Forms of the triskelion in 3D of potential significance

Cognitive Implications in 3D of Triadic Symbols Valued in 2D (Part #4)

Rotation of basic form: A point of departure is to present the 3D triskelion as it is normally perceived in 2D, as shown (left image below). The three-dimensional nature of the form is evident through its rotation (central image). In a virtual reality context, movement of the spheres may be enabled along the pathway constituted by the form (as indicated subsequently).

Screen shots of 3D triskelion as normally viewed in 2D (variously associating coloured spheres with focal points; and rotation on vertical axis)
Stationary view	Rotation of form (slow)	Rotation of form (fast)

Although such presentations are but a small step beyond what can be achieved in 2D, it is appropriate to note the flexibility inherent in the 3D variant. The X3D (or VRML) formats basic to the images and animations can be readily modified with a text editor. Thus the cross-section of the tubular path can be increased or decreased, as with the presence or absence of the added spheres. Similarly the rate of rotation of the form, or any movement of the added spheres (shown below), can be increased or decreased.

With respect to any new level of significance, the addition of the spheres to the presentations can be explored in the light of the challenging dichotomy between the "red pill" and "blue pill" metaphor highlighted by the popular movie The Matrix (1999), as separately discussed (Psychosocial Transformation by "Pill Pushing"? Model-making, strategic advocacy and the myth of the "red pill", 2017). The latter notably includes a discussion of Values and principles as pills? and Visualization in 3D of a trinity of connotations as a cognitive pill. In the absence of any symbolization of the relationship between the mutually hostile Abrahamic religions, the pattern provides an interesting basis for discussion -- especially with the role of any third in relation to the whole, which has proven so problematic to imagine.

Inversion and duplication: Much controversy is typically associated with the inversion of triadic forms, namely the problematic significance of having the upper circle pointing downwards. This was previously addressed in the rotation and counter-rotation of the triskelion (as mentioned above).

Inversion and duplication of the triskelion (with rotation)
Upright version with inverted version	Vertical rotation of two versions (video)	Horizontal rotation of two versions (video)

Movement of spheres along triskelion pathway: The coloured spheres can be readily moved along the spiral pathways, as shown below. It is notable that the form of the pathway does not result in the spheres colliding. The relative speed of movement of the spheres can be modified, as well as their relative sizes; made smaller, they could move "within" the tubular pathway. Both could b varied along the pathway, perhaps to the point of rendering a sphere invisibly small in portions of the pathway.

Screen shots of 3D triskelion with movement of spheres (coloured spheres moving between focal points; video)

These images are potentially useful as a means of holding contrasting views of a triadic pattern, notably that on the left. This could involve rotation around a horizontal axis indicating the emergence of an inverted form, and "reabsorption" of one contrasting form. The rotation of perspectives which are potentially orthogonal to one another is indicated with respect to the vertical axis by the central and right-hand images (and as animated). Again it can be stressed that the relative rate of rotations of the two forms can be readily modified.

Triple rotation around binary axes: Namely 3-fold over rotation around axes through centers (forthcoming)

Non-tubular solid: In quest of a "solid" form of the triskelion, one approach is to use the 2D pattern in a simple linear extrusion between two points. The result is problematic, as indicated by the image on the left below, about which Sergey Bederov remarks that the cross-section is highly non-convex, and such cross-sections may not be readily supported by some viewers (even with convex="false") -- as is the case with H3DViewer. Using one option (solid="false") he was able to render visible at least some of the intended shape. All VRML viewers display non-convex polygons correctly and some X3D viewers may do so without such obvious deficiencies.

Following further manipulation, as an alternative (to enable rendering with H3DViewer), Bederov made use of the IndexedFaceSet option, with all faces explicitly triangulated to enable use of that viewer (as in the alternative image on the left). The deficiences of any viewer suggest consideration of the possible cognitive deficiencies in the "view" of any higher order form, notably as potentially associated with any particular cognitive model. The recourse to triangulation merits consideration of its more general implications, especially in relation to any trinity (Triangulation of Incommensurable Concepts for Global Configuration, 2011; Holiness framed by a triangulated configuration of holes, 2014).

Linear extrusion into 3D of the triskelion pattern (wireframe video; interactive rotating variants: x3d; vrml/wrl)
"viewer failure"	"alternative"	solid	wireframe

Exploratory exercises kindly prepared by Sergey Bederov of Cortona3D

3D-printable variants are variously marketed, notably in the form of a pendant (Triple Spiral - Triskele - Sacred Geometry - Celtic; Triple Spiral)

Triangular extrusion of the triskelion pattern: (in preparation)

Circular extrusion of the triskelion pattern: A linear extrusion usefully raises the question as to the greater integrity implied by an extrusion along a circular spine, if that is possible. Of particular relevance is then the diameter of the circle, with one such exercise (giving of toroidal form) variously presented below. A smaller diameter would necessarily increase the degree of closure. Without using the wireframe rendering no hint of the spiralling inner structure would be possible in the views shown, however that inner structure can be explored using 3D viewers. It should be emphasized that the triskelion is the cross-section of the toroidal tube depicted below -- understood as extruded from the triskelion, or as a rotation of it around the centre of the torus.

