Toroidal polyhedral ring array

Cognitive Embodiment of Patterns of Governance of Higher Order (Part #10)

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

The drilled truncated cube is recognized as approximating a torus. The argument can be developed further given recognition that fullerenes of carbon can take toroidal form (as mentioned above). This can be related to exploration of its psychosocial implications (Imagining Toroidal Life as a Sustainable Alternative: from globalization to toroidization or back to flatland? 2019).

A spreadsheet application has been developed by Sergey Bederov of Cortona3D to explore the possibility of a potentially more stable toroidal fullerene (Toroidal fullerenes as a complement to the global form, 2022). This introduces pentagons in accordance with research on toroidal carbon nanotubes (Florian Beuerle, et al, Optical and Vibrational Properties of Toroidal Carbon Nanotubes, Chemistry: a European Journal, 17, 2011, 14; Pakhapoom Sarapat, A Review of Geometry, Construction and Modelling for Carbon Nanotori, Applied Sciences. 9, 2019, 11). Such research recognizes that in order to maintain their Euler characteristic (zero for a torus), an equal number of heptagons (7-gons) needs to be added (coloured blue and red in the screen shots below to distinguish them by size). Whether pentagons or hexagons, these are in each case by hexagons; mathematically all vertices lie on a perfect torus and edge lengths differ by only 24%.

Screen shots of a toroidal fullerene with pentagons and heptagons surrounded by hexagons
Screen shots derived from a 3D model kindly developed by Sergey Bederov of Cortona3D. X3D model

Such experiments suggest the possibility of configuring the 12 Archimedean polyhedra together in the form of a ring as variously illustrated below -- in relation to a torus of otherwise. Alternatives could use the Catalan variants.

Configuration of 12 Archimedean polyhedra in relation to a torus
Animations prepared using X3D Edit and Stella Polyhedron Navigator

Such a configuration is consistent with earlier exercises focusing on the design metaphor of an archetypal Round Table -- together with its numeric relationship to the Last Supper and the communication issues implied by both:

• Increasing the dimensionality of the archetypal Round Table? (2018)
• Implication of the 12 Knights in any Strategic Round Table (2014)
• Clarifying the Unexplored Dynamics of 12-fold Round tables (2019)
• Dialogue: towards the dynamics of the archetypal "Round Table"? (2011)

The strategic focus on 12-fold patterns suggests that these are intuitively recognized as representing 12 distinctive compromises between unity and diversity -- exemplified by their various approximations to a sphere as the set of semi-regular Archimedean polyhedra. The challenge of the 13th is evident in various forms. The relation to the 5 more regular Platonic polyhedra is another matter.

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