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Toroidal fullerenes as a complement to the global form
Sustainability through Global Patterns of C-60 Organization (Part #10)
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The focus on globalization and its spherical representation does much to obscure the implications of the toroidal form. This is somewhat ironic since the Earth as a globe can be understood as tracing out a torus in its revolution around the Sun (ignoring the manner in which the solar system is moving as a whole, such that the pathway can also be understood as being of helical form). In this sense there is a case for recognizing the degree to which people effectively live on a torus, as discussed separately (Imagining Toroidal Life as a Sustainable Alternative: from globalization to toroidization or back to flatland? 2019) with respect to the following:
Of particular relevance to human memorability is the degree to which toroidal configurations are valued in the design of memory in supercomputers, as discussed separately (Framing Cognitive Space for Higher Order Coherence: toroidal interweaving from I Ching to supercomputers and back? 2019) with respect to the following:
Research on fullerenes (or toroidal polyhexes) now extends to the possibilities of their toroidal form, as noted in the following:
- B. T. Ortega and M. McGuigan: Visualization and Simulation of Carbon Structures with Higher Genus (Brookhaven National Laboratory. 2018)
- C. Chuang, Y. C. Fan and B. Y. Jin: Systematics of Toroidal, Helically-Coiled Carbon Nanotubes, High-genus Fullerenes, and Other Exotic Graphitic Materials (Procedia Engineering, 14, 2011)
- Chern Chuang and Bih-Yaw Jin: Hypothetical toroidal, cylindrical, and helical analogs of C60 (Journal of Molecular Graphics and Modelling 28, 2009, 3)
- Chern Chuang, Yuan-Chia Fan and Bih-Yaw Jin: Generalized Classification Scheme of Toroidal and Helical Carbon Nanotubes (Journal of Chemical Information and Modeling, 49, 2009, 2)
- Michael C. Parker and Chris Jeynes: Fullerene Stability by Geometrical Thermodynamics (Chemistry Select, 5, 2020, 1)
- Ming-Hsuan Kang: Toroidal Fullerenes with the Cayley Graph Structures (Discrete Mathematics, 311, 2011, 21)
- M. Baca, et al.: Total Edge Irregularity Strength of Toroidal Fullerene (Mathematics in Computer Science, 7, 2013).
- E. C. Kirby, et al: Toroidal Polyhexes (Journal of the Chemical Society -- Faraday Transactions, 1993, 89).
There are necessarily far fewer depictions of toroidal fullerenes, especially in 3D -- as is potentially appropriate to the psycho-social implications explored here and the issues of comprehension. It is therefore appropriate to note the remarkable facility offered by a spreadsheet application developed by Sergey Bederov enabling an extensive range of toroidal forms to be generated in X3D format by modification of several parameters -- irrespective of whether they are to be considered as fullerenes of chemical significance. Screen shots of selected 3D examples of the output are presented below.
| Screen shots of generated toroidal fullerenes composed entirely of hexagons (polar and side views) | ||
| 8-2 (64 vertices); X3D model | 15-6 (360 vertices); X3D model | 20-5 (400 vertices); X3D model |
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| 30-3 (360 vertices); X3D model | 30-4 (480 vertices); X3D model | 60-2 (480 vertices); X3D model |
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| Screen shots derived from a spreadsheet application kindly developed by Sergey Bederov of Cortona3D | ||
Closed convex fullerenes of spherical form are of particular interest when they conform to the to the isolated pentagon rule (IPR) which is indicative of their stability and the possibility of their synthesis. Those of toroidal form are questionably convex and do not necessarily obey that rule. As noted by Sergey Bederov with respect to those generated by his spreadsheet application, those above lack pentagons -- being entirely composed of hexagons. It is questionable whether these are an optimal structure from a chemical, since the lengths of edges (corresponding to chemical bonds) are different.
In developing that spreadsheet application, Bederov explored the possibility of a potentially more stable toroidal fullerene through the introduction of 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 | |
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| Screen shots derived from a 3D model kindly developed by Sergey Bederov of Cortona3D. X3D model | |
From a psycho-social perspective, the construction of a toroidal fullerene is remarkably reminiscent of the design challenges faced in the toroidal construction of a nuclear fusion reactor. There the key concern is the toroidal configuration of magnets to ensure that plasma at very high temperature does not come in contact with the walls of the container -- according to the tokamak design principle. Toroidal fullerenes may offer insights into the design of a reactor for "cognitive fusion" fundamental to viable global strategy, as discussed separately (Enactivating a Cognitive Fusion Reactor: Imaginal Transformation of Energy Resourcing (ITER-8), 2006).
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