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10 June 2019 | Draft

Coherent Representation of the European Union by Numbers and Geometry

Mapping structural elements and principles onto icosahedron and dodecahedron

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Table of strategic structural attributions by number of elements
Symmetry elements of dual polyhedra enabling integrated mapping
Structural questions and commentary

Annex of Experimental Visualization of Dynamics of the European Parliament in 3D (2019)


The European Union is a very complex structure. It is unclear who has clear insight into that complexity and by what means they see it as coherent. The difficulty is presumably all the greatr for the Europeans who are governed by that structure and who are expected to indicate their policy preferences. The same may well apply in the case of other structures, whether international, regional, national or even local.

In the case of institutional complexes like the European Union it is therefore surprising to note that a degree of coherence is supplied through numbering parts of the stucture, notably the most fundamental to their principles and strategic operations. Thus much is made of the "three pillars" and the "four freedoms", but more articulated sets also evident. Whilst such sets may be readily cited, as with "10-point Action Plans", it is unclear how thus multiplicity of numbered articulations is rendered comprehensible, and how it may be understood as indicative of the coherence of the endeavours of the European Union.

It could be said that most with any degree of familiarity with the European Union have no need of other ways of comprehending its coherence as a whole -- to which they may in practice be indifferent. The matter can be understood otherwise where attention is drawn to any democratic deficit and to the widely noted rise of euroscepticism. Of what are people sceptical if the coherence of the structure is only conveyed in lengthy speeches and wordy texts -- some extending to thousands of pages?

The following is a simple exploratory exercise to cluster the organizing and strategic principles by number -- and to associate those numbers with polyhedra which exemplify coherence. Presumably this may primarly appeal to those who are confused (and unconvinced) by lengthy speeches by important people -- and by wordy texts which they have little inclination to read. Expressed otherwise, might some such mapping enable and enhance credibility in circumstances in which it is clearly lacking to an unfortunate degree.

The procedure in elaborating the following table has been simply to locate via the web any texts associated with "EU" or "Europe" which indicated an N-point articulation (eg "9-point Action Plan"). In illustrating the process, the table is incomplete in various ways. As an exploratory exercise, it excludes:

One hypothesis discussed in the main paper is that there is a natural preference -- governed by cognitive constraints -- for patterns of "N-foldness" of a certain size. From that perspective, this exploration considers the possibility of mapping the articulations onto several polyhedra, most notably the icosahedron and its geometric dual (the dodecahedron). Their many symmetry properties are considered significant to the memorability of relatively high degrees of complexity. A case was made in the main paper for the value of representations in 3D, especially with the rapid development of interest in the associated technology and its availability.

The contents of the table invite a variety of comments indicated thereafter, as with the indicative mapping onto polyhedra. Of particular interest is the manner in which equivalent "N-foldness" between polyhedra allows the representation on one to be smoothly morphed into representations on another.

Table of strategic structural attributions by number of elements

Coherent representation of the European Union by numbers and geometry
Numbers Structural elements of European Union Icosahedron Dodecahedron Cuboctahedron Other polyhedra
1 Flag of Europe
Anthem of Europe
3 Pillars of the EU
3 categories of European Pillar of Social Rights
  3 10-fold axes 3 4-fold axes  
4 Four freedoms in Europe
Four principles underlying cohesion policy
    4 3-fold axes  
5 Pentagonal symbolic stars
Securing the Energy Union: five pillars and five regions
Five constitutional principles of EU law
EU's Russia policy: five guiding principles
5 Scenarios for future of EU (white paper)

  12 pentagonal faces    
6 6 Principles for further developing EU-Turkey Cooperation in tackling the Migration Crisis (2016)
6 Privacy Principles of the GDPR: data protection, for personal data processing (2019)
6 Principles of EU democracy support (2009)
Sixth Scenario for the Future of Europe (2017)

6 vertex axes
6 5-fold axes
6 face axes
6 5-fold axes
6 vertex axes
6 2-fold axes

Principle EU decision-making institutions
7-point Solidarity Package for agriculture (2016)
7 Principles of the GDPR (2019)
7 Guidelines for AI Ethics (2019)

    7 face axes 7 Szilassi faces
7 Csazar vertices
8 8 Principles of integrated coastal zone management (2002)
8 Principles of Data Protection and GDPR (2017)
8 principles for a fair and humane approach to migration (Oxfam, 2017).

