Refining appreciation of distinctions by refining pattern geometry

Radical Localization in a Global Systemic Context (Part #4)

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Geometry and design: There is a very long history to playing cards and their design (Catherine Perry Hargrave, A History of Playing Cards and a Bibliography of Cards and Gaming, 2012; A History of Playing Cards: looking at the style and type of the suits, 2010). Less evident are any studies of why the most common suit designs "work" in their cognitive appeal, as indicated by Joshua Johnson (Design History: the art of playing cards, Design Shack, 7 November 2011). There is little trace of the considerations governing the proportions of the heart, spade, diamond or club patterns.

As noted above, a particular choice was made in exploring the set of overlapping outlines. In retrospect, the rejected alternative might have been equally instructive -- especially to the extent that the curvature of its heart form appears to approximate more closely to that of the Gaussian distribution curve as represented. It might be suspected that those in common use "work" because of classic proportions embodying the golden ratio, for example.

Golden diamond? This could be evident in the case of the diamond, although it is noteworthy that the sides are slightly bent inward in some variants -- matching curiously the "diamond" patterns engendered by the fourfold overlapping of hearts or spades in the experiments above. Reference is commonly made to the golden rectangle as a basis for proportion in design, most notably in architecture. With respect to the diamond shape, in contrast to the rectangle, there is some corresponding reference to a golden diamond (Koji Miyazaki, An Adventure in Multidimensional Space: the art and geometry of polygons, polyhedra, and polytope, 1986; Jay Kappraff, Connections: the geometric bridge between art and science, 2001; Steven L. Griffing, The Golden Section: an Ancient Egyptian and Grecian proportion, 2007, pp. 161-164).

Griffing notes that a diamond shaped figure can be constructed using two isoceles triangles -- inverted so that they share a base. Additionally it is noted (p. 69) that the variant of the isoceles triangle, known as the golden triangle, has a ratio of either leg to the base equal to thegolden ratio and is used in the formation of a logarithmic spiral. It has been known historically as the Sublime Triangle, the Triangle of Plutarch, and the Triangle of the Pentalpha. Each corner of a pentagram is a golden triangle. With respect to any refinement to the design of the spade or club pattern, consideration could be given to use of a golden triangle as the base.

Comparison of diamond pattern designs in the light of the golden rectangle
Isoceles-based golden rectangle (golden ratio between side of rectangle and base of triangle)	Playing-card design	Diagonal-based golden rectangle (golden ratio between longer side of rectangle and shorter side of rectangle)
Major distinctions are evident on rotation. The implication is that sensitivity to extremes with regard to the ideological axis are matched by minimal sensitivity to the extremes represented by the vertical axis of subjectivity/objectivity). Correspondingly, when there is greater sensitivity to extreme distinctions on the vertical axis, there is relative lack of distinction on the ideological axis.	On rotation, the card suit design makes little distinction between the horizontal and vertical axis as these might distinguish extreme conditions	This variant suggests less contrast on rotation between the sensitivity to extremes on the vertical and horizontal axes.

The contrasting "stories" implied by the pattern designs above recall some of the distinctions made in appreciation in classical Greece of the golden mean between extremes, notably featuring in consideration of proportions of relevance to governance in an ideal state as envisaged in dialogues by Plato (Republic, 619; Laws, 691c, 756e-757a). In that respect, many aspect of the arguments regarding ordering patterns in space have their equivalent with respect to patterns of tones over time, as notably articulated by Ernest G. McClain (The Myth of Invariance: the origins of the Gods, Mathematics and Music from the Rg Veda to Plato, 1976; The Pythagorean Plato: prelude to the song itself, 1978; Meditations Through the Quran: tonal images in an oral culture, 1981). This consideration features in a separate discussion (Designing Global Self-governance for the Future: patterns of dynamic integration of the netherworld, 2010).

With respect to comprehension and its extremes (especially with regard to "radical" at the present time), of relevance from that classical Greek perspective is the understanding of hyponoia), as discussed separately (Transforming from Paranoia through Metanoia and Hyponoia? 2013). A conventional use of the term is associated with deficient or sluggish mental activity or imagination. This pathological condition is also called hypopsychosis. Controversially this might be seen as as the condition deliberately sought through the dumbing down of the population, especially via the media, more effectively to ensure its exploitation.

