# 7. Platonic solids

Article 7 Module 3

*Let it be agreed, then, both according to strict reason and according to probability, that the pyramid is the solid which is the original element and seed of fire; and let us assign the element which was next in the order of generation to air, and the third to water.*

* The four elements*, Timaeus,

(Plato’s Dialogues, c. 360 BC)

## Plato's wisdom

Here is Plato’s wisdom. He merged the Ancient Greek’s fascination with the 5 “Platonic Solids” with their understanding of the Universe into a single treatise. In the *Timaeus*, Plato has an elderly Greek Philosopher (Timaeus) describe how the elements of air, fire, earth, and water mix and match to create new entities.

The five Platonic solids (tetrahedron, octahedron, cube, icosahedron, and dodecahedron) are regular polyhedra that are perfectly symmetrical. They have equal faces, angles, and their vertices *lie on a sphere*. It was around Plato’s time (400 b.c. or so) that his contemporary Theaetetus was able to prove that there are only 5 regular polyhedra, ever.

Mathematically, points on a sphere are always far away from each other when they lie at the vertices of one of the Platonic Solids. There is something to be said for Geometric figures that are perfectly symmetrical whichever way you look at them.

### Plato’s Solids and Nature’s Geometry

Plato was a visionary. He could see the far-reaching touch of Platonic Solids on almost everything in Nature. Platonic solid configurations are seen in crystals, molecules, viruses, cellular structures, and in innumerable instances in the fiber of Natural systems. From hexagonal snowflakes, to icosahedral viral shapes, to cubic crystals, to dodecahedral protein configurations, to our very common octahedral NaCl salt, and our tetrahedral carbon forming an octahedral diamond.

On the ocean floor there are the remains of a creature whose sole purpose seems to be proving the supremacy of Platonic Solids. These 0.1 mm one-celled organisms - the Radiolaria - have silica skeletons. When the cell dies, the silica coat is shed onto the ocean floor. Some areas of the deep ocean floor have as much as fifteen percent of their ocean bottom layered with the dead remains of this ancient species of silica-cloaked creatures. It takes a while for any measurable siliceous layers to develop: about 1 cm every millennium.

In the late 1800s, Ernst Haeckel studied the Radiolaria, which had been collected during the first oceanographic expedition in the mid 1870s. These specimens were dredged up from the Mariana Trench, the deepest part of our oceans at ~ 11 kms (~6.8 miles). Haeckel drew thousands of pictures of Radiolaria, as well as hundreds of other species. In his book, *Art Forms in Nature*, he drew multiple organisms, from bright orange to blue, from microscopic to larger species, from smooth to intricately spiny. Here’s the fascinating part: for the Radiolaria, their hard envelope comes in a variety of shapes.

Five of those are our typical Platonic Solid configuration.

### Pauling, Kepler, and Moon

Like Haeckel, Linus Pauling (1901 - 1994) believed in the fundamental unity of natural things. In 1948, on a mild, unusually warm February evening in London, Pauling gave a talk at the Royal Institute of Great Britain where he said:

"Hence we may say that life has borrowed from inanimate processes the same basic mechanism used in producing these striking structures that are crystals, with their beautiful plane faces, their unfailingly constant interfacial angles, and their wonderfully complex geometrical forms.”

Others have looked into mapping the Platonic solids into our natural systems. For example, in 1596, Kepler related the 6 planets that were known into a system that followed a Platonic Solid orbit. Kepler believed that Earth’s orbit followed a sphere encircling a dodecahedron.

Robert Moon, a nuclear physicist, was born in 1911. This was the same year that Ernest Rutherford developed his model for the atom, which is the currently held model of a centrally highly concentrated positive charge containing the bulk of the atomic mass: the nucleus of atom. Seven decades later, Moon proposed an organization for the nuclear protons and neutrons. He suggested a Platonic Solid model.

Yet others (see Notes and References) have looked to Platonic Solid geometry to explain the Periodic table (tetrahedral model), Electron arrangements (Dan Winter’s model of the atom) , and our Natural systems (Geometry, All around us and The Hidden Geometry of Life).

