estimated to be of that order (and widely depicted), as illustrated separately (Cognitive Engagement with Spike Dynamics of a Polyhedral Coronavirus, 2020; Spike-endowed Global Civilization as COVID-19, 2020).

In the quest for the relevance of such patterns to new approaches to knowledge architecture and the organization of communication patterns in virtual conferences, there is clearly an advantage to be able to switch dynamically between distinctive patterns of N-foldness, whether as required by the gathering as a whole or as preferred by individual participants. Of particular relevance to how and why such selections are preferred, the cognitive arguments of George Lakoff and Rafael Nuñez merit consideration (Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2001).

Potential indication of governance communication pathways in a zome configuration

Requisite complexity of integration of 9 planetary boundaries: Great significance is attributed (in principle, at least) to the environmental constraints of the "planetary boundaries" (as noted above), and as recognized in the doughnut model of economics. There is therefore a strong case for recognizing the geometrical patterns of the edges of a 9-gonal zome (in contrast to the emphasis placed above on the distinctive possibilities of its face colouring). These patterns are indicative of the integrative complexity which merits consideration in the dynamics of any related discourse and in the relevant knowledge cybernetics (Maurice I. Yolles, Knowledge Cybernetics: a metaphor for post-normal science, Cybernetics and Systems Theory in Management, 2010).

Animations of 9-gonal zome with edges coloured according to different conventions
Commentary on the colouring distinctions is offered in the Stella Manual.
Each edge type is given its own colour
(but not mirror reflections)
Each edge type is given its own colour
(including reflections)
Edges are coloured the same only if they lie in the same planes through the centre
(as great circles)
9-frequency zome with edges coloured by type 9-frequency zome with edges coloured by type 9-frequency zome with edges coloured by great circle
Edge colours set by averaging the colours of connected faces. Edge colors set by averaging the colours of connected faces (and taking its complement) Edges coloured the same only if they are parallel.
9-frequency zome with edges coloured by averaging connected neighbours 9-frquency zome with edges coloured by averaging connected neighbours 9-frequency zome with edges coloured by parallel
Animations made using Stella Polyhedron Navigator

A polyhedral form used for ordering issues of relevance to governance, such as the 9-fold zome above, can be understood as being in dynamic relationship with is geometric dual, namely where the where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. The dual of the 9-fold zome is indicated below with the edges coloured according to various conventions (as above).

Geometric duals of 9-gonal zome with edges coloured according to different conventions
Each edge type is given its own colour
(including reflections)
Each edge type is given its own colour
(but not mirror reflections)
Edges are coloured the same only if they lie in the same planes through the centre
(as great circles)
Edges coloured the same only if they are parallel. Edge colours set by averaging the colours of connected faces.
Dual of 9-frequency zome with edges coloured by type Dual of 9-frequency zome with edges coloured by type 9-frequency zome with edges coloured by great circle Dual of 9-frequency zome with edges coloured by parallel Dual of 9-frequency zome with edges coloured by averaging connected neighbours
Duals derived using Stella Polyhedron Navigator

With respect to the quest to reflect the pattern of protein spikes on a coronavirus, the 9-fold zome -- a zonohedrified 9-gonal antiprism -- is of some interest given the illustration earlier of 72-fold and 74-fold patterns of such spikes on that virus, as previously discussed (Cognitive Engagement with Spike Dynamics of a Polyhedral Coronavirus, 2020; Spike-endowed Global Civilization as COVID-19, 2020).

As noted there the 9-fold zonohendron has 74 vertices (5 types), 72 faces (4 types), and 144 edges (9 types). Its geometric dual has 74 faces (5 types), 72 vertices (4 types), and 144 edges (9 types). Both forms suggest a degree of approximation to the pineapple form, whether or not further investigation highlights one in which patterns of pentagonal or hexagonal zones prove relevant. The potential alternation between the two patterns is itself of interest.

Despite widespread depiction of artists impressions of protein spikes on the virus (readily available on the web), it should be emphasized that details of the virus are in fact too small to be seen by any means and can only be inferred. Confirmation as to the number of spikes is not evident in the literature, however realistic the depictions may be assumed to be. However, if the depicted form of the virus (with which the world is "at war") is essentially imaginary, there is every case for imagining an appropriate response to whatever form it may take.

