Mutually orthogonal Abrahamic symbols from the perspective of projective geometry

Reconciling Symbols of Islam, Judaism and Christianity (Part #11)

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Polyhedral "trinity": There is an intriguing sense in which the "geometry" of the relationship between the Abrahamic religions as an essentially incomprehensible trinity is explicated to a degree in the elusive relationship between the points, lines/edges and faces/sides by which polyhedra are formed -- potentially engendering a volume of higher dimensionality. Viewpoints and perspectives offer a pattern of understanding of a polyhedron, as of the models above.

This metaphor (possibly only of potential interest) necessarily avoids associating point, line/edge or face/side with a particular religion. Each may indeed make points, develop lines of argument, or be associated with a particular side in any engagement with otherness. It is the elusiveness of how those geometric elements are to be understood more generically that is analogous to how the Abrahamic religions might be reconciled. Their interrelationship in psychosocial terms clearly engenders the space within which many move and have their being -- whether or not each such religion s to be compared with a planet which believers inhabit and may never leave -- or do so at a cost.

Projective geometry: The rendering comprehensible of higher dimensional configurations through orthogonal projection, as indicated above, provides a reminder that any question of viewpoint and perspective in interfaith discourse may well be clarified by the mathematical discipline of projective geometry. It is the study of geometric properties that are invariant with respect to projective transformations (H. S. M Coxeter, Projective Geometry, 2003; Lawrence Edwards, Projective Geometry, 2004; David Hilbert and S. Cohn-Vossen, Geometry and the Imagination, 1999).

There have seemingly been few interpretations of its significance for comprehension of the "higher dimensional" subtleties valued by the Abrahamic religions -- with the exception of its adaptation to the insights of Rudolf Steiner, as described by Olive Whicher (Projective Geometry: creative polarities in space and time, 1971).

Mutual orthogonality: In these terms a strikingly simple possibility is to consider that the symbols of the Abrahamic religions can be reconciled through a mutually orthogonal configuration, namely with their 2D planes at "right angles" to one another -- perhaps consonant with the sense in which each perceives itself to be uniquely "right". Consistent with the difficulty of each to attach significance to the others, the perspectives then offered reduces others to a line "crossing" that preferred. The variety of perspectives offered by the 3D model of that configuration allows different narratives to be elaborated, thereby serving to justify the understandings which may be associated with the individual symbols in 2D. The manner in which each may be "crossed" is indicative of the challenge each experiences from the other in practice.

The simplicity of the model is more readily comprehensible through the videos (accessible below with interactive variants).

Screen shots of perspectives on a single configuration of mutually orthogonal Abrahamic symbols (Configuration of Star of David -- Islamic Star -- Christian Cross, best understood through the central column of images below, with the others only offering a 2D perspective -- hinting a 3D perspective. Second row presents the controversial inverted images also encompassed by the configuration)
Interactive model: x3d, wrl/vrml. Video (thin variant): mp4. Video (thick variant): mp4. Prepared with X3D-Edit.

The model could be elaborated further with features such as the following:

• allowing the symbols to rotate with respect to one another in a more dynamic configuration -- with the cross offering the possibility of rotation on either axis or with respect to its end points.
• increasing or reducing the visibility (transparency) of the individual symbols relative to one another
• increasing or decreasing the size of one or other symbol with respect to the others
• unfolding the geometry of the symbols from a common element (a single line) and collapsing them back to such an element -- effectively folding them into one another
• sliding the 2D symbols along their respective orthogonal axes -- away from their contiguous position, or passing through it
• incorporating such changes into a cycle -- possibly privileging one or other symbol during the cycle.

Emphasizing two-dimensionality of symbols: In exploring the significance that might be associated with such "special effects", the cross-section of the linear elements of a symbol can be modified such that that it becomes so "thin" that it is virtually invisible from a particular direction, even though it may be much "thicker" from another perspective. This procedure was used in a second variant which gave rise to a sharper configuration. It is from this that the images above were then derived.

To a greater extent than with the first variant, the 3D model as a whole is best understood through a second video. Perhaps significantly, it is quite difficult to illustrate the model with any single image and even the set above detracts from full understanding of its integrity. It is potentially significant that the conventional view of each symbol -- free of "interference" by the two others -- is only possible along the axis through the origin, a perspective from which the others are invisible.

This approach suggests that the first variant (as indiciated in the first video) is inherently misleading -- as with the earlier models in which the elements of the symbols were all given a circular cross-section, however small the radius. The cognitive implications of the assumption of "thickness" can be usefully challenged.

Empasizing the thinness in 3D invites narratives to the effect that none of the 2D symbols has "depth" of significance, although that may indeed be inferred. Curiously that depth is traditionally associated to a degree with the "weight" of the cross, typiclly not a feature in its depiction. Similarly, the Star of David, as the Shield of David, implies three-dimensionality; this too does not feature in depictions.

This process could be related to use of lighting to ensure that the thicker lines cast a shadow even though a given symbol is invisible through direct observation from a particular angle. This supports any narrative suggesting that some patterns can only be "seen" by the shadow they cast. Other trompe-l'oeil effects might be explored.