Screen shots of extrusion of the triskelion pattern along a circular path (video; interactive versions: x3d; wrl/vrml)
"Upright" view	"Side" view	"Overview"

Exploratory exercise kindly prepared by Sergey Bederov of Cortona3D

Some impression of navigation within the 3D form above in virtual reality can be obtained from the following screen shots obtained using the VRML variant, using the viewer Cortona3D. Presumably the views could be enhanced for greater clarity with other techniques.

Screen shots of inner structure of extrusion of the triskelion pattern along a circular path (left image is a wireframe rendering of that in the centre; background changed for image on right)

As rendered in the images above, the circularly extruded 3D triskelion invites reflection in terms of the following:

Although not visible in the absence of 3D viewer capacity, the spiral coilings around the toroidal form (but contained within it) are consistent with the earlier explorations noted above (Visualization in 3D of Dynamics of Toroidal Helical Coils: in quest of optimum designs for a Concordian Mandala, 2016; Concordian Mandala as a Symbolic Nexus, 2016).
The spiral form is reminiscent of the marine nautilus valued as emblematic of desirable forms of growth and development. It is notably used both as a symbol of educational development and of strategic appropriateness (New Zealand Curriculum Nautilus, Nautilus Institute for Security and Sustainable Development). The question is how to benefit from this dynamic, as discussed separately (Enabling Governance through the Dynamics of Nature: exemplified by cognitive implication of vortices and helicoidal flow, 2010).
The form of the triskelion is also somewhat reminiscent of that of standard renderings of the fractal Mandelbrot set, whose psycho-social implications can be explored (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). It is therefore appropriate to note explorations of 3D renderings of the set, known by the term Mandelbulb.

As noted above, the triskelion is composed of three interweaving Archimedean spirals, and is of the same class as the hyperbolic spiral and Fermat's spiral. All static spirals appearing in nature are however logarithmic spirals, not Archimedean, although many dynamic spirals (such as the Parker spiral of the solar wind, or the pattern made by a Catherine's wheel) are Archimedean. Such distinctions suggest reflection on contrasting modes of psycho-social development, as can be variously considered (From global to helicoidal -- from pi to phi?, 2010; "Seeing" as implied by the Fibonacci spiral, 2011; Spiralling around "nothingness" and "pointlessness": the implication of phi, 2012).

As might be expected, being of interest to mathematicians, there are various accessible examples of Archimedean spirals, extruded to 3D, presented dynamically and interactively.

From "biskelions" to triskelions? The collection of images of polyskelions (presented above), includes what might be termed a "biskelion" or a "diskelion" -- namely two interweaving Archimedean spirals (image on left below). More correctly these are Fermat spirals, namely a type of Archimedean spiral. This may be compared with a variety of mathematically distinct spirals, including: an Euler spiral (spiros, clothoid or Cornu spiral) and a spiral tiling (as presented in the MathWorld entry on tiling, from the cover illustration of Branko Grünbaum and G. C. Shephard, Tilings and Patterns, 1987). It can also be compared with the classic symbol of Taoism (image on right).

Comparable two-fold symbolic indications of challenging attraction of dyadic thinking?
"Biskelion"	Euler spiral / Clothoid	Spiral tiling	Tao symbol

This raises the question of the comparability of these patterns, if only in a cognitive sense (beyond fascination with a single spiral) -- and of how this may extend to the triskelion with further implications for comprehension. They suggest various exercises relating the Fibonacci spiral, the Clothoid and the Tao symbol (Designing Global Self-governance for the Future: patterns of dynamic integration of the netherworld, 2010; Tao of Engagement -- Weaponised Interactions and Beyond: Fibonacci's magic carpet of games to be played for sustainable global governance, 2010).

Of current relevance are the various anouncements of the detection of gravitational waves associated with interacting black holes, notably by the CALTECH Laser Interferometer Graviational-Wave Observatory (LIGO), as discussed by Graham Templeton (What are gravitational waves, and where does physics go from here now that we've found them? Extremetech, 12 February 2016), Whether presented as photos, simulation screen shots, or imagined artwork, the dynamics bear a surprising resemblace to the twin spiral biselion discussed above.

Spiral dance of black holes	Colliding black holes	Black hole merger

Source: LIGO/T Caltech. Pyle; also video of artist's animation showing merger of two black holes and gravitational waves rippling outward during the event (another video).	Detail of image of Gravitational wave detection showing shape of ripples from black hole collision (The Guardian, 27 September 2017) Source: alamy_JHP71R.jpg	Black hole merger and space-time reverberations visualized. (T. Dietrich, Max Planck Institute for Gravitational Physics / R. Haas, NCSA, 27 September 2017); and video

Especially intriguing is both the reversal of direction of the spirals and the manner in which the centres of the spirals may be linked through another dimension (as cognitive "wormholes"), potentially to be recognized only intuitively, as in the case of the triskelion. More intriguing is the cosmic significance of the 2-spiral process and the possible cognitive functionality of a 3-spiral process -- understood as the classic three-body problem of concern to physics and to quantum mechanics, and celebrated in popular science fiction by r Liu Cixin (The Three-Body Problem, 2008).