Details of 9-fold zome pattern
Indication of 9 of the 18 edge loop patterns (with others present but uncoloured) Animation clarifying the relation between 9 loops and their spiralling form Highlighting 74 vertices (of which 2 are 9-valent
9 edge loop patterns of 9-frequency zome Animation of 9 spiralling edge loop patterns of 9-frequency zome Indication of 74 vertices  of 9-frequency zome
Made using Stella Polyhedron Navigator and X3D-Edit

Of further interest with respect to the pattern of 144 edges, is the distinctions that can be appropriately and interrelated by that pattern. This is discussed separately (Challenge of mapping 144 distinct forces in 3D; Mutual entanglement of two patterns of 72-foldness?; Reframing binary governance as minimally a fourfold challenge?; Reframing the mapping challenge of 144 distinctions in terms of 288, 2020).

Experimental rotation of spirals: As indicated above with respect to the pineapple model, of interest is the manner in which spirals in opposite directions interweave or interlock. In a preliminary exercise (below left), the 21-fold zome configuration with coloured faces was nested within another of the same configuration in wireframe form. The two forms move in counter-rotation as a means of indicating the dynamics of spiral interweaving. Different speeds could be chosen.

In a second exercise, a 13-fold configuration (in green) was added to explore multi-configuration effects, With the coloured version static, this could be indicative of shifting patterns of communication between fixed positions. The animations (below) are crude and jerky because of limitations in the recording technology, web technology, and the computer processing load. They are far better viewed using interactive virtual reality technology.

Indicative animations of 21-frequency zome -- a zonohedrified 21-gonal antiprism with counter-rotating spiral configuration
Animation of 21-frequency zome  with counter-rotating spiral configuration Animation of 21-frequency zome  with counter-rotating spiral configuration Animation of 21-frequency zome  with counter-rotating spiral configuration
Animations made using Stella Polyhedron Navigator and X3D-Edit

The approach and the effects recall the separate experiments in nesting 20 rotating rings of 50 "lotus-petals" (Satellite Constellation and Crown Chakra as Complementary Global Metaphors? Experimental representation of crown chakra in virtual reality, 2020). The question is whether such nested rotation is indicative of the manner in which empowering insight is stepped-up or down in systems of communication in a manner resembling systemically the mechanical gear transmissions (as mentioned above).

Nesting of multiple zomes, with suitable adjustment of relative speed of rotation, could give rise to Moiré patterns of some relevance. An indication of this possibility with respect to the governance of 30 disabling and 30 enabling trends is discussed and illustrated separately (Spiraling trends: cyclones in a climate of change?; Interweaving "cyclones" and "anti-cyclones" in a global system, 2012).

There is also a degree of similarity with antenna design (Italo Mario Fabbri, (The Spiral Solenoids and the Leaf Antenna in Phyllotaxis Differential Geometry, University of Milan, 2018; Jean-Christophe Angevain, et al, Phyllotactic arrangements of reflector mesh facets to decrease grating lobes9th European Conference on Antennas and Propagation, 2015). A related possibility is the subject of a patent (Aidan Rhys Dwyer, et al, Photovoltaic array utilizing phyllotaxic architecture, US20120260967A1, 2012).

From the perspective of governance, knowledge architecture, and communication processes -- notably in virtual gatherings -- virtual zome configurations could be usefully explored in terms of time management, namely a dynamic emphasis rather than from a static understanding alone.

Simulation of flows along communication pathways: 17-fold UN SDGs: Given the surprising degree of symmetry in the 17-fold zome configuration, this suggests an exploration of its role with respect to the 17 Sustainable Development Goals, now framed as so fundamental to global governance. As previously noted, it remains unclear how that number of goals was derived, especially given its seeming lack of memorable symmetry -- especially in contrast to a 16-fold pattern (noted above as engendering a more complex pattern in a zome configuration).

In the following animations use is made of small spheres moving along distinctively coloured spiral pathways in the zome configuration.  Only a limited set of coloured pathways is shown for that purpose, with a further set of white spheres moving along selected pathways spiralling in the reverse direction -- namely cutting across the coloured pathways.

Such animations offer numerous possibilities for mapping contrasting understandings of communication processes. These include colour choices, speeds of movement, number of spheres on a pathways (and their direction of movement), thickness of pathways. There are clearly aesthetic design considerations to enhance memorability.

Animations of communications suggested by movement of coloured spheres
Non-transparent Wireframe
Animations of communications suggested by movement of coloured spheres over a zome configuration Animations of communications suggested by movement of coloured spheres over a zome configuration
Animations made using Stella Polyhedron Navigator and X3D-Edit

Especially intriguing is any sense that the movement of spheres, or the development of other features, could evolve over time -- especially during the course of a virtual conference -- in order to reflect a greater understanding of complementarity, if not consensus. Musical metaphors are of relevance in this respect.

