Hyperbolic space: realm of hypothetical entities?

Engaging with Hyperreality through Demonique and Angelique? (Part #7)

Exploring triangular curvature: A useful contrast can be made between conventional focus on the Euclidean triangle and two non-Euclidean equivalents distinguished as by Riemannian manifolds of constant curvature. Through the process of triangulation, the first is clearly fundamental to surveying. It has its implications in considering the interlocking of networks in systems and cyberspace. So depicted, the triangle is fundamental to various symbol systems and associated sets of categories, as explored by Paris Arnopoulos (Sociophysics: cosmos and chaos in nature and culture, 2005) and separately discussed (Triangulation of Incommensurable Concepts for Global Configuration, 2011; Triangulation, Connectivity and Stitching: enabling coherent global system dynamics, 2012).

The most obvious challenge to use of Euclidean triangles is evident in the navigation of the globe. This requires insight into spherical triangles and those in elliptic space.

Contrasting spaces of constant curvature
Hyperbolic space (negative curvature)	Euclidean space (zero curvature)	Elliptic space (positive curvature)

Beyond conventions of the familiar, flat Euclidean triangle, a global civilization could be expected to require equivalent elliptic insights for its navigation, as discussed separately with respect to the mathematical discovery of the Pentagramma Mirificum (Global Psychosocial Implication in the Pentagramma Mirificum: clues from spherical geometry to "getting around" and circumnavigating imaginatively, 2015).

In distinguishing the three forms of triangle depicted above, it is typically noted that the angles of the Euclidean type sum to 180 degrees, and those of the elliptic type to more than 180 degrees. It is the third variant which is of interest here, namely the hyperbolic triangle, in which the angles sum to less than 180 degrees (possibly to be understood as zero, given the asymptotic convergence at the vertices). The first is characterized by zero curvature, the second by positive curvature and the third by negative curvature. With respect to any "agreement", it could be said that in the first case the results of triangulation "add up". More complex insight is required to recognize such coherence in the second. However in the hyperbolic variant, meaningful coherence and consensus dependent on such forms is far less obvious.

Negative curvature of hyperbolic space: Such negative curvature may well have 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). These clues were used in relation to the aforementioned discussion of cognitive epicycles -- on the assumption that a more elegant "heliocentric" focus may emerge (Cognitive epicycles -- prefiguring more fundamental insight? 2012; Identification with a sustaining "heliocentric" locus? 2012)

The recent results of astrophysics rely 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.

Reframing the focus of integration: As presented schematically below, a hyperbola may be used to offer an alternative means of expressing the complementary nature of incommensurables -- effectively framing the paradox of the challenge of modernity,

Use of the hyperbola to frame cognitive paradox

Re-presentation of symbols framing paradox: The mystery attributed to the Trinity and to the Star of David may derive primarily from the hyperbolic equivalents they imply rather than from their conventional Euclidean form. Given the argument above for mathematical theology, the fundamental importance of this paradox is remarkably illustrated by the use of the set of 3 entangled 3 Borromean rings for both a medieval depiction of the Christian Trinity and through its recent selection as the most appropriate symbol for the International Mathematical Union.

Borromean rings in 2D and 3D
Christian Trinity in 2D (image reproduced from Wikipedia)	3D Logo of the International Mathematical Union

Of relevance to this argument is the cognitive implication in such patterns, as discussed separately (Representation of Creative Processes through Dynamics in Three Dimensions: global insight from spherical reframing of mandalas, the zodiac and the enneagram, 2014). The potential of those patterns -- as articulated and tested with respect to patterns in "external" material form -- thus derives from "internal" patterns of thinking. This perspective has been extensively argued, from a cognitive psychological perspective with respect to mathematics more generally by George Lakoff and Rafael Núñez (Where Mathematics Comes From: how the embodied mind brings mathematics into being, 2001).

5-pointed star versus 6-pointed star: It is curious to recognize the extent to which global dynamics at this time is significantly determined by principles associated with such distinctive star symbols. The schematic above using Dante's distinctions can be usefully adapted as below.