      Stellated octahedron
9 9 political priorities of the Joint Research Centre
9 Political groupings in European Parliament (2018)
9-point Action Plan to increase compliance with and improve governance on EU environmental rules (2018)
9 principle underlined by social partners in European Pillar of Social Rights (2018)
9 GDPR principles relating to processing personal data
9 General Principles of EU Administrative Procedural Law? (2015)
9 principles for meaningful social innovation (2019)
9 principles to achieve quality social and health services (2019)

Embedded enneagram
(main paper)

EU Policy priorities (2015-2019)
10 Point Action Plan to further reinforce the European Parliament.(2012)
10 point action plan on migration (2015)
10-point plan for future relations with China (2019)
10-point plan for improving the EU's public profile (2001)
10-point plan for Indstry and Europe
10-point plan for European tax justice (2019)
10 Principles for integrated child protection systems

10 face axes
10 3-fold axes
10 vertex axes
10 3-fold axes
11 11 Common Basic Principles for Immigrant Integration Policy in the EU.(2004)        

12-starred Flag of Europe
12-Point EU Action Plan to support the Millennium Development Goals (2010)
12-Point Socialists' Action Plan (European Parliament, 2007)
!2-Point Plan to bring the EU close to the citizen (2017)
12-point commitment to respect the landmark 2015 nuclear agreement (2019)

12 vertices 12 pentagonal faces 12 vertices  
13 13-point Call for Action to the European Union by the European Network Against Racism (2012)        
14 14-Point Action Plan for European coastal and maritime tourism: jobs, growth and sustainability (2014)
14-Point Critical Summary: what the EU Constitution does (2005)
14-Point No-deal Brexit Contingency Plan (2018)
14-Point Humanitarian Interventions System (HOLIS reporting system to inform of new bilateral aid initiatives)
    12 faces 14 Szilassi vertices
14 Csaszar faces?
15 15-point plan to make EU trade and sustainable development more effective (2018)
15-point action plan: Civil Society Dialogue on Trade and Sustainable Development (2019)
15 great circles
15 2-fold axes
15 2-fold axes    
16 16-point strategy to improve technology policies across the EU (2015)
16-point Digital Single Market initiative (2015).
16-point statement on issues of obesity, nutrition and physical activity and endorsing specific actions
17 17-point plan of action on Western Balkans Migration Route (2015)
17-point checklist of European Local Government Workers and Employers (2009)
18 18-point Road Map for adhesion of Bosnia (2000)        
19 19-point plan emerges from EU G8 heads of state summit (2008)        
20 European Pillar of Social Rights in 20 principles
20 General Principles of EU Administrative Procedural Law (2015)
20 triangular faces 20 vertices    
30 28 Member states (plus or minus future accessions)? 30 edges 30 edges    

Symmetry elements of dual polyhedra enabling integrated mapping

Symmetry axes of Icosahedron
Colours of
symmetry axes
Symmetry axes of Dodecahedron
Animation showing symmetry axes of icosahedron

Yellow = Vertex 20
Animation showing symmetry axes of dodecahedron


Cyan = Face (centered)



Mauve = Edge (centered) 30

Produced using Stella Polyhedron Navigator

Further indications of symmetry can be added to such depictions, as suggested by the following.

Coherence suggested by animation of polyhedra showing both symmetry axes and reflection planes
Icosahedron Dodecahedron
Coherence suggested by animation of icosahedron  showing both symmetry axes and reflection planes Coherence suggested by animation of dodecahedron showing both symmetry axes and reflection planes
Produced using Stella Polyhedron Navigator

Text can be readily added to such polyhedral representations, the problem being to avoid cluttering the depictions in this case. Examples are given separately of the use of such an approach with respect to the European Convention on Human Rights, Arab Charter on Human Rights, and the Universal Declaration of Human Rights (Dynamic Exploration of Value Configurations: polyhedral animation of conventional value frameworks, 2008).

Animation of arbitrary mapping of 28 member states of the European Union on polyhedra showing reflection planes
Icosahedron Dodecahedron
Animation of a mapping of 28 member states of the European Union onto icosahedron Animation of mapping of 28 member states of the European Union onto dodecahedron
Produced using Stella Polyhedron Navigator

Structural questions and commentary

Possible insights relating to N-foldness of conceptual organization more generally are discussed seprately (Patterns of N-foldness: comparison of integrated multi-set concept schemes as forms of presentation, 1980; Examples of Integrated, Multi-set Concept Schemes, 1980; Patterns of Conceptual Integration, 1984).

Hidden coherence? It is particularly interesting to note how many of the multi-point articulations of the European Union are usefully "captured" in mapping terms by the geometry of either the icosahedron or the complementary dodecahedron -- widely considered as exemplars of symmetry.

Arguably there is a "geometry" to the European Union as a complex of strategic initatives -- a geometry which has been effectively "hidden" such as to diminish appreciation of its coherence. This is unfortunate in a period of institutional crisis for the European Union with the rapid emergence of euroscepticism. The current approach to the representation of its complexity clearly encouragea misrepresentation.

