Plato and Pythagoras were right: numbers and geometry rule the world. Two articles tend to show this: Matthias Brack, “Metal Clusters and Magic Numbers”, Scientific American, December 1997, pp. 50-55, and Donald E. Ingber, “The Architecture of Life”, ibid, January 1998, pp. 48-57.
Brack shows that small clusters of metal atoms (e.g. sodium, Na), up to 1000 atoms or so” are stable when they have configurations of regular icosahedra (20 triangular sides), cubes (6 square sides), or octrahedra (8 triangular sides).. This happens when the mini-clusters are cold. WC;n hot, the mini-clusters become spherical; they melt, a phase change from solid to liquid. But of course, spheres are only n-sided polyhedra when n approaches infinity.
Ingber shows that most structures, from the tiny to the huge, self-assemble into structures of “tensegrity”, in which tensions and compressions are balanced. Ingber borrowed ‘the concept from architecture. Buckminster Fuller’s geodesic dome is an example. Kenneth Snelson used a similar concept in his statues.
The cytoskeleton of cells is composed of micro-filaments (producing tension), microtubules (bearing compression, like struts), and intermediate filaments (tying them together). This is analogous to the macro-level skeleton of vertebrates, compcsed of muscles, bones, ligaments, tendons, and sinews.
The shape of cells can be changed from flat to globular through intermediate stages, and this affects the cells’ behaviour. Flat cells tend to divide repeatedly (mitosis), round cells tend to undergo apoptosis (a death program), while cells with in-between shapes tend to behave normally, fulfilling the function of the organ in which they are situated. This is in accordance with the Goldilocks effect: the golden mean is better than either extreme. It also illustrates the balance between two opposing tendencies common in living systems, e.g. the balance between the sympathetic and the parasympathetic nervous system. This effect of the shape of cells on their behaviour is relevant to the initiation of cancer: flat cells receive signals to keep on dividing and become immortal, round cells receive signals to stop dividing and commit programmed suicide; in-between cells are instructed to do their normal job.
The tensegrity model applies universally, eg to a buckyball (60 carbon atoms, 90 C-C bonds), an adenovirus, a protein-enzyme complex, a pollen grain, and multicellular Volvox. Most of these structures have radial symmetry, like an atom, like the benzene ring, like the Earth, like the Sun, like any star. These are Mandalas or prayer wheels. (A spiral galaxy does not quite have radial symmetry).
In folded proteins, the alpha-helix domains are the compression-bearing struts, the beta-regions the tension-bearing filaments. There is no radial symmetry. Neither is there in RNA and DNA, which have not even axial symmetry, only spiral symmetry. Yet both folded proteins and nucleic acids are also tenseqrity structures. Nucleic acids have a ribophosphate spine as a strut (like the spine in a vertebrate skeleton), and the purine-pyrimidine bridges (steps in the circular staircase) as the tension-bearing links that stabilize the backbone spiral, even if it gets over-twisted, as it does sometimes. When DNA unwinds to replicate or transcribe to RNA, each single strand looks like a charm bracelet with the coding bases hanging down.
Now let us go back to the structures with radial symmetry. Brack’s Platonic solids (metal mini-clusters) do not have struts and filaments; they are close-packed heaps of atoms, piled up like apples or oranges at the grocery, which naturally form regular polyhedra. Each polyhedron has a “magic number” of atoms, determined by geometry. For example, the series for hot sodium metal mini-clusters is 8, 20, 40, 58, 92, 138, 198, 264, 344, 442, 554. Magic numbers also occur for completed electron shells of the rare gases in the periodic table of chemical elements (2, 10, 18, 36, 54) and for the numbers of protons and neutrons in atomic nuclei (2, 8, 20, 28, 50, 82, 126). Magic numbers are reminiscent of the frequency intervals in a musical scale, or the combining ratios of atoms to molecules in simple compounds. (However, intermetallics and complex macromolecules have different rules.)
Each Platonic solid can be geometrically imagined as having compression-bearing struts, in the various hypotenuses., to hold up the outside perimeter.
The oligo-atomic clusters of Brack fit into the Ingber size series (see above) at its beginning, at the scale of the buckyball or a clathrate made of water ice and hOlding a rare gas atom in the middle. A buckyball is an oligo-atomic structure too, but made of carbon instead of metal.
The size scale can be extended to structures of a single atom with orbiting electrons held at a distance from the nucleus by “struts” of quantum restrictions, thus being prevented from spiralling into the nucleus as classical physics would predict. The struts have become non-material, mere force-f1elds, but the principle is the same. This structure. is repeated again, at an even smaller scale, in the structure of the atomic nucleus, with its various shells of enhanced stability. At the other end of the size scale is the solar system (Sun and planets), with sub-systems of planets and their moons, held in place by the balance of qrav1tational and centrifugal forces. The Sun and other stars are held in place by-the balance of gravitational collapse and fusion-heat-generated outward pressure:
And perhaps each planet, including the Earth, is held in place by its own spin, static structures being less stable than dynamic ones (like a moving versus a stationary bicycle). Finally, the universe at the macro scale has huge voids and walls, which could somehow uphold the whole structure.
The entire size scale then is: nucleus, atom, nano clusters including bucky-balls and clathrates, nucleic acids and proteins, viruses, the cell with its cytoskeleton, Volvox, pollen grain, vertebrate skeleton, planet, star, solar system, universe. Not all are classical polyhedra, but all are tensegrity structures. Close-packed structures sometimes substitute for geodesic dames, and sometimes force-fields replace material struts. .
This architectural view of structures from the sub-atomic to the cosmological, via the physical and the biological, confirms the idea that structure is more important than matter, not only in organisms but in the whole “cosmic zoom“emphasizing the formal over the material cause of Aristotle. Note that some of the struts are non-material forces or fields, at both ends of the size scale.
An even more complex view is that process supersedes structures, as Fritjof Capra states in “The Web of Life”. Under “process”, Gregory Bateson would include the two “great stochastic processes”, development (embryology) and evolution.
This whole view of Popper’s “World I” (things in themselves), derived from our phenomenological perceptions and our Platonic mathematical insights (Penrose’s physical and Platonic spheres feeding into our mental sphere) gives us a physical yet non-material theory of how things come together. We cannot know World I, but we can infer, from evidence and intuition, its basic structure and process. What we construct (Popper’s World II) shares some basic properties in common with World I. We still see only the shadows on the cave wall (Plato), but we discern at least the two-dimensional outlines of the real world.
This is only a theory, but complex and interesting enough to have the ring of truth. This is how God would create it, if only we could approach the mind of God.