Modifying the cyclic symmetry of the star torus

Global Coherence by Interrelating Disparate Strategic Patterns Dynamically (Part #12)

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The emphasis above is on the mapping possibilities of the dual of the simplest torus alone, especially in relationship to 16-fold, 12-fold and 8-fold patterns of strategies. The Antiprism application offers a means of increasing the cyclic symmetry of that basic polyhedron. This is of interest in various respects, succinctly suggested by the following animations from 4-fold to 20-fold and from 4-fold to 18-fold.

Animations of different patterns of N-foldness -- varieties of star torus
Animation 4-fold to 20-fold					Animation 4-fold to 15-fold
	N	faces (Nx4)	edges (Nx7)	vertexes (Nx3)		N	faces (Nx4)	edges (Nx7)	vertexes (Nx3)
	4-fold	16	28	12		4-fold	16	28	12
	8-fold	32	56	24		5-fold	20	36	15
	12-fold	48	84	36		7-fold	28	49	21
	16-fold	64	112	48		9-fold	36	63	27
	18-fold	72	126	54		14-fold	56	98	42
						15-fold	60	105	45
	20-fold	80	140	60
Images for animations generated from Antiprism (freely downloaded). Single command line instruction for the generation of those above, and immediately below, for interactive 3D viewing (change NN to number required, eg 8 or 16): off_util null -A v2,1,0 -A v2,-1,0 -A v1,1,1 | poly_kscope -s D4 | off_util -Ma -A f2,4,6,3,7,9 -A f0,2,9,1 | poly_kscope -s D4h | off_util -Ma | pol_recip | to_nfold NN | off_color -f U | antiview

Screen shots from a selection of 3D star torus models, with and without indication of symmetry elements
5-fold	8-fold	12-fold	16-fold
Images generated from Antiprism

The models above can be exported from Antiprism in VRML format. However, in order to benefit from some facilities in Stella Polyhedron Navigator, they were exported in OFF format from Antiprism and imported into the Stella application. Images of both the model and its dual are presented below in each case.

Screen shots from a selection of 3D star torus models, with indication of symmetry elements (dual variant above)
5-fold	8-fold	12-fold	16-fold
20 faces, 36 edges, 15 vertexes	32 faces, 56 edges, 24 vertexes	48 faces, 84 edges, 36 vertexes	64 faces, 112 edges, 48 vertexes
15 faces, 36 edges, 20 vertexes	24 faces, 56 edges, 32 vertexes	36 faces, 84 edges, 48 vertexes	48 faces, 112 edges, 64 vertexes
Images generated with Stella Polyhedron Navigator

The animations below illustrate use of the models for mapping purposes:

• 5-fold: Use is made of the Chinese WuXing pattern in this case. The case for 5-foldness is discussed separately with other examples (Patterning Intuition with the Fifth Discipline, 2019). This includes sections on: Possible cognitive implications of 5-foldness; Requisite fifth-order thinking?; Clues to systemic implications of 5-fold cognitive organization? The argument is a critical review of The Patterning Instinct: a cultural history of man's search for meaning (2017) by Jeremy Lent.
  
  
• 8-fold: The set of labels derives from the Chinese BaGua pattern, as one example. The labels in this case, and as used for the 5-fold, are of particular relevance to the current global environmental crisis and its comprehension, notably as separately argued (Weather Metaphors as Whether Metaphors, 2015; Enhancing Strategic Discourse Systematically using Climate Metaphors, 2015; Cognitive implication in a Chinese system articulated through weather-related metaphors, 2015). The BaGua pattern offers a more systematic pattern of interrelationships than that of the UN's Millennium Development Goals.
  
  
• 12-fold: Many sets of labels could have been used as examples (Checklist of 12-fold Principles, Plans, Symbols and Concepts: web resources, 2011). Given the emphasis here on interrelationship between the functions mapped, the example used here is that derived from the study by Arthur Young (Geometry of Meaning, 1976), as separately discussed (Geometry of Meaning: an alchemical Rosetta Stone? 2013; Typology of 12 complementary dialogue modes essential to sustainable dialogue, 1998; Typology of 12 complementary strategies essential to sustainable development, 1998; Characteristics of phases in 12-phase learning / action cycles, 1998). Unfortunately no functional distinctions are made between the 12 stars of the Flag of Europe which would have been an obvious alternative. More symbolic examples, in the light of the above argument, would have been the 12 Knights of the Round table, or the 12 Apostles at the Last Supper -- if their functions could be distinguished as with the 12-fold pantheons of Greece and Rome.
  
  
• 16-fold: Many possible examples have been listed separately (Deprecation of potential correspondences: 16-fold patterns? 2019). Preference was given here to the set of UN Sustainable Development Goals. An obvious alternative ould have been the set of 16 logical connectives discussed earlier. Note that the 17th SDG is potentially more appropriately implicit in the vertica central axis in the 16-fold images above.

Illustrative use of geometry of star torus for mapping purposes (use browser facilities to enlarge animations and labelling)
5-fold	8-fold	12-fold	16-fold
Animations generated with Stella Polyhedron Navigator

In terms of systemic viability, there is a case for comparing such models in the light of multi-armed starfish. Research has noted the advantages of the 5-armed variety (Liang Wua, et al, The Advantages of the Pentameral Symmetry of the Starfish, 2012). Curiously in the light of the extant strategic sets above, some species have six or seven arms and others have 10-15 arms. From the perspective of systemic viabiity, consideration could also be given to the 8-armed octopus and the 7-armed variant.

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