Cognitive heart dynamics framed by two tori in 3D

Visualization in 3D of Dynamics of Toroidal Helical Coils (Part #15)

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As a cardioid, Haskell's coaction compass featured in a separate discussion of the mathematical derivation of the "heart pattern" potentially characteristic of the widespread cognitive acceptability of playing cards (Radical Localization in a Global Systemic Context Distinguishing normality using playing card suits as a pattern language, 2015). The distinctions between a "cardioid" and the conventions of "heart" depiction were highlighted. That discussion also included interactive demonstration of 3D heart shapes based on different equations for cardioid renderings (Equations for Valentines, Wolfram Demonstrations Project). The argument can be taken further in this context by considering the pattern implied by the following as presented there.

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)

2D animations: These suggest he possibility of 2D animations of the (circular) dynamics around which implicitly frame a heart pattern, as shown below -- and potentially suggestive of a cognitive dynamic analogous to that between the four "chambers of the heart".

Various 2D animations of dynamics defining the 4 conditions of the heart-pattern using juxtapositioned cross-sections of two 3D tori

3D animations: The image on the left suggests the possibility of a 3D rendering using the juxtaposition of two tori with their major radius related by phi.

The resulting heart pattern differs significantly from the conventional cardioid but it does recall the domes of many mosques -- with the much appreciated cognitive implications these may elicit. Curiously only a small proportion of constructed mosques closely approximate that form (more especially those in Malaysia, but including the Taj Mahal), whereas many Islamic greeting cards do so to a far higher degree in their idealized representations of a mosque (for Ramadan and Eid). The domes of most constructed mosques bear a closer resemblance to the conventional cardioid -- ornamented with a minimally tapering spire.

In anticipation of more appropriate animation possibilities to highlight the heart pattern in 3D, the following justaposition of horn tori frames such a pattern to some degree, given the difficulty of ensuring contiguity. The movement into (or out of) the central vortex is consistent with the animations in the 2D cross-sections above -- meshing correctly between the upper and lower tori.

Animation of two horn tori of major radius in proportion of phi (2 of the 4 variants presented in the 2D animations above)
implied upright 3D heart pattern	implied inverted 3D heart pattern
Adaptation, with permission, of animation by Wolfgang Daeumler (Horn Torus)

The other two variations noted in the 2D animations are inversions of those above. The clockwise and counter-clockwise rotational dynamic (not indicated in the 2D animations) suggest that there are 8 distinctive patterns that could be considered.

A much clearer understanding of the heart pattern embedded within, or framed by, the two contiguous horn tori is evident from the animations below, although these lack the longitudinal and latitudinal rotations of those above -- which could be added. Note that switching between filled and wrireframe renderings is achieved when viewing.

Animations indicative of embedding of heart pattern within contiguous tori (Variants: X3D: without tori / with tori; VRML: without and with tori)
Upright heart pattern (filled rendering)	Inverted heart pattern (wireframe rendering)
X3D and VRML models kindly prepared by Sergey Bederov of Cortona3D

Lissajous curves: In relation to the horn torus, Wolfgang Daeumler comments on the presentation of Lissajous curves on the surface, as illustrated by the animation (below). The visual form of these curves is often suggestive of a three-dimensional knot, and indeed many kinds of knots, including those known as Lissajous knots, project to the plane as Lissajous figures. The animation below is somewhat reminiscent of the hypersphere animations suggestive of higher-dimensional brain functioning (as presented above).

Animation of Lissajous curve on horn torus

Reproduced, with permission, from Wolfgang Daeumler (Horn Torus)

Visual renderings of Mandelbrot set: As noted in the earlier discussion, it is widely recognized that the cardioid corresponds closely to the larger "bulb" of the many contrasting visual renderings of the Mandelbrot set fundamental to complexity science -- with all its potential cognitive implications (Psycho-social Significance of the Mandelbrot Set: a sustainable boundary between chaos and order, 2005). The variety of renderings presented there can be generated using the interactive XaoS fractal zoomer and morpher. The more comprehensive heart pattern may correspond more closely to the curve which encompasses both the larger "bulb" and the asymptotic extension beyond the smallest "bulb". This possibility is illustrated as one variant of the 2D animations (above).

