Necessary cognitive twist: star symbols as bladed propellers -- for propulsion in 3D?

Engaging with Elusive Connectivity and Coherence (Part #5)

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Misleading comprehension via 2D: Although interest in the Triple Helix model necessarily calls for the helical form, the restrictions of academic publishing typically force any depictions into a 2D format. Arguably this has cognitive implications with regard to comprehension of the nature of n-tuple helices. The issue can be highlighted in the case of the provocatively improbable structure known as the Discordian Mandala. This is depicted in 2D with the explanation offered by Wikipedia that some knot-theoretic links contain multiple Borromean rings configurations; one five-loop link of this type is used as a symbol in Discordianism, based on a depiction in the Principia Discordia.

However, as with the illusory depiction in 2D of the simplest Borromean example (as shown above), the question is how the form of such a Discordian Mandala might be depicted in 3D. This specific example framed a more general challenge explored separately with multiple images and animations in 3D (Concordian Mandala as a Symbolic Nexus: insights from dynamics of a pentagonal configuration of nonagons in 3D, 2016; Visualization in 3D of Dynamics of Toroidal Helical Coils: in quest of optimum designs for a Concordian Mandala, 2017).

Requisite dynamics in 3D: The argument with respect to star symbols as requiring comprehension through their dynamics in 3D (introduced in the previous discussion) then calls for experimental depiction as shown below -- recognizing the constraints in presenting them in this mode. The point to be emphasized is that seen from "above" their helical nature is far from evident. Each is wrapped around a hidden torus -- of which only a pale blue tracing circle is an indication. This helical wrapping is far more evident in the second row of images offering a slanted view of each..

"Top" and "Side" views of helical loops with from 2 to 6 winds
Two	Three	Four	Five	Six

The argument here is that each symbol can be rotated on the central axis -- whether at a slow (and elegant) rate or at a speed recalling that of a propeller, namely the speed required for that rotation to exert a propulsive force (whether in air or in water). As seen from a perspective down the axis, however, it is far from evident why rotation should exert any force. The side view makes it clear that the windings (if completed as a surface, rather than left as an outline) function as vanes or propeller blades. As such they then recall the argument for a degree of equivalence with the propulsion achieved by 2-wnged birds or 4-winged insects -- transformed here into rotary motion of blades functioning as wings (as with the helicopter).

Slanting symbolic "blades" to a point: Of particular interest is how wings and blades are adjusted in response to various requirements of flight -- to be able to "cut it". This is in addition to the actual design of the blades, which is typically the focus of intense research. The visual renderings in 3D, as depicted above, are governed by design options regarding the dimensions of the helix. The formula for wrapping a helix around a torus to create the effect of blades is as follows.

Parameters for a helix of n winds wrapped around a torus of major radius R and minor radius r with t based on the number of sets of xyz coordinates.
(x coordinate)	(y coordinate)	(z coordinate)
x=(R+rcos(nt))cos(t)	y=(R+cos(nt))sin(t)	z= r(sin(nt))

The degree of slant to any "blade" is achieved by adjusting the relation between the major and minor radius of the implied torus. For the purpose of the above exercise, this was specified in terms of the golden ratio -- as offering an aesthetic result of potential significance to comprehension. Other effects can be achieved by making the major and minor radius virtually identical -- if the same, the helix would be effectively wrapped around a horn torus.

The golden ratio gives rise to symbolic stars which have seemingly rounded "points" (as shown above), whereas the horn torus would engender sharp points. There would appear to be an intermediary condition more closely corresponding to the manner in which symbolic stars are depicted in 2D. The sharper the apparent point, the more that point corresponds to a vertical line in 3D (along the missing dimension of a 2D depiction).

Further investigation of such proportions would determine how such variously "bladed symbols" would offer a third dimension to the star-symbols so readily misinterpreted in 2D: 2 (Tao?), 3 (Trinity?), 4 (NATO?), 5 (Pentagon?, Wu Xing?), 6 (Star of David?).

"Cognitive twist": Such misinterpretation can also be explored in terms of a requisite "cognitive twist" to achieve empowerment in psychosocial systems (Sphere eversion as guide to the cognitive twist of global introversion?, 2013; Toroidal configurations as fruitful loopery, 2015).

This can be understood as addressing the disconnect between "linear" strategic thinking in 2D, in contrast with a variously recognized need for "curvature" -- potentially consistent with any "global" understanding. Intriguingly, this is consistent with the need for spherical geometry to enable navigation (Global Psychosocial Implication in the Pentagramma Mirificum: clues from spherical geometry to "getting around" and circumnavigating imaginatively, 2015).

Stars and wheel spokes: Reflection on the number of "blades" required could usefully draw on the number required in propeller design. More generally it relates to the number of spokes required in a wheel for appropriate load distribution.

Of interest is the contrast in the number of spokes between recent designs for bicycle wheels (3, 4, 5 spokes) with the optimum number (15) found to be required for war chariots of the past. The cognitive organization of the octopus also merits consideration.

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