Given the importance of the interlocking feedback loops associated with the 9 planetary boundaries, the above approach can be repeated with the 9-fold zome of spiralling loops, as speculatively indicated below. As presented, this can be recognized as somewhat reminiscent of the doughnut model. The central torus is however also reminiscent of the design challenges of the ITER nuclear fusion reactor and the insights it offers, as discussed separately (Enactivating a Cognitive Fusion Reactor: Imaginal Transformation of Energy Resourcing (ITER-8), 2006). The challenge of the latter is to stabilize the circulating plasma by rings of electromagnets -- of which the feedback loops indicated are indicative in the governance analogue.

Indication of selected feedback loops associated with 9 planetary boundaries framing a coherent global dynamic
  9 balls circulating around 9 of the 18 spiralling loops
(with 9 others present but uncoloured)
Animations made using Stella Polyhedron Navigator and X3D-Edit

The coloured loops, seemingly unrelated, could however be seen as forming a continuous helix of 9 windings -- especially if the colour is understood as transformed (in the central vortex) to that of the next loop in the sequence. The uncoloured set of 9 loops, which interlock with the first set, would then constitute a complementary set of helical windings. Such patterns of winding recall the electromagnetic operations of a solenoid.

The possibility of related illustrations of 9-foldness is discussed separately (Visualization in 3D of Dynamics of Toroidal Helical Coils -- in quest of optimum designs for a Concordian Mandala, 2016; Concordian Mandala as a Symbolic Nexus: insights from dynamics of a pentagonal configuration of nonagons in 3D, 2016)

Elongation into dome configuration: 17-fold UN SDGs framed otherwise: As indicated above, the zome configuration can be "stretched" vertically to correspond to a "dome" -- emphasizing that this is a configuration in virtual reality. The vertical elongation could evolve over time. This may have the the merit of associating the communication architecture with that of some traditional domed edifices. 

Animations of communications in configurations of heightened domed form
Non-transparent Wireframe
Animations of communications suggested by movement of coloured spheres over a zome configuration Animations of communications suggested by movement of coloured spheres over a zome configuration
Animations made using Stella Polyhedron Navigator and X3D-Edit

Use of a zome configuration in practice and as a focus of symbolic significance

The argument above focuses on the potential of zomes for the more effective configuration of communication in the future. There is however a more immediate application as an adaptation of messaging processes in virtual gatherings and otherwise.

Participant interaction: Prior to the explosive development of social media and related facilities, extensive experiments were undertaken in the use of simpler technologies to enable and order communications among participants in large conferences -- bypassing the channels of communication via a podium (History of Participant Interaction Messaging 1979 to 1995, 2007; Nadia McLaren, Participant Interaction Messaging: manual and guidelines, 1992).

With the development of social media, there is now clearly the possibility of allowing participants in a virtual gathering to tag messages in relation to the structure of a zome of choice -- its faces, edges and vertices. The choice of zome could be based on the structure of the agenda of the gathering.

Alternatively, with more effort, use could be made of algorithms to attribute tags to messages to associate them with features of a zome -- itself open to visualization in virtual reality. Available facilities could be used to enable viewing of messages on any particular feature of the zome, or along thematic pathways interconnecting portions of the structure. There is every reason that such facilities could be offered in the near future as an extension of existing social media and of collaborative software such as Zoom (Multi-option Technical Facilitation of Public Debate: eliciting consensus nationally and internationally, 2019).

More challenging is the possibility of using such software to switch to a more appropriate zome in the light of trends in the incoming communications from participants. Emphasis might then be placed on evolution of the zome configuration through the course of the virtual gathering -- or the ability of participants to choose to switch between alternative zomes through which the pattern of communications could be configured.

Symbolic value: In symbolic terms it is appropriate to recall an initiative on the occasion of the UN's Earth Summit (Rio de Janeiro, 1992). The production of the film One Child - One Voice (1992) as the centerpiece of the UN's Save the Earth media campaign (distributed to some 100 countries) allegedly evoked more than one million "leaf postcards" with messages from children indicating their hopes for the future. These were received in Rio to be attached to a "Tree of Life", as associated with various mythologies, including the Yggdrasil or "World Tree" (Axis Mundi, Yggdrasil, Omphalos and Sahasrara? 2020)

At the Earth Summit in 1992, the Tree of Life had been specially sculpted. No such process was initiated for the subsequent Earth Summits (2002, 2012), nor was use made of the interactive messaging process developed for the 1992 event. Given its symbolic associations, a suitable zome could be used on which to hang "postcards" -- or a virtual analogue could be used with which to associate virtual messages. There is a charming irony to the fact that symbolic trees continue to be used in various cultures to hold messages from a community, notably termed prayer trees or wishing trees.

