Combination of both alternative configurations in a single model with greater potential dynamics

Reconciling Symbols of Islam, Judaism and Christianity (Part #7)

---

[Parts: First | Prev | Next | Last | All] [Links: To-K | From-K | From-Kx | Refs ]

---

As complementary configurations in their simpler forms, each of the above alternatives is effectively a dual of the other in geometrical terms. The question is whether a more complex geometrical framework would allow the 5-fold and 6-fold symbols to be configured together more fruitfully. As an indication of possibilities, one experiment towards that end was made separately and gave rise to the following animation (Middle East Peace Potential through Dynamics in Spherical Geometry, 2012). Stars of David (blue) and Islamic pentagonal stars (red) are embedded in a network folding into a truncated icosahdron with 12 regular pentagonal faces, 20 regular hexagonal faces, 60 vertices and 90 edges.

Animation of a reconciliation of 5-fold and 6-fold frameworks within a truncated icosahedron

Animation developed with Stella Polyhedron Navigator

More fruitful possibilities could be explored in geometries of higher dimensionality. As shown above, the various stars do not rotate with respect to one another around their shared edges. This feature could of course be added.

Of relevance as a preliminary step is recognition of the complementary "cognitive cages" formed by various combinations of the Islamic pentagonal star and the Star of David hexagonal patterns, as shown below. In each case a shared edge is used to form the frame. The models elaborated in the earlier sections above were derived from those used to generate the images in the last two columsn on the right. In those below no rotation around a shared edge as been activated (although the technical quality of the earlier models has been partially improved in those rendered accessible below).

"Side" and "Axial" views of "cognitive containers" -- whether "cages" or "fortresses"
Hexagonal / Hexagonal (2 Stars of David framing 6 Stars of David)	Pentagonal / Pentagonal (2 Islamic pentagons framing 2x5 Islamic pentagons)	Hexagonal / Pentagonal (2 Stars of David framing 2x6 Islamic pentagons)	Pentagonal / Hexagonal (2 Islamic pentagons framing 5 Stars of David)
A combination of 8 Stars of David x3d: static; rotations	A combination of 12 Islamic pentagons (x3d)	12 Islamic pentagons framed by 2 Stars of David (x3d)	5 Stars of David framed by 2 Islamic pentagons (x3d)
3D Images developed using X3D-Edit.

The question raised by this set of four models is whether and how they might be understood as combined in any way, given the manner in which they result from shared edges between hexagonal and pentagonal stars. They can be explored in terms of geometric duality and the alternation between necessarily complementary forms. They might all emerge from a configuration in 4D or more.

Clearly the patterns can be further elaborated by ensuring that all edges are shared in some way to encompass greater insights into complexity. It is especially intriguing that the complexification is rendered comprehensible primarily through rotation on a shared edge in 3D. As shown in the animations accessible above, it is the cycle of phases which provides a sense of coherence, otherwise obscured by a multiplicity of seemingly minimally related lines.

Of relevance to further exploration is the unicursal hexagram that can be traced or drawn unicursally, in one continuous line rather than by two overlaid triangles, In relation to the rotation on a shared edge, the images below render comparable the manner in which the hexagonal Star of David is composed of two triangles (not distinguished in the rotational explorations above), the doubling of the pentagonal Islamic star (as currently featured in the models above), and the unicursal hexagram constitutes a form of "compromise" between the two (meriting exploration through rotation of its elements).

Hexagonal star	Unicursal hexagram	Overlapping star pentagons

As a knot (with overlapping lines), the unicursal hexagram has evoked speculation unrelated to the symbolism of the Star of David. Drawn as a knot it is a specific instance of the far more general shape which features in Pascal's Hexagrammum Mysticum Theorem (1639). Mathematically if six unordered points are given on a conic section, they can be connected into a hexagon in 60 different ways, resulting in 60 different instances of Pascal's theorem and 60 different Pascal lines. This configuration of 60 lines is termed the Hexagrammum Mysticum and has proven to be a particular confluence of significance with respect to:

• Kirkman points: The 60 Pascal lines intersect three at a time in 60 points known as Kirkman points.
• Steiner points: The 60 Pascal lines intersect three at a time in 20 points known as Steiner points.
• Cayley lines: Each Steiner point lies together with three Kirkman points on a total of 20 lines known as Cayley lines.
• Plücker lines: The Steiner points also lie, four at a time, on 15 Plücker lines.
• Salmon points: The 20 Cayley lines pass four at a time through 15 points known as the Salmon points

Given the "points" made in interfaith discourse (possibly even as "bullet points" in a presentation), and the associated "lines" of argument, what relation might these bear to these points and lines distinguished mathematically? How do either relate to "perspectives" and "visions" as discussed below?

Of corresponding relevance in spherical geometry, and vital to global navigation, has been the Pentagramma Myrificum of Gauss, as discussed separately (Global Psychosocial Implication in the Pentagramma Mirificum: clues from spherical geometry to "getting around" and circumnavigating imaginatively, 2015).

---

[Parts: First | Prev | Next | Last | All] [Links: To-K | From-K | From-Kx | Refs ]