The possibilities merit careful consideration given that the Islamic symbol is widely recognized as double -- star and crescent. How indeed can the crescent, with its implication of three-dimensionality, be meaningfully integrated into the spherical geometry of the above model? The inclusion of a circumsphere (as discussed above) is of relevance. Another approach is to use a rotating circular band -- implying the sphericity variously evident in relation to the iconography of the 3 orthogonal symbols.

Given that the model enables recognition of each Abrahamic symbol along one of three orthogoanl axes, this raises the question as to how symbols of other religions (say 8) might be uniquely recognized along axes in a higher dimensional space.

Controversial orientations: The visibility of the controversial inversions of the symbols from some perspectives offers a coherent means of integrating understandings valuable to some. These are notably associated with acknowledgment of the psychological shadow, namely a part of the unconscious mind consisting of repressed weaknesses -- potentially consistent with the arguments of John Ralston Saul (The Unconscious Civilization, 1999). As noted above, the inverted variant of the cross recalls use of the Cross of St Peter, just as the (omitted) horizontal cross recalls the Cross of Saint Philip and Saint George's Cross, and the angled version that of the Cross of Saint Gilbert.

Given the association of the Cross of St Peter with the symbolism of the hanged man, it is appropriate to note the effort by Aleister Crowley, highly controversial in his own right, to enable the design of a set of Tarot cards such that they could be placed contiguously so that their symbols formed a larger pattern.

Multidimensional implications of the Abrahamic "Fall": The narratives of the Abrahamic religions share a degree of understanding of the Fall of Man. There is the fascinating possibility that this may be understood as a progressive constraint on comprehension as the number of dimensions of awareness is reduced. There is a fall "through the dimensions" to the lowest dimensionality -- that of 2D and 3D experience.

Mathematicians have produced classic studies of the experience of higher dimensionality at that interface (Edwin Abbott Abbott, Flatland: a romance of many dimensions, 1884; Ian Stewart, Flatterland, 2001). A number of presentations make technical arguments in that respect (Sophie Weiner, Here's a Cool Way to Visualize Higher Dimensions, Popular Mechanics, 13 August 2017; Maureen J. St. Germain, Waking Up in 5D: a practical guide to multidimensional transformation, 2017; Rob Bryanton, Imagining the Tenth Dimension: a new way of thinking about time and space, 2007; Daniel Smilkov, et al., Visualizing High-Dimensional Space). Mystical experience has offered insights articulated otherwise.

Arguably those favouring a particular belief follow a particular pathway in their "fall". This suggests that the symbols they favour in 2D, as orthogonal projections of higher dimensionality, could be called into question as effectively different forms of "cognitive flat pack". These offer the potential of being "unpacked" into configurations of higher dimensionality -- through the imaginative engagement they elicit as sacred forms. Notions of any corresponding "ascent" then imply an increase in the dimensionality of awareness -- in which the role of time, process philosophy, and movement may be relevant. (Clues to 'Ascent' and 'Escape', 2002; Clues to Movement and Attitude Control, 2002). There is a sense in which movement and its embodiment imply the experience of 4D and 5D through "body thinking" (Mark Johnson (The Meaning of the Body: aesthetics of human understanding, 2007; Maxine Sheets-Johnson (The Primacy of Movement, 1999; The Corporeal Turn: an interdisciplinary reader, 2009). Radicalism may be justified when the dimensions articulated through movement are precluded -- as in the binary thinking which currently characterizes global policy-making.

It is especially intriguing the manner in which the unitary simplicity of conventional symbols is held to imply and reflect a unitary experience of the highest dimensionality. This is all the stranger in that the frequent appeals for unity by politicians and religious authorities take little account of its multidimensional form and the intimate cognitive experience with which that may be associated. Given the extent of religious pluralism -- each framing its perspective as "right" -- how might the "mutually orthogonal configuration" of 8 religions be understood in 4 or 8-dimensional space, for example? Is the tesseract indicative in that respect?

Geometrical entrapment: The challenging cognitive nature of any trinity -- whether between the Abrahamic religions or within the cognitive framework of any of them -- can be rendered more explicit through a triangular pattern basic to the mathematical argument of q-analysis, as developed by Ron Atkin (Multidimensional Man; can man live in 3-dimensional space?, 1981). Subsequent to his work on the relevance of mathematics to social structures (1974, 1980), Atkin is most widely reputed for his work on chess computing -- anticipating the development of artificial intelligence.

As separately summarized (Comprehension: Social organization determined by incommunicability of insights), Atkin illustrates the challenge of comprehension in relation to experience "within" the geometry of a triangle -- especially with regard to the perspective necessary to comprehend the geometry of the triangle as a whole -- namely the trinity. He demonstrates how cognitively one can be trapped by:

• a single point in the triangle, with inadequate comprehension of the lines linking them
• a single line, or the sequence of lines forming the triangle, but without any comprehension of the triangle as a whole
• the side framed by the points and lines, enabling comprehension of the triangular form -- but not, by implication of the polyhedral form of which it is a part

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