Role of 17 2D tiling patterns in ordering SDGs?

Systemic Coherence of the UNs 17 SDGs as a Global Dream (Part #4)

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We are surrounded by geometric patterns. These can be seen in the tiles on floors, in wallpaper patterns on our walls, and in the clothes we wear. They are also found in decorative artwork that adorns the outside of buildings and in signs, advertising, and in graphic and web design.... All major cultures have produced and enjoyed them.... What is the allure of these patterns? Why have we been compelled to create and view them over the course of human history? (The Perceived Beauty of Regular Polygon Tessellations, Symmetry, 2019, 11, 984)

Arguably the allure of such patterns extends, if only unconsciously, to preferences for the arrangement of concepts -- and to the clustering of strategies such as the SDGs.

Friedenberg examines the questions he highlights by looking at a group of tessellations, also known as tilings, whose geometric and mathematical properties have long been well defined (What are the conditions for a polygon to be tessellated? Mathematics Stack Exchange). Their properties are used to predict and help explain their aesthetic appeal. Different types of tessellations can be identified. They can be categorized by the types of polygons they contain, the way they are arranged around defining vertices, and their symmetry properties. Mathematics also explores a generalization of 2D tilings into higher dimensional tessellations with a variety of geometries.

Some special kinds of tesselation include Euclidean tilings by convex regular polygons. There are only  15 types that tesselate; no convex polygon with seven or more sides can tessellate. There are only 21 combinations of regular polygons that will fit around a vertex. And of these 21 there are there are only 11 that will actually extend to a tessellation:

• Regular tilings: Every regular tiling is defined by a vertex point. There must be 6 equilateral triangles, 4 squares or 3 regular hexagons at that vertex, yielding the 3 regular tessellations. A regular polygon can only tessellate the plane when its interior angle (in degrees) divides *** (this is because an integral number of them must meet at a vertex). This condition is only met for equilateral triangles, squares, and regular hexagons.

  Indication of the 3 Archimedean 1-uniform semi-regular tilings

• Archimedean, uniform or semiregular tilings: In this case regular tiles of more than one shape are combined with an identical arrangement around every vertex. For every pair of vertices there is a symmetry operation mapping the first vertex to the second. This gives rise to 8 uniform tilings.

  Alternative indications of the 8 Archimedean 1-uniform semi-regular tilings

  Reproduced from Wikipedia

  Reproduced from Tessellations by Polygons (Math and the Art of M C Escher)

As noted separately (Uniform tilings, Wikipedia), further distinctions can be made between:

• the 11 1-uniform tilings (as above), with their inclusion of 3 regular tilings, and 8 semiregular ones (with 2 or more types of regular polygon) faces.
• the 20 2-uniform tilings
  
• the 61 3-uniform tilings
• the 151 4-uniform tilings
• the 332 5-uniform tilings and
• the 673 6-uniform tilings

Such patterns frame the question as to whether and how their recognition governs the organization of patterns of concepts and strategies. What is the cognitive role of pattern recognition with respect to sets of concepts? What degrees of order in concept articulation are considered appropriate and "well-formed" -- and why?

There are exactly 17 two-dimensional space (plane symmetry) groups. These are sometimes called wallpaper groups, as they represent the seventeen possible symmetry types that can be used for a wallpaper (as noted below). It is the two dimensional case of a more general problem: the 3D case, for example, can be interpreted as the number of different crystalline structures.

This little known property is suggestive of a set of fundamental cognitive patterns implied by the distinctive nature of each SDG -- offering the possibility of an unusually distinctive image by which it could be identified (reminiscent of the role of Scottish tartans). Unfortunately the term "wallpaper" also invites more cynical interpretation as it might relate to SDGs -- in "papering over the cracks" of global governance.

Yet to be clarified is why such symmetrical patterns are cognitively satisfactory and how the geometric constraint on their formation as a tesselation offers an implication of closure somehow vital to a sense of the completeness and closure of a strategic set. Whereas tesselation requires that only 2*- worth of polygonal angles can be arranged around each point, it is unclear how this functions as a constraint on strategic coherence.