Much more interesting are the historical uses of the term as indicated by this comment cited in AlphaDictionary.com (from the Etext Center of the University of Virginia Library):

... Hyponoia was the term which, Plutarch tells us (De audiendis poetic 4.19), the "ancients" had used, and it implies a hidden meaning, a conjectural or suppositious sense, buried under the literal surface. Plato (Republic II. 378d), Euripides (Phoenicians 1131-33), Aristophanes (Frogs 1425-31), Xenophon (Symposium III, 6), all use hyponoia to mean what is later subsumed under allegory (Pépin, pp. 85-86). Hyponoia furthermore has a noetic character; the reader or listener will have to think his way through a semantic barrier, beyond which lies a realm of mystic knowledge. Thus Philo Judaeus may equate the hyponoia of a text with its latent theme, its mystery, its secret, its unexpressed, unseen, nonliteral, or simply intelligible meaning.

Recent commentary notes that hyponoia, as used in the classical period, referred to hidden or allusive meanings, what is now termed allegory. It is indicative of the "veiling function of language" or "an allusion to". Rosario Garc?a Del Pozo (The Mirror of Interpretations and Husserlian Discourse, Analecta Husserliana, 29, 1990, pp. 309-321) notes the suspicion that underneath language, in the shadow of what is said, lies the most important meaning; what the Greeks called "allegory" and "hyponoia". This calls for further exploration in relation to any understanding of "radical", "extreme" and "fundamental".

Engendering the heart pattern using phi: It is to be suspected that the design of the heart pattern has such lasting appeal because it is based in some way on the golden ratio (denoted by phi). The following images are an illustration of this possibility.

Defining the heart pattern using the golden ratio (phi)
Heart pattern framed by 4 circles (phi is the ratio of separation of centres of the smaller circles to that separating the larger)	Heart pattern on the left overlayed (highlighting the relation to the form of the Gaussian distribution curve above)

The image on the left invites reflection on its relation to the chambers of the heart and to the dynamics basic to the function of the heart as they might be represented by expansion or extraction of the various circles, possibly with phi as a norm. Also of interest is the sense in which the smaller circles define the pattern from within, whereas the larger circles define it from without. These considerations then invite reflection on the image on the right (notably in relation to the form of the final animation below).

Heart curve: Many approaches to drawing a heart with mathematical assistance are listed by Jürgen Köller (Heart Curve, 2004) with numerous examples and links. A useful overview of heart variants and the functions generating them is provided by Eric W. Weisstein (Heart Curve, MathWorld). Notable is the freeware available for generating many such variants (Rick Parris, Winplot for Windows; application examples) .

One mathematical formula for rendering of a more standard variant of the heart is presented by Hans-Jürgen Caspar (Draw a Heart; Program script; Ausgewählte, in der Analysis untersuchte Kurven).

Heart drawn with a mathematical function (screen shot of animation by Hans-Jürgen Caspar)

Reframing the heart pattern in terms of the cardioid : It is also tpo be suspected that the form of the heart pattern might conform to a particular mathematical function. One such function, the cardioid (meaning "heart") does not engender the heart as commonly designed. The manner of its generation in geometry -- as shown by the animation from the Wikipedia description below -- is however of potential interest to further investigation of systems dynamics.

Animation of generation of a cardioid (reproduced from Wikipedia)

As separately discussed (Cardioid Attractor Fundamental to Sustainability: 8 transactional games forming the heart of sustainable relationship, 2005), such a pattern, suggestive of the heart pattern design, merits further exploration in the light of the extensive work of Edward Haskell on the coaction cardioid, as schematically summarized below. This usefully distinguishes the mathematical form of the cardioid (and the manner by which it is generated) from the typical representation of the heart suit pattern. Note that the cardioid is nested within a pattern with which Haskell has associated a form more closely recalling that of the heart suit -- by supplementing the cardioid with a colour-shaded area (and its complement).

Coaction compass Adapted from Ed Haskell (Full Circle: the moral force of unified science, 1972)

The implications of the coaction cardioid have notably been a preoccupation of Timothy Wilken (UnCommon Science, 2001). Also of relevance is the form of the evolute of the cardioid.

Mandelbrot heart or set of clubs? The cardioid merits much further attention in that it is the principal feature of the visual rendering of the Mandelbrot set, as discussed separately (Sustainability through the Dynamics of Strategic Dilemmas -- in the light of the coherence and visual form of the Mandelbrot set, 2005; Psycho-social Significance of the Mandelbrot Set: a sustainable boundary between chaos and order, 2005).

To the extent that "radical localization" requires consideration of non-linear functions, the Mandelbrot set fractal corresponds to the simplest nonlinear function -- but is also as complicated as a fractal can get. It distinguishes the simplest boundary between chaos and order. This sense is clearly central to conventional strategic security preoccupation with the chaotic challenge of any radical extreme -- as a manifestation of disorder disruptive of ordered normality. With respect to the relationship between the form of the cardioid and that of the heart, and missing from the cardioid approximation to the heart, is the manner in which the Mandelbrot rendering (appropriately oriented) could be understood to have a significant "tail" complementing the "cleft" at the other end.