### Why is this important?

Seeing things with their electrons at the vertices of a Platonic Solid helps predict structure and function.

Nowhere is this seen as clearly as in water. In 2000, Professor Martin Chaplin proposed an icosahedral network of water molecules. Fourteen molecules balanced out in a tetrahedral formation; 20 of those groups would cluster into a 280 icosahedral package. To make things more fascinating, these large water molecule clusters contain dodecahedron configurations.

When we see atoms as the ancient Greeks (and Pauling) saw them, with vertices, edges, and faces, it is easier to imagine them connecting with each other. Visualizing electron configurations in a Platonic Solid format helps us understand the unusual, unpredictable parts of nature (the chaos) as well as the order (the systems). When atoms increase in number, one gets mathematical systems of Tessalations, Fibonacci numbers, and E8 Lie groups . These are patterns that create order out of chaos.

The goal is to see Platonic Solids as part and parcel of atoms. As we will see in the next chapter, a model of Platonic Solid arrangements explains atomic orbital structure.

### Notes/references:

•Pauling, L. The Nature of the chemical bond. Application of results obtained from the Quantum Mechanics and from a theory of paramagnetic susceptibility to the structure of molecules. J. Am. Chem. Soc., 1931, 53 (4), pp 1367–1400. (seminal paper by Linus Pauling on the structure of the chemical bond leading to descriptions of hybrid orbitals; Publication Date: April 1931).

•Prkic L. Geometry, all around us. https://www.linkedin.com/pulse/geometry-all-around-us-lada-prkic-ceng. Published Sept 15, 2016.

•French, KL. The Hidden Geometry of Life: the science and spirituality of nature. Watkins, 2015.

•Haeckel, E. Art forms in Nature. The Prints of Ernst Haeckel.

•Pauling, Linus. The nature of forces between large molecules of biological interest. Royal Institution of Great Britain weekly evening meeting, Feb 27, 1948.

•Pauling L. The chemical bond; a brief introduction to modern structural chemistry. Ithaca, NY: Cornell University Press, 1967.

•The life and work of Dr. Robert Moon. *A previously unpublished transcript of a presentation by Dr. Robert J. Moon, Jr., Sept. 4, 1987, in Leesburg, Virginia. *(*Dr. Moon was introduced by Laurence Hecht, saying, “I asked Dr. Moon to give two lectures on the development of his model. The question I asked him to address tonight is: ‘How did he do it?’ ”). *21st Century Science.

•Hecht L, Stevens CB. The Moon Model: Towards a new model of the nucleus, based on the pioneering work in physics of Robert J. Moon. 21st Century, Fall 2004, 58-73.

•Solà-Soler, J. Sacred Solids in the Atomic Nucleus. Sacred-Geometry.es. Updated March 21, 2013.

•Wichernik J. Souls Distortion Awakening. 2008, 73-86.

•Attiyah M and Sutcliffe P. Polyhedra in Physics, Chemistry, and Geometry. Milan Journal of Mathematics, May 2003., 71 (1).

•Tsimmerman V. Derivation of mathematical expression of Mendeleev’s Periodic Law and its implications. International Society for the Philosophy of Chemistry summer symposium, August 2016 (from perfectperiodictable.com)

•Weisstein, Eric W. "Platonic Solid." From *MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/PlatonicSolid.html

•Bending, T. Platonic solids inside and outside a unit sphere. *Last modified on 25 October 2017*

•Marrin DL. Water and Nature’s Geometry. Water Sciences and Insights, May, 2008.

•Chaplin, Martin. A Proposal for the Structuring of Water. *Biophysical Chemistry* 83, (2000): 211-221.

Picture credits:

- fdecomite. Platonics. Flickr.com, taken on June 20, 2008.
- michael kooiman. Plato. Flickr.com, taken August 30, 2015.
**•Image**: Die Radiolarien (Rhizopoda radiaria) : eine Monographie**Year**: 1888 (1880s)**Authors**: Haeckel, Ernst Heinrich Philipp August, 1834-1919**Subjects**:*Radiolaria*;*Radiolaria*, Fossil; Rhizopoda- By adison pangchai. Model of Abstract Atom Structure. Shutterstock, ID: 550452931.