August 12

Understanding energy: A simple Platonic solid model

Atoms, molecules, and life


  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.                                                                                                                                                                                                                                                                        Timaeus in Plato’s Dialogues

                                                                                                                                                                (The Four Elements)

                                                         c. 360 BC

A Platonic solid is a regular polyhedron, or a structure in 3-dimensional space. It has polygonal faces that are identical in shape and size. The angles and sides of the polyhedron are equal; there are the same number of faces meeting at each vertex. Five solids meet these criteria: Tetrahedron, Octahedron, Hexahedron (cube), Dodecahedron, and Icosahedron.

Platonic solids

Platonic solids

Platonic solids

These are the geometric shapes that we don't think twice about. They are all around us. 

From icosahedral viral shapes, to cubic crystals. Also, dodecahedral protein configurations. Then, our very common octahedral NaCl salt. And, our tetrahedral carbon forming an octahedral diamond. 

As you can imagine, there has been a tremendous amount of information written about Platonic solids. Not only because of their universal presence. But also they are innately fun. Stacking cubes, creating crystal structures, and looking at geometric patterns in nature. In addition, it turns out that much of art and architecture revolves around Platonic solid rules.

Sierpinski cube

Sierpinski cube

I have placed a number of references below for further review. 

Haeckel’s Radiolaria, 1888

Haeckel’s Radiolaria, 1888

Creatures made of Platonic solids

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. This harbors 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; and 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.

Atoms, electrons

Atoms, electrons

Do Platonic solids explain kinetic energy of electrons?

Systems in Nature will always veer to the lowest energy that they can exist in. A ball on a table has a higher potential energy than one on the floor.  Each state will move towards a more stable situation, to equilibrium. It is then that the system has the lowest energy. 

Electrons are not immune to the Laws of the Universe, even though they travel at an estimated rate of 2 million meters/second in a hydrogen atom. At that speed, it would take an electron about 3 minutes to reach the moon from Earth. Even at such speeds, electrons are still vying for lower energy levels and stability within atoms.

Thus, for a given energy level, electrons have a choice of 4 orbitals they can "travel" in. The s, p, d, and f orbitals. Whenever possible, an electron will choose lower energy configurations. For example, opting for a 4s orbital will give the electron a lower energy than if it were to move up to the 4p, 4d, or 4f orbitals.  

Why don't electrons crash into each other?

How do electrons do this? And why don't they crash into each other?

The answer lies with the organization of Platonic solids. Using a Platonic solid model, one can mathematically prove a number of points:

  • why s, p, d, and f orbitals exist
  • How electrons choose to fill energy levels and orbitals in a particular way
  • The reason the s orbital can have only 2 electrons,  
  • and so on (chemical bonds, water's odd structure, etc)

In the next series of articles/posts, I will walk you through a Platonic solid model that helps place the Mathematics of Platonic solids at the center of electron decision making.

Bottom line: 

In summary, electrons decide what energy level and orbital pattern they need using Platonic solid rules. It is only in this way that they can reach their most-stable-lowest-energy levels.

References on Platonic solids: 

  1. Platonic solids and the Periodic table ((tetrahedral
  2.  Platonic solids and electron arrangements (Dan Winter’s model of the atomTowards a new model of the nucleus, based on the pioneering work in physics of Robert J. MoonThe life and work of Dr. Robert Moon)
  3. Geometry of Natural systems (Geometry, All around usThe Hidden Geometry of Life).
  4. Polyhedra in physics; Attiyah M and Sutcliffe P. Polyhedra in Physics, Chemistry, and Geometry. Milan Journal of Mathematics, May 2003., 71 (1). 
  5. Icosahedral and tetrahedral structure of water (water and nature's geometrysupercooled water/ice)
  6. Metaphysical aspects of Platonic solids ( Sacred Solids in the Atomic Nucleus).

Picture credits: 

  1. NASA Universe. Platonic Solids, taken on July 31, 2013.
  2. fdecomite. Platonic box, taken on Sept 14, 2008.
  3. By F. ENOT. Render of the inside of a Sierpinski fractal object, ID: 64586176.
  4. Haeckel’s Radiolaria, 1888.
  5. Kevin Dooley. Atom: Protons, Neutrons, Electrons, Probability, taken on Dec 7, 2013,

About the author 

Juman Hijab

Juman is a retired physician after having been in clinical practice for more than four decades. Her lifelong interest has been in the chemistry of life.

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