Prime number and curvature implications for global governance?

Meta-pattern via Engendering and Navigating Pantheons of Belief? (Part #7)

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Comprehension of sustainable development: Rather than idle curiosity, the questions evoked by the "fundamental equations" above acquire far greater pertinence through the manner in which the United Nations has engendered a set of Sustainable Development Goals. The set could be understood as the global constitution of a form of secular pantheon in strategic terms. With no explanation as to the systemic nature of that pattern of 16 Goals (with a 17th coordinating Goal), it is the successor to the 8-fold set of Millennium Development Goals. Similar questions could be asked of the latter with regard to the perceived systemic inadequacies which resulted in its replacement.

Intended as they are to change the world (as noted above), it is only a mathemtician who could comment on the coherence of a prime number set of 17 goals in changing the world -- in the light of the coherence of a 17-fold set of equations held to have had a similar function, as claimed by Ian Stewart (In Pursuit of the Unknown: 17 equations that changed the world, 2012; Jumping Champions: leaping over the gaps between prime numbers, Scientific American, December 2000).

Wallpaper group: Writing on prime numbers, the challenge is framed otherwise by Stewart's colleague, Marcus du Sautoy (The Music of the Primes, 2003), variously subtitled: Why an Unsolved Problem in Mathematics Matters and Searching to Solve the Greatest Mystery in Mathematics. With a specific mandate to enhance public understanding of science, Marcus du Sautoy has initiated one project Maths in the City aiming to highlight the fundamental role that maths plays in society by viewing the urban environment in a mathematical way; another is a BBC Two series The Code. Both note the unsuspected role of the 17-fold "wallpaper group", as does another study by Ian Stewart (Professor Stewart's Cabinet of Mathematical Curiosities, 2009).

Although no such indication is offered, ironically this is seemingly one of the very rare ways in which the 17-fold set of UN Goals might be recognized as coherent (Anna Nelson, et al, 17 Plane Symmetry Groups; Frank A. Farris. Creating Symmetry: the artful mathematics of wallpaper patterns, 2015). Others are variously presented (Prime Curios: 17; Tanya Khovanova (Number Gossip: 17).

These include the fact that 17 distinct sets of regular polygons (triangles, squares and hexagons) can be packed in combinations around a point (Counting how many regular polygons combinations can form 360 degrees around a point, Math StackExchange, 2019). Understood as a tesselation, this is otherwise expressed in terms of the 17 possible ways that a pattern can be used to tile a flat surface with a common single vertex. Used separately the three polygons make a total of 3

The set of 17 derives from the fact that a graph can be viewed as a polygon with face, edges, and vertices, which can be unfolded to form a possibly infinite set of polygons which tile either the sphere, the plane or the hyperbolic plane. If the Euler characteristic is positive then the graph has an elliptic (spherical) structure; if it is negative it will have a hyperbolic structure; but if it is zero then it has a parabolic structure. When the full set of possible graphs is enumerated it is found that only 17 have Euler characteristic 0, namely a wallpaper group. As noted by Marcus du Sautoy, the Alhambra palace in Granada contains examples of all 17 patterns.

A further lead to any intuited sense of 17-fold coherence in 4 dimensions is offered in by the 64 convex uniform 4-polytopes of which 5 are polyhedral prisms based on the Platonic solids and 13 are polyhedral prisms based on the Archimedean solids. One is however duplicated with the cubic hyperprism (namely a tesseract), reducing the set to 17.

Cognitive implications of tesselation? Of potential interest in relation to the degree of preference for the coherence of 15-fold strategic articulations, is the recent discovery of the 15 tilings of convex pentagons (Olena Shmahalo, Pentagon Tiling Proof Solves Century-Old Math Problem, Quanta Magazine, 11 July 2017). Of similar relevance to other clustering preferences are those variously described as:

• regular tesselations (3) made of either triangles, squares or hexagons separately
• isogonal, non-edge to edge tilings (7) of convex regular polygons
• Archimedean, uniform or semiregular tilings (8)  with two or three triangles, squares or hexagons; only eight combinations of regular polygons create semiregular tessellations.
• semiregular or uniform tilings (11) with one type of vertex, but two or more types of face, namely 3 regular tilings, and 8 semiregular (List of Euclidean uniform tilings)
• nonconvex tilings (14) using star polygons, and reverse orientation vertex configurations (in addition to the 11 convex uniform tilings).
• 2-uniform (20), or 2-isogonal tilings, defined by two vertices (), also called or demiregular tilings)
• 3-uniform tilings (61)

If such tiling patterns are indeed a key to comprehending cognitive clustering preferences, this immediately raises the question of whether more appropriate clusters would result from consideration of tilings on a sphere (positive Euler characteristic). Seemingly constrained as it is to planar tilings (zero characteristic), does this suggest that humanity has trapped itself unknowingly in a "flat Earth" strategic perspective rather than exploring "global" or other possibilities (Irresponsible Dependence on a Flat Earth Mentality -- in response to global governance challenges, 2008).