3-fold and 4-fold: The questions are given a degree of focus by the widespread preference for 3-fold sets of concepts, as can be variously discussed (Triangulation of Incommensurable Concepts for Global Configuration, 2011). This explored the following themes:

An analogous argument could be made for 4-fold sets of concepts, given the extensive use of explanation in terms of quadrants of categories:

• Dana Fusek: Explaining Four Quadrants of Data Visualization Charts
• John Lee: The 4 Quadrant Model: a co-occurring treatment framework (Choose Help, 13 September 2019)
• Daniel Mullins: The 4 Types of Visualisation and the Role They Play in Market Research (B2B International)
• Marten Kuilman: The Quadralectic Theory; Quadralectic Architecture.
• Boston Consulting Group (BCG) Matrix (Corporate Finance Institute)
• The Four Quadrants of Integral Metatheory: on the value of integrating multiple perspectives (Psychology Today, 8 October 2020)
• SWOT 2x2 matrix: a strategic planning technique

The relation between 3-fold and 4-fold articulations is however a challenge -- perhaps most usefully highlighted by tiling patterns. The relation has notably been a preoccupation for Carl Jung and those influenced by his perspective  -- framed by the leitmotiv of the Axiom of Maria. (Marie-Lousie von Franz, Number and Time: reflections leading toward a unification of depth psychology and physics, 1986). As noted by Lance Storm:

In regard to the problem of three and four, the Jungian scholar Marie-Louise von Franz once wrote: "It becomes evident that a psychological problem of considerable importance is constellated between the numbers three and four". (From Three to Four: the influence of the number archetype on our epistemological foundations, Quadrant: the journal of the C G Jung Foundation, 33, 2003, 1)

5-fold? In the light of the specific exclusion of the pentagonal polygon from the possible tiling patterns described above, the status of 5-fold patterns merits careful consideration in relation to the 3-fold and the 4-fold, especially given its fundamental role in the Hygeia of the Pythagoreans and the Wu Xing model of Chinese philosophy (Memorable dynamics of living and dying: Hygeia and Wu Xing, 2014).

From a 2D planar tiling perspective, the 5-fold is not "compatible" with 3-fold, 4-fold or 6-fold. It is however curious to note its special navigational role in 3D spherical geometry as the Pentagramma Mirificum (Global Psychosocial Implication in the Pentagramma Mirificum: clues from spherical geometry to "getting around" and circumnavigating imaginatively, 2015). Of potential relevance is the particular importance given to 5-foldness by Peter Senge (The Fifth Discipline: the art and practice of the learning organization, 1990). This has been used as an exploration a critique of the patterning process (Patterning Intuition with the Fifth Discipline: critical review of the conclusion of the 5-fold Patterning Instinct, 2019; 5-fold Pattern Language, 1984).

6-fold: The mathematics of tiling are closely related to the constraints governing the distinctive symmetries of crystal systems (Crystal symmetry)

The crystal classes may be sub-divided into one of 6 crystal systems 6 crystal systems. Space groups are a combination of the 3D lattice types and the point groups (total of 65). Each of the 32 crystal classes is unique to one of the 6 crystal systems: Triclinic, monoclinic, orthorhombic, tetragonal, hexagonal and isometric (cubic) hexagonal and isometric (cubic) Interestingly, while all mirror planes and poles Interestingly, while all mirror planes and poles of rotation must intersect at one point, this point may not be a center of symmetry

The question is then how (and why) such 6-foldness might "translate" into coherent thinking in other domains, as for example:

• John Greathouse: Six Fundamental Laws That Control Your Business, (Forbes, 5 July 2012)
• Anne Marie Helmenstine: Six Steps of the Scientific Method  (ThoughtCo, 18 February 2020)
• Edward de Bono: Six Thinking Hats (1985); Six Frames For Thinking About Information (2008)
• Raymond Abellio: La Structure absolue. essai de phénoménologie génétique (1965)

Language appropriate to distinguishing the underlying nature of any fundamental operations is offered in the case of the 6-fold set of Grothendieck's six operations. This is a formalism in homological algebra which originally arose from the relations in étale cohomology that arise from a morphism of schemes f : X â** Y. The basic insight was that many of the elementary facts relating cohomology on X and Y were formal consequences of a small number of axioms. These axioms hold in many cases completely unrelated to the original context, and therefore the formal consequences also hold. The six operations formalism has since been shown to apply to contexts such as D-modules on algebraic varieties, sheaves on locally compact topological spaces, and motives. The operations are six functions, usually between derived categories and so are actually left and right derived functors.

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