Comprehension of complex dynamics: It must be stressed that the argument here is preoccupied with how complex dynamics are to to be comprehended and what constraining approximations may be made in framing intuitive recognition of it. The use of the upright heart pattern offers the suggestion that the dynamics associated with the larger torus "beneath it" are in some way implicit or virtual. Use of the inverted heart pattern offers the complementary suggestion that the implicit or virtual dynamics are in some way "above it" (as might be characteristic of the use of that orientation in temple architecture).

Same complex dynamics of higher dimensionality viewed through distinctive "toroidal projections"
concentric	nested / lemniscate	interlinked / knotted	stacked / interlocked

The fractal organization of any Mandelbrot set rendering, and the complex dynamics implied, are necessarily a major challenge to comprehension. The widespread appeal of the heart pattern may constitute a very simplistic recognition of this -- an approximation to a more complex insight, as suggested by the various "projections" above. As may then be imagined, the dynamics of the Concordian Mandala which is the focus of this exercise, cannot be readily mapped through a 2D or 3D rendering.

There is then a case for recognizing the extent to which the central "valley" of the heart pattern could be associated with the entropic focus in Haskell's cardioid (depicted above), with the opposite asymptotic feature associated with his negentropic focus. Framed by two horn tori (as in the left image above), the entropic focus is defined by the smaller and the negentropic focus by the larger. There is the further advantage to this framing in that the smaller torus frames the mystery of a hole (perhaps in its most problematic sense?), whereas the larger frames a larger (more positive?) insight associated with the infinite.

Such a framing offers the possibility of juxtaposing a pattern inspired by 3D rendering of Haskell's "cardioid", understood in cognitive terms, with Poza's toroidal proposal for the periodic table (as presented above).

Symbolic mind maps and learning stories? As a much valued symbol in a society in need of integrative symbols, there is a case for exploring how the heart pattern might be integrated into a more systemically complex learning story with a variety of pathways and dynamics -- a more extensive mind map. As a prelude to a suitable interactive model, the following is such an exercise, using a configuration of 9 heart patterns -- whether configured in a circle or vertically stacked. Any such configuration is best understood as a projection of a complex system.

Indicative systemic mind maps?

The stacking is suggestive of a hierarchy of systems, possibly of progressive greater subtlety -- the lowest possibly associated with first order cybernetic processes, the highest with the subtlest envisaged by cybernetics and the wisdom sciences. Positioning one heart pattern midway between the two extremes enables a distinction to be made between the two immediately contiguous (a 3-fold set), between the neighbouring 4 (a 5-fold set), the neighbouring 6 (a 7-fold set), in contrast with the totality (a 9-fold-set). Appropriate polygons are used to reinforce such patterning. The smaller sets could be understood as having constraints on their subtler and/or more fundamental sensitivity.

Of particular interest to fruitful complexification of the systemic story is recognition of the rotational effects. The variants are distinguished in the images below. Each 2-tori variant can be used with others of the same (opposing) directionalities, meshing together when stacked. There are thus two kinds of stack, whether inverted or not. However a further distinction is possible in terms of the rotation around the common axis. Meshing then necessarily requires rotation in the same direction for the whole stack, clockwise or counter-clockwise (as illustrated earlier) -- making a total of 4 upright patterns and 4 inverted patterns.

A further point of interest is the perspective on any stack along the axis. As shown in the central image below, the smallest heart pattern then appears at the centre of what is highly reminiscent of a conventional bullseye target -- for archery, darts, etc -- some of which have 9 rings. Whether as a need hierarchy or otherwise, this can be understood as the focus for the subtlest aspiration. Viewed otherwise the 9 levels of the stack are appropriately reminiscent of the 9-level temples design of some cultures (cf. Nine Story Stupa, Chichen Itza, Tikal) and their association with understandings of 9 levels of consciousness. Such levelling now also features in some online games.

Views of conjoined 2-tori dynamics
Circulation through the vortex (Variant A)	View down common axis (9 rings)	Circulation through the vortex (Variant B)

Framing the heart pattern by two implicit tori suggests that this framing could as well have been understood in terms of the helical windings around an implicit torus as rendered visually (in the earlier argument) -- or possibly between neighbouring tori (as suggested by the Lissajous curves). Hence the insights to be gained from gimbal dynamics (forthcoming paper). Especially intriguing is the sense in which the helical windings -- as a twisted sine wave -- could be considered as the dynamic resolution of the longitudinal and latitudinal rotation dynamics illustrated in the animations above.

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