Configurations of symbols within a sphere and implications of higher dimensionality

Reconciling Symbols of Islam, Judaism and Christianity (Part #9)

Circumspheres: Religious symbols are frequently circumscribed by a circle when depicted in 2D. Clearly this suggests the merit of embedding 3D symbolic configurations (like those above) within a circumsphere. Consideration can also be given to inscribing a suggestive insphere within any such configuration -- when it takes the form of a hollow container. Those above already include a small (yellow) sphere as a central point of reference. Several further examples are shown below.

Embedding of 3D model of Stars of David and Islamic pentagonal stars within a circumsphere (variously rendered visible)

3D Images developed using X3D-Edit

A circumsphere is especially interesting in relation to the number of vertices in the configuration of symbols. One model, with 5 Stars of David (configured around 2 Islamic pentagonal stars), potentially gives rise to 30 vertices (5x6). The other alternative presented, with 6 Islamic pentagonal stars (configured around 2 Stars of David), also potentially gives rise to 30 vertices (6x5). If the number of such sets is doubled by including the mirror elements, this gives rise to 60 vertices.

Polyhedra with relevant characteristics (as illustrated below) include:

the icosahedron, a regular polyhedron with the dodecahedron as its dual (each with 30 edges)
the icosidodecahedron, a semi-regular polyhedron with the rhombic triacontahedron as its dual
the truncated icosahedron (60 vertices), a semi-regular polyhedron with the pentakis dodecahedron (60 faces) as its dual

The last, indicated above with respect to reconciling 5-fold and 6-fold stars, is especially well known in the form of the stitching pattern on the traditional soccer ball.

Icosidodecahedron (30 vertices; 12 pentagonal faces; 20 triangular faces)		Rhombic triacontahedron (dual of icosidodecahedron) (32 vertices; 30 faces all of rhombic form)

Images produced with Stella Polyhedron Navigator

These polyhedra feature in various relevant mappings previously discussed (Embodying Global Hegemony through a Sustaining Pattern of Discourse, 2015; Geometrical configuration of Alexander's 15 transformations, 2010; Configuring Globally and Contending Locally: shaping the global network of local bargains by decoding and mapping Earth Summit inter-sectoral issues, 1992).

Nesting spherically symmetrical symbols: The role of the icosidodecahedron as a "container" is especially intriguing in that its dual provides a unique container for the dynamics of the set of 5 Platonic polyhedra which are so fundamental to the form of some symbols in 3D due to their symmetry, as separately discussed (Nesting polyhedra to enable comparison of patterns of discourse, 2015).

Rhombic Triacontahedron (green)
as a nesting framework for the dynamics of Platonic polyhedra
with Dodecahedron (blue), Icosahedron (red), Cube (grey), Octahedron (yellow), with Tetrahedron (cyan) and Tetrahedron (magenta)
(virtual reality variants static: vrml or x3d;
mutual rotation: vrml or x3d; "pumping": vrml or x3d;
videos: "pumping" mp4; "rotation" mp4)

Platonic polyhedra nested within Rhombic triacontahedron

The rhombic triacontahedron has proven to be fundamental to a new and unexpected state of matter termed a quasiperiodic crystal (quasicrystal) with local five-point symmery (Guy Inchbald. A 3-D Quasicrystal Structure, 2002; Walter Steurer and Sofia Deloudi, Crystallography of Quasicrystals: concepts, methods and structures, 2009). For this discovery of a pentagonal network that acted like hexagonal nets, Daniel Schechtman received the Noble Prize in chemistry in 2011 -- following an extensive period of hostility dating from the annoucement of the discovery in 1984. A remarkable articulation of the relation between the icosidodecahedron and its dual in 8-dimensional geometry is provided by Tony Smith (E8 Physics and Quasicrystals: Icosidodecahedron and Rhombic Triacontahedron, 2013). As images, the orthogonal projections of that geometry are strangely reminiscent of those of Islamic patterns.

Is the reconciliation between the religions only to be comprehended through 8-dimensional geometry -- through an unforeseen pattern of order? Given the manner in which Jerusalem is presented as a "territorial" challenge, should this rather be explored as a tiling problem of tessellation? This would point to the possibility of reconciliation in "higher" dimensions, given that it is has long been demonstrated mathematically that certain tiling configurations are not possible in two dimensions -- where only orthogonal projections offer symbols of such higher dimensionality.

Temples and crowns: It is of course the case that the most obvious rendering in 3D of valued symbols is through the architecture of temples and the design of crowns -- variously featuring "domes", typically hemispherical. The images and animations above are variously suggestive of that form -- with the added perspective as to how either temple or crown might take a dynamic form of cognitive significance.

An indicative metaphor in that respect is the modern planetarium in which configurations of constellations are variously highlighted and named, with the possibility of representing the patterns seen in previous and future millennia, or from distant perspectives. This technology suggests the possibilities of cognitive temples in which patterns of cognitive constellations could be variously projected -- notably reminiscent of the patterns in Islamic mosques (Keith Critchlow, Islamic Art and Architecture: system of geometric design, 1999: Keith Critchlow and Seyyed Hossein Nasr, Islamic Patterns: an analytical and cosmological approach, 1976).

With respect to any reconciliation of the Abrahamic faiths, the possibilities for an appropriate temple have been discussed separately (Imagining the "architecture" of the temple of Jerusalem, 2017). This included sections on: the Third Temple?, Scriptural ambiguity, Temple as a form of container?, and Modes of imagining "temple design".

The implications with regard to crown design have also been discussed separately (Engaging with Globality through Cognitive Crowns, 2009). This included sections on: Engaging with globality through "triple crown"cognition?, Understandings of "crowning experience", and Organization of memory.

Imagining a "hyperdimensional" Jerusalem: With the remarkable development of fundamental physics and its exploration of the dimensionality of physical reality, it is appropriate to ask how such thinking could be reconciled with the insights framed by religion in terms of the subtleties of mystical insight. This was discussed in a previous document (Comprehending the nature of a potentially hyperdimensional Jerusalem, 2017).

Of particular relevance, as noted there, are the possibilities of facilitating such comprehension to some degree through virtual reality technology. It is appropriate to recall that the polyhedra in 3D are understood in terms of geometry as 3-polytopes and merely part of increasingly complex forms of higher dimensionality (List of regular polytopes: 4-polytopes, 5-polytopes, 6-polytopes, 7-polytopes, 8-polytopes, 9-polytopes, 10-polytopes, and many more). The renderings of many of these bear a striking resemblance to patterns of religious significance (mandalas, rose windows, and the like).

It is potentially remarkable that so little is heard of 4-polytopes ("4D polyhedra"). for example, otherwise known as polychora. Their potential relevance and comprehensibility, through understandings of climate dynamics, is discussed separately (Enhancing Strategic Discourse Systematically using Climate Metaphors: widespread comprehension of system dynamics in weather patterns as a resource, 2015). The latter includes sections on:

Eliciting new symbolic configurations using artificial intelligence: With respect to reimagining a Jerusalem of higher dimensionality, this is discussed separately (Artificial intelligence and the Middle East peace process? 2017). Given the remarkable recent demonstrations of such facilities, the question is whether they can be used to elicit new patterns and render them comprehensible from a variety of perspectives.

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Configurations of symbols within a sphere and implications of higher dimensionality