Fundamental cognitive patterns assumed to be characteristic of belief systems

Middle East Peace Potential through Dynamics in Spherical Geometry (Part #2)

The assumption here is that an extremely fundamental cognitive identification is associated with the distinct patterns below -- thereby constituting an expression of quite different faiths. These distinct patterns may imply a yet more fundamental identification which cannot be readily expressed through symbols in two dimensions.

Whilst these symbols are indeed distinctly favoured in the flags of the cultures in conflict as considered here, the nature of their fundamental relationship to some form of cognitive identity could be questioned. The case of the Star of David is more straightforward. The use of the five-pointed star by Islamic nations is however subject to reservations -- especially given its use in the flags of non-Islamic cultures and for other purposes. The Red Star has been widely used as a symbol of both Socialism and Communism. Use of the six-pointed star by Muslims -- derives from a period when the Star of David and the Seal of Solomon were more harmoniously related in accord with medieval Jewish, Christian and Islamic legends.

Of relevance to the argument here, the 5-pointed star was a focal symbol of the Ottoman Empire. through which the crescent moon and star became affiliated with the Muslim world. This type of flag has become the de facto Islamic flag, and is used, with variations, by multiple Muslim lands such as Algeria, Azerbaijan, the Comoros, Malaysia, Maldives, Mauritania, Pakistan, Tunisia, Turkey, Turkmenistan, the Turkish Republic of Northern Cyprus, Uzbekistan, and the Western Sahara. As the crescent and star have no religious significance however, some Muslim scholars are against attaching these signs on mosques and minarets or using them to denote Muslim societies. The 5-pointed star as used by Islam is not limited to red in colour, green may also be used, or white against a background such as red or green.

Stars considered symbolic of cultural identity
Star of Islam	Star of David

Mathematically these patterns cannot be readily combined. This issue is described in mathematics in terms of tiling, described generically as tesselation (cf Craig Kaplan, The Trouble with Five, Plus Magazine, 2007). This of course has immediate relevance to the attribution and layout of territory -- with all the implications for politics and violence which have characterized the Middle East (And When the Bombing Stops? Territorial conflict as a challenge to mathematicians, 2000).

It is noteworthy that in the quest for a key metaphor for the structural challenges of the immediate future, the tessellation of disparate orders was identified by the sesquicentennial symposium of Boston University (Lance Morrow, Metaphors of The World, Unite!, Time, 16 Oct. 1989). Such tilings of zones of order, might also be understood metaphorically as paving stones separated by cracks of varying sizes. For optimists these might be hairline cracks; for pessimists the degree of separation might be such as to prevent movement and to offer little protection from the disorderly 'gunge' rising from beneath.

The approach taken here is to treat each of these patterns as "carried" by a polygon. This gives rise to the following, which still cannot be "fitted" together as a pattern in two-dimensions..

Symbolic stars embedded in corresponding polygons
5-fold Star	6-fold Star

A set of hexagons and pentagons can however be uniquely fitted together as a particular three-dimensional polyhedron, namely the truncated icosahedron. This has a total of 32 faces: 20 hexagons and 12 pentagons. It has 60 vertices with 90 edges of the same length. These are of two different types according to their position in the structure: 60 of one type, and 30 of another.