As indicated by the images below, of further interest is the possibility of rendering the set with colours so as to highlight zones both inside and outside the main boundary of the form. These suggest a means of exploring degrees of both normality and extremism in a new way -- whilst offering another way of framing the nature of the boundary between them. The emergence of what are termed "bulbs" around the central cardioid correspond to a curious degree to the design of the clubs pattern (if the period does not exceed 3). The rendering thus combines associations to both heart and clubs.

Selection of renderings of the Mandelbrot set using distinctive colouring conventions Reproduced from Imagination, Resolution, Emergence, Realization and Embodiment: iterative comprehension ordered via the dynamics of the Mandelbrot set (2005) Images generated by Xaos: realtime fractal zoomer

Given that the preoccupation with radical location (and the "misbehaviour" of extremists) is framed in terms of risk analysis, it is appropriate to note the recognized relevance of the Mandelbrot set to risk analysis in the financial markets (Benoit Mandelbrot and Richard L. Hudson, The Misbehavior of Markets: a fractal view of financial turbulence, 2006; Justin Fox, Why didn't people in finance pay attention to Benoit Mandelbrot? Reuters, 18 October 2010; Martin Hutchinson, What We Can Learn From The Stock Market Genius That Wall Street Loves to Ignore, Money Morning, 20 October 2010). Given the normal distribution by which the above argument has been developed, it is noteworthy that Mandelbrot introduced an understanding of seven states of randomness with respect in probability theory, fractals and risk analysis as an extensions of the concept of randomness as modeled by that normal distribution. His classification builds upon the three main states of randomness: mild, slow and wild. Understanding of radical extremism merits exploration in such terms, as suggested by Judith K. Boyd (Solving homegrown violent extremism through fractal geometry? Homeland Security Watch, 14 May 2010).

Experimental use of Fibonacci spiral to "reverse engineer" the heart and other patterns: In the absence clear indications of the origin of the semi-standard suit designs, the question is whether there is a geometrically "purer" design based on proportions of known aesthetic significance. In that respect it is the Fibonacci spiral which is of special interest as one of the approximations to the golden spiral with its particular embodiment of the golden ratio, indicated by the Greek letter phi (Mario Livio, The Golden Ratio: the story of phi -- the world's most astonishing number, 2002). Although contested, the heartbeat has itself been related by some to phi and the Fibonacci pattern (Gary Meisner, Human Heartbeat and Fibonacci Patterns, GoldenRatio.net, 13 May 2012).

Stages in progressive pairing of mirror images of Fibionacci spiral to derive a cardioid-like variant of the heart pattern

Of further interest is the degree to which approximations to standard suit designs could be derived from the pairing of the Fibonacci spiral in the image on the right above. The question is whether relevant curves could be used from within the paired pattern. Possibilities are indicated below without rescaling any constituent portions of the pattern.

Experimental derivation of suit designs from Fibonacci spiral
Heart pattern	Spade pattern	Club pattern

Of interest in relation to this argument are the proportions of the Fibonacci spiral with their implications for the heart pattern. These are illustrated by the following. The golden ratio is is derived from the proportions a/b from the image on the left, namely 1.618. How might such a proportion be of significance to the appropriate proportions of "radicals" in any group?

Proportions of the Fibonacci spiral with implications for the heart pattern

The value of the Fibonacci spiral has also been discussed separately with respect to designing a mapping of a Chinese metaphorical pattern language (Adaptive Hypercycle of Sustainable Psychosocial Self-organization, 2010). It is noteworthy that one of its manifestations in nature, the marine nautilus, is valued both as a symbol of educational development and of strategic appropriateness (New Zealand Curriculum Nautilus, Nautilus Institute for Security and Sustainable Development).

Exploration of the pairing of the spirals can be seen as consistent with the existence of both clockwise and counter-clockwise variants, namely the issue of their chirality (Chaorong Li, et al, Stressed Fibonacci spiral patterns of definite chirality, Applied Physics Letters, 90, 2007, 164102; Lisa Zyga, Scientists find clues to the formation of Fibonacci spirals in nature, Phys.org, 1 May 2007; Chaorong Li, et al, Stressed Triangular Tessellations and Fibonacci Parastichous Spirals, Advanced Materials, 28 August 2009). The relevance to social systems is indicated by related commentary, included valuable illustrations by David Petch (Stock Market Cycles Chirality and Chiral Inversions, Market Oracle, 22 October 2013; The Contracting Fibonacci Spiral, Technical Analysis of Stocks and Commodities, April 2013; Markets Trapped in a Contracting Fibonacci Spiral, Point of Singularity in 2019, Financial Sense, 13 July 2011).

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