Questions: Basic questions with regard to the European Union include:

Of potentially greater relevance are the following:

Problematic cases? A particular case study is suggested by examples of 17-fold articulation (17-point plan of action on Western Balkans Migration Route, 2015), notably in the light of the commentry by Katarina Kosmina and Lana Radovanovic (Lessons Not Learned Commentary on the EU's 17-Point Plan and Its Alternative, European Policy Centre, 3 Novenber 2015). Of related interest is the 17-point set of UN Sustainable Development Goals (2015), notably criticized as being a "mess" (The 169 Commandments, The Economist. 26 May 2015). This superseded the UN's 8-fold set of Millennium Development Goals (2000). The two sets of goals are the subject of a separate commentary (Interplay of Sustainable Development Goals through Rubik Cube Variations: engaging otherwise with what people find meaningful, 2017).

Especially of interest, however, is the seeming "awkwardness" of a 7-point articulation, given that the very structure of the European Union iis defined by the Treaty on European Union (2007) as a 7-fold complex of EU's principal decision-making bodies. This was discussed in the main paper in terms of their mapping onto the highly unusual Szilassi polyhedron -- unusual in that it has 7 faces, each in contact with the others. As shown in the table above, there is a a form of "transition" through the cuboctahedron (with 7 face axes) to the Szilassi polyhedron.

Transitional coherence and variable geometry? The possibilities of so-called "variable geometry" are discussed in the main paper, notably in relation to the process of morphing between structures with properties in common, as illustrated there between the icosahedron and the dodecahedron (as polyhedral duals).

Of interest are the circumstances under which greater coherence is offered by mapping features onto faces, vertices or edges -- and what morphing to the corresponding perspective then implies. As pects of this question are discussed with respect to Engaging with Globality -- through cognitive lines, circlets, crowns or holes (2009).

More provocative is the sense in which the dual perspective may be a challenge to the mindset associated with its complementary form -- whether the challenge is as an "alternative" framing or as highlighting otherwise neglected patterns of systemic significance. Potentially even more provocative is any sense in which a dual form is the "shadow" of the other (metaphorically understood), with all this may possibly imply in terms of unconscious patterns. It was in this sense that the main paper drew attention to the dynamics between representational forms rather than any static focus on one or the other. A more general pattern of dynamic transformations between polyhedral forms of potential significance has been presented separately (Memetic Analogue to the 20 Amino Acids as vital to Psychosocial Life? 2015).

Of interest is whether the transformations between various patterns through which the European Union can be perceived lend themselves to its representation in terms of an analogue to the mappings of metabolic pathways in biochemistry.

Especially significant as an intermediary in such variable geometry is the cuboctahedron, as extensively studied by Buckminster Fuller -- and widely presented through the dynamics of the so-called Jitterbug Transformation (Buckminster Fuller's Jitterbug, YouTube, 2007; Joe Clinton, R. Buckminster Fuller's Jitterbug: its fascination and some challenges, YouTube, 2011). This features the transformation between a set of concentrically arranged polyhedral forms through an unusual twisting-contracting, inside-outing dynamic (H. F. Verheyen, The complete set of Jitterbug transformers and the analysis of their motion, Computers and Mathematics with Applications, 17, 1989; Buckminster Fuller Demonstrating the Jitterbug Movement of the Vector Flexor, 1 September 2014; Vector Equilibrium and its Transformation Pathways, 1980).

Complementary perspectives? It is clear that articulated sets may well be very carefully reviewed from a legal perspective. Less evident is to what degree they are reviewed in terms of wider comprehensibility and memorability -- a "public relations" perspective, especially with what that may imply for decision-makers. More problematic is the question whether they are effectively reviewed in terms of their systemic implications, notably how any one element of the articulation may impact another in practice. These requriments clearly call upon different degrees of competence and expertise of which only the first is especially obvious during the process of articulation. The challenge is well-defined by the title of a UNHCR document: Bringing the New York Declaration to Life: Applying the Comprehensive Refugee Response Framework (2016). To such complementary perspectives might be added a symbolic perspective, potentially fundamental to the communicability and viability of any strategic framework.

Arguably a well-formed and memorable set of N-fold points is appreciated -- and rendered credible in operational terms -- to the extent that there are more symmetry effects interrelating the elements and effectively reinforcing one another. This effect is evident in a memorable poem or song. The aesthetics of a set are argued separately (A Singable Earth Charter, EU Constitution or Global Ethic? 2016). The main paper idicates the role of the golden ratio its relation to the golden rectangles characteristic of icosahedron and dodecahedron ( Gary Meisner, Golden Ratio in Art Composition and Design, Phi 1.618: the Golden Number, 4 May 2014;Jordi SolÀ-Soler, Phi in Sacred Solids, Sacred Geometry). Given the well-recognized role of phi in architecture and design in general, it could be asked whether this offers clues to appropriate organization design within an institutional complex like the European Union.

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