Three-fold and two-fold patterns -- by contrast?

Global Coherence by Interrelating Disparate Strategic Patterns Dynamically (Part #14)

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This argument has focused on patterns of 4-fold configuration and higher. Given the facility offered by the Antiprism application, any contrast with a 3-fold and 2-fold configuration is therefore potentially of interest.

Three-fold: Separate consideration has been given to 3D representation of triadic symbols (Cognitive Implications in 3D of Triadic Symbols Valued in 2D: representations of the triskelion in virtual reality and implications for quantum consciousness, 2017; Psychosocial Learnings from the Spiral Form of Hurricanes: implications of the triple helix and the 3-fold triskelion as "cognitive cyclones"? 2017). As noted above, such consideration merits particular attention in a period of preoccupation with the Triple Helix model of innovation.

The following animations result from generating models according to the procedure described above.

Animations 3-fold patterns derived from the 4-fold simplest torus (faces visible and transparent)
3-fold star torus (12 faces, 21 edges, 9 vertexes)		3-fold dual of star torus (9 faces, 21 edges, 12 vertexes)
Generated by Antiprism as an OFF file, with animations displayed by Stella Polyhedron Navigator

Two-fold pattern: The possibility of a 2-fold pattern consistent with the process explored above is far from evident. It was therefore a surprise to discover that a "non-polyhedral" form could be generated by the Antiprism application as a VRML model, which could then be manipulated by a 3D viewer. The model as shown below appears "flat" -- as might be expected. The most instructive feature of the model is that two of the faces are coplanar -- meaning that they lie in the same plane, even though distinct colours have been attributed to the four faces of the form (however that is to be comprehended).

As a consequence, when the pattern is moved in any way, the four colours (red, blue, green, yellow) generate unpredictable interference patterns in the central portion of the model. This phenomenon only applies to the centre of the form, not to the extremes whose colours are not modified. Note that the edges, like the face planes, do not intersect as might be otherwise inferred from superficial inspection of the model -- but only at the vertex points. Note also that edges bounding the central (red) zone as it appears, are each coincidental (co-linear) -- hence the total of 10 edges rather than 8 (as appears).

A further issue of particular interest is the manner in which the software generates such a form for display in 3D (such as VRML). Since the display expects convex polygons, by default it will triangulate polygons. For a crossed quadrilateral (as in this case), this will involve adding an extra vertex, and hence more edges -- thereby modifying their indicated counts. With or without triangulation, display issues may be evident. It is for this reason that three display variants are presented as animations.

As a focus for further discussion, also presented below is the dual of the star torus -- of box-like form (as was evident in the 4-fold and other variants above).

Experimental animations of 2-fold star torus pattern showing interference effects when displaced (note variation in numbering according to remarks above; numbering starts at 0, not at 1) Dual variant shown below in each case
Triangulation before displaying (with no extra triangulation)	Triangulation with display (labels with original edge numbers, but displaying a triangulated model)	No triangulation (does not display correctly)

Images in animations developed using Antiprism with the guidance of its developer Adrian Rossiter

Psychosocial significance of representational constraints and interference effects? Display issues, and any modifications to the count of faces, edges and vertexes, can be understood as valuable to the use of such models as an illustration of challenges to comprehension -- the underlying purpose of the above argument. Perhaps remarkable is the fact that much metaphorical use is made of such terms from geometry in strategic discourse (points, lines, sides, etc), as discussed separately (Engaging with Globality -- through cognitive lines, circlets, crowns or holes, 2009; Engaging with Globality through Cognitive Realignment: making points and aligning a target, 2009; Engaging with Globality through Cognitive Circlets: learning/action cycles, 2009; Engaging with Globality through Cognitive Crowns: all-encompassing, well-rounded experience, 2009; Engaging with Globality through Knowing Thyself: embodying engagement with otherness, 2009).

In considering the challenge of representation in 2D (as so widely preferred), with respect to any constraints and "effects", there is a case for rcalling the much-cited remark of Marshall McLuhan: the medium is the message.

Curiously the interference effects across the central zones of the 2-fold star torus help to make a basic point of this argument in stressing the value of a three-dimensional configuration of strategies rather than using 2D modalities. The confusing interference effects could be said to be characteristic of the chaotic relationships between distinctive strategic initiatives usefully representative of "violence".

This is evident when the pattern as a whole is seen from different orientations -- despite vigorous denial of any possibility of confusion on the part of their advocates (when committed to a binary perspective). The strategies can indeed be recognized as coherent "locally" (at the extremes of the form) -- when not subject to such interference in the "global" context to which they are claimed to apply. This is consistent with the argument above regarding conventional reinforcement of a "Flat Earth" perspective with respect to issues otherwise claimed to be "global". The merit of 4-fold and higher articulations is that such "violent" interference effects are reduced in ways which merit further consideration.

The presentation of the interference effects above is only indicative (and deprecated by developers of visualization software). However those effects recall the extensive studies, originated by Ernst Chladni, of so-called Chladni patterns. Given the potential relevance of wave-based theory for international relations (as indicated above), such figures are indicative of further possibilities for investigation (Theory behind patterns formed on Chladni plates? Physics Stack Exchange).

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