Contrasting conventional interpretations of 5-star and 6-star symbolism

.For the purpose of this argument, the locus of the mysterious integrative insights implied by the contrasting star symbols can be explored in terms of configurations of hyperbola. Each configuration frames that central focus -- but from an unconventional perspective. Note that in "fitting" the 2 curves of a hyperbola to the stars:

the 5-pointed variant uses one set of half-hyperbola for the up-pointing star, and a second down-pointing half-set to complete the hyperbola, this can be recognized as forming and a 10-pointed star, also significant in Islamic symbolism
in the 6-pointed star the centre is framed differently by the paired halves of the hyperbola and no inverted variant emerges from the animation

Fundamental star symbols speculatively reframed using hyperbola (animations)
5-pointed Star of Islam (featured on many Islamic flags)	6-pointed Star of David (featured on the flag of Israel)

Given the problematic perceptions between those associating with the symbolism of each, it is intriguing to note that both are intimately related to life processes. The hexagonal structure is fundamental to organic matter through the so-called benzene molecule. The pentagon is fundamental to Eastern and Western traditional understanding -- through Hygeia and Wu Xing -- of 5-fold organization of life processes (Memorable dynamics of living and dying: Hygeia and Wu Xing, 2014).

The possibility of reconciling such seemingly incompatible cognitive geometry was the focus of a separate exercise (Middle East Peace Potential through Dynamics in Spherical Geometry: sngendering connectivity from incommensurable 5-fold and 6-fold conceptual frameworks, 2012). This noted that the contrasting patterns could be uniquely reconciled within the spherical form of the truncated icosahedron -- a seemingly obscure geometrical form which is ironically familiar worldwide as the common football.

Recognizing the truncated icosahedron stitching pattern of a soccer ball (adapted from Wikimedia entry, to highlight the alternative colour variant)

Using the above hyperbola-based adapations of the 5-star and 6-star forms, these can be applied to the truncated icosoahedral pattern. As shown below, the stitching pattern of the common football then suggests a possible means of exploring a form of reconciliation between the contrasting symbolism and the conflicts to which their interaction otherwise gives.

Application of hyperbola-based 5-star and 6-star forms to the truncated icosoahedron
3D configuration	Unfolded representation in 2D

Reconciling Heaven and Hell?: Given the important role that the the art of M. C. Escher played in early promotion of explorations of hyperbolic configurations, a valuable introduction to hyperbolic geometry is provided, with extensive illustrations, on the Math and the Art of M C Escher Wiki (at Hyperbolic Geometry and at Euclidean and Non-Euclidean Geometry).

Those insights are extensively explored by others (B. Van Dusen and R. P. Taylor The Art and Science of Hyperbolic Tessellations Nonlinear Dynamics, Psychology, and Life Sciences, 17, 2013, 2, pp. 317-323) notably by Douglas Dunham (M.C. Escher's Use of the Poincaré Models of Hyperbolic Geometry; Transformation of Hyperbolic Escher Patterns; H.S.M. Coxeter and Tony Bomford's Colored Hyperbolic Rugs).

Experimental animations indicating the "dance" between angelic and demonic forms Based on the classic Circle Limit IV (Heaven and Hell) (1960) of M.C. Escher
Coloured to highlight demons (dark)	Coloured to highlight angels (dark)

Angels and Demons "dancing on a pin head" in negative space? The introduction mentioned the question How many angels can dance on the head of a pin? -- as long used to deprecate and dismiss scholasticism and angelology. The following animations suggest a way of thinking about intuitions regarding the manner of existence of the kinds of insubstantial entities indicated above (whether understood "positively" or "negatively").

Animations suggestive of the dynamics of hyperbolic space
Dynamics of the Angelique ?	Dynamics of the Demonique ?

At this stage it is appropriate to interpret the shapes as sets of categories or concepts, whether 3-fold, 4-fold, 5-fold, or more -- in each case functioning as a form of integrative cognitive container. The schematics are then indicative of the manner in which such sets are "invoked" -- for a period -- for purposes of explanation, or as offering guiding principles, as previously discussed (Representation, Comprehension and Communication of Sets: the role of number, 1978; Patterns of N-foldness: comparison of integrated multi-set concept schemes as forms of presentation, 1980; Spherical Configuration of Categories to reflect systemic patterns of environmental checks and balances, 1994; Comprehension of Numbers Challenging Global Civilization, 2014).

These animations anticipate further comment below, especially with regard to their asymmetry as polygons in hyperbolic space. Their animation is suggestive of the dynamics of global communication in which such categories are called into play for a time, only to be replaced by other sets -- also for a time only.