Negative curvature -- a hyperbolic structure (negative characteristic)? The case for topological complexification in the quest for more fundamental order can be made otherwise in terms of the significance accorded by astrophysicists to recognition of negative curvature and its implications for understanding the shape of the universe, as discussed separately (Eliciting a Universe of Meaning -- within a global information society of fragmenting knowledge and relationships, 2013). Recent research by Stephen Hawking and colleagues (Accelerated Expansion from Negative Lambda, 2012) has shown that the universe may have the same surreal geometry as some of art's most mind-boggling images (Lisa Grossman, Hawking's 'Escher-verse' could be theory of everything, New Scientist, 9 June 2012). This offers a way of reconciling the geometric demands of string theory, a still-hypothetical "theory of everything", with the universe as observed -- through a negatively-curved Escher-like geometry (essentially a hyperbolic space).

The insight relies on a mathematical twist previously considered impossible, namely the use of a negative cosmological constant rather than a positive one. The new approach provides a description of "all the possible universes that could have been -- including ones in which the solar system never formed, or in which life might have evolved quite differently". Making conventional use of a positive cosmological constant, it had proven impossible to describe universes that were "anything more than clunky approximations to reality". A plethora of universes have now been generated from wave functions with negative cosmological constants.

Arguably, whether discovered by artificial intelligence or otherwise, analogous topological breakthroughs may have significance for connectivity in the ways of knowing, as argued separately in relation to deprecated symbol systems (Engaging with Hyperreality through Demonique and Angelique? Mnemonic clues to global governance from mathematical theology and hyperbolic tessellation, 2016; Quest for a "universal constant" of globalization? Questionable insights for the future from physics, 2010). Might viable global governance require some analogue to negative curvature to render global order coherent?

Sustainable Development Goals and "God's number" of 20? Despite the 17-fold pattern of the UN Goals, it is curious to note the extent to which worldwide enthusiasm for Rubik's Cube has been interpreted in that light (Recognition of Rubik's Cube as a relevant strategic development metaphor, 2017). It is then especially curious from a mathematical perspective, in the light of the 20-fold argument above, that there is considerable focus on the minimal number of moves required to resolve a scrambled Rubik's Cube. As noted by Wikipedia with respect to optimal solutions for Rubik's Cube, There are two common ways to measure the length of a solution to Rubik's Cube. The first is to count the number of quarter turns. The second is to count the number of outer-layer twists, called "face turns".

The maximum number of face turns needed to solve any instance of the Rubik's Cube is 20, and the maximum number of quarter turns is 26.These numbers are also the diameters of the corresponding Cayley graphs of the Rubik's Cube group. That diameter is known as "God's Number" (Tomas Rokicki, et al., The Diameter of the Rubik's Cube Group Is Twenty, SIAM Journal on Discrete Mathematics, 2013; God's Number is 20, 14 August 2010). There are many algorithms to solve scrambled Rubik's Cubes. An algorithm that solves a cube in the minimum number of moves is known as God's algorithm. Ironically it is to that capacity that cubing enthusiasts aspire.

Potential relevance of insights of neuroscience? Complementing insights into "polyhedra as systems" are the sresults of recent neuroscience research which indicate the remarkable possibility of cognitive processes taking up even up to 11-dimensional form in the light of emergent neuronal connectivity in the human brain:

Using mathematics in a novel way in neuroscience, the Blue Brain Project shows that the brain operates on many dimensions, not just the three dimensions that we are accustomed to. For most people, it is a stretch of the imagination to understand the world in four dimensions but a new study has discovered structures in the brain with up to eleven dimensions - ground-breaking work that is beginning to reveal the brain's deepest architectural secrets..... these structures arise when a group of neurons forms a clique: each neuron connects to every other neuron in the group in a very specific way that generates a precise geometric object. The more neurons there are in a clique, the higher the dimension of the geometric object. ...

The appearance of high-dimensional cavities when the brain is processing information means that the neurons in the network react to stimuli in an extremely organized manner. It is as if the brain reacts to a stimulus by building then razing a tower of multi-dimensional blocks, starting with rods (1D), then planks (2D), then cubes (3D), and then more complex geometries with 4D, 5D, etc. The progression of activity through the brain resembles a multi-dimensional sandcastle that materializes out of the sand and then disintegrates. (Blue Brain Team Discovers a Multi-Dimensional Universe in Brain Networks, Frontiers Communications in Neuroscience, 12 June 2017) [emphasis added]

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