Engendering zomes and "populating" them with significance: The preceding paragraphs offer simpler pointers to practical possibilities. Of potentially greater significance in engendering coherence are the implications mentioned with regard to the use of a artificial intelligence to filter sets of issues into clusters then to be allocated to a framework -- namely into the form of a zome.

A precursor to this possibility is evident in the interconnected online databases of the Encyclopedia of World Problems and Human Potential. These continue to be developed by culling, profiling and interrelating the thousands of problems and strategies identified by international constituencies. The organization of the data invites use of algorithms to cluster it into patterns of N-foldness. The question is whether "N" is predefined -- as with the 9 planetary boundaries, or the 17 Sustainable Development Goals -- or whether the number of clusters is determined otherwise, possibly interactively (or possibly as preferred alternatives for particular purposes).

A zome configuration appropriate to the number of clusters can then be readily engendered (as illustrated above). The further issue is how the clusters are to be most appropriately mapped onto the features of the zome. Here advantage can be taken of the extensive network of relationships already identified in those databases between problems, between strategies, and between both -- in addition to links to a very extensive set of human values (by which goals and problems are engendered).

An early anticipation of the possibility formed the subject of a presentation (Simulating a Global Brain -- using networks of international organizations, world problems, strategies, and values, 2001). Somewhat ironically, the techniques required are already evident in so-called persuasive technology, namely the sophisticated profiling of users of social media in order to target marketing messages.

Imaginative inspiration: There is some irony to the spiralling features of the zomes variously presented above, given the etymological origins of "spire". It is notably associated with understandings of "breathing" and "coiling", as variously reflected in terms such as inspiration, respiration, and expiration. As shown the spirals could be "collapsed" together into two-dimensional form -- suggesting the geometrical possibility of higher dimensional analogues of relevance to more complex forms of communication.

There is a further irony to the fact that the popular imagination has long been stimulated by so-called "flying saucers", of which some of the zome configurations are reminiscent. It could be argued that one of the challenges of global governance is the development of an inspiring new style of vehicle in communication terms -- "flight-enabled" such as to ensure that its projects "fly" and "get into orbit", rather than being effectively "grounded". The reform of the United Nations might be understood in such terms.

Curiously the challenge of achieving a sense of direction and coherence in virtual conferences -- of whatever scope -- has been remarkably caricatured in poetic form in the traditional Persian tale by Farid ud-Din Attar (The Conference of the Birds, 1177). As noted in the Wikipedia description:

In the poem, the birds of the world gather to decide who is to be their king, as they have none. The hoopoe, the wisest of them all, suggests that they should find the legendary Simorgh, a mythical Persian bird roughly equivalent to the western phoenix. The hoopoe leads the birds, each of whom represent a human fault which prevents man from attaining enlightenment. When the group of thirty birds finally reach the dwelling place of the Simorgh, all they find is a lake in which they see their own reflection.

The poem names and characterizes 18 of the birds of which many offer distinct "excuses" for not participating in the journey -- to which the hoopoe offers persuasively insightful responses. The poem continues with respect to the other 12 uncharacterized birds of that avian ecosystem as follows:

The other birds in turn received their chance
To show off their loquacious ignorance.
All made excuses - floods of foolish words
Flowed from these babbling, rumour-loving birds.
Forgive me, reader, if I do not say
All these excuses to avoid the Way;
But in an incoherent rush they came,
And all were inappropriate and lame.
How could they gain the Simorgh? Such a goal
Belongs to those who discipline the soul.
The hoopoe counselled them: The world holds few
As worthy of the Simorgh's throne as you,
But you must empty this first glass; the wine
That follows it is love's devoted sign.
If petty problems keep you back -- or none --
How will you seek the treasures of the sun?
In drops you lose yourselves, yet you must dive
Through untold fathoms and remain alive.
This is no journey for the indolent --
Our quest is Truth itself, not just its scent!
Farid ud-din Attar (The Conference of the Birds, 1984)


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