For over two millennia—from the time of Plato (428–384 BC) to the dawn of the Renaissance—the prevailing wisdom held that the universe was composed of just five elements. Four were familiar: earth, air, fire, and water. These were believed to form the very fabric of all life—plants, animals, and humankind alike. But there was a fifth, more mysterious element: a divine substance, reserved for the heavens, used by God Himself to craft the universe.
Each of these five elements was paired with one of the five perfect, regular, three-dimensional forms known as the Platonic solids. These shapes—so elegant, so mathematically pure—were seen not merely as geometry, but as the very architecture of creation.
To modern minds, raised on the periodic table and the discoveries of atomic science, this worldview may seem quaint. We now know of more than a hundred chemical elements. Yet the Platonic solids endure. Their symmetry, their balance, their beauty—they continue to inspire artists, architects, and scientists to this day.
For the Operative Masons of old, these forms were not abstract curiosities. They were essential knowledge—an extension of the planar geometry taught in the Craft Degrees into the three-dimensional realm of Royal Arch architecture. The Platonic bodies were the bridge between symbol and structure, between the earthly and the divine.
The theory of the elements, ascribed to the Greek philosopher Empedocles (430–409 BC), was later refined by Plato, who believed that mathematics was the key to unlocking the secrets of the cosmos. Each element, he taught, had a geometric soul:
- Fire was the tetrahedron—sharp, piercing, and energetic.
- Air was the octahedron—light and expansive.
- Earth was the cube—solid, stable, and enduring.
- Water was the icosahedron—fluid, adaptable, and deep.
- And the fifth element? The dodecahedron—twelve pentagonal faces forming a celestial vessel, too sacred for the mundane, reserved for the divine.
Even Aristotle, though he diverged from Plato, acknowledged the elemental qualities—hot, cold, wet, and dry—and linked them to these same forms. Yet he omitted the dodecahedron, perhaps sensing its power lay beyond the physical world.
Now, let us turn to the solids themselves:
- The Tetrahedron: Four equilateral triangles, three meeting at each vertex.
- The Octahedron: Eight triangles, four at each of six identical vertices.
- The Icosahedron: Twenty triangles, five converging at each of twelve vertices.
- The Cube: Six square faces, three at each of eight corners.
- The Dodecahedron: Twelve pentagonal faces, three at each of twenty vertices.
These are the only five regular solids that can exist in three dimensions. Why? Because only certain combinations of regular polygons can meet at a point without collapsing into a flat plane. Try to join six equilateral triangles? You get a hexagon. Four squares? A flat sheet. More than three pentagons? They overlap. No regular solid can be formed from hexagons or higher-sided polygons. The proof is elegant—and final.
Other polyhedra exist, of course—thousands of them—but none share the Platonic solids’ unique properties: identical faces, identical vertices, and equidistant points from the centre. Only the Platonic solids can nest perfectly within a sphere, each vertex touching the inner surface. This led Kepler to imagine a model of the solar system built from these forms—planets orbiting in nested Platonic shells. A beautiful idea, though ultimately eclipsed by more accurate models.
For 2,500 years, these five forms have captivated the minds of geometers and philosophers. Plato described them in his dialogue Timaeus. Hippocrates, the father of medicine, linked them to the four humours of the body: red tetrahedron for blood, yellow octahedron for bile, black cube for melancholy, and white icosahedron for phlegm. The dodecahedron, again, was left untouched—perhaps too divine to be sullied by mortal ailments.
And here, dear Companions, we return to Masonry.
There is a profound geometric link between the Platonic solids and the Triple Tau. Let us examine the external angles of each solid, reduced to right angles:
- Tetrahedron: 4 triangles × 2 right angles = 8 right angles
- Octahedron: 8 triangles × 2 = 16 right angles
- Icosahedron: 20 triangles × 2 = 40 right angles
- Cube: 6 faces × 4 right angles = 24 right angles
- Dodecahedron: 12 faces × 6 right angles = 72 right angles
Each total is divisible by eight—the number of right angles in the Triple Tau, as described in the Chapter’s Book of Work (p. 130): “two right angles on each of the exterior lines and two others by their union in the centre.”
Thus:
- Tetrahedron = 1 Triple Tau
- Octahedron = 2 Triple Taus
- Cube = 3 Triple Taus
- Icosahedron = 5 Triple Taus
- Dodecahedron = 9 Triple Taus
This is no coincidence. It is a symbolic harmony—a sacred geometry that binds the material to the metaphysical, the Craft to the Chapter, the human to the divine.
In summary, these five Platonic solids are not merely mathematical curiosities. They are ancient symbols of cosmic order, of divine craftsmanship, and of the eternal quest for understanding. They have stirred the minds of philosophers, physicians, and Masons alike. And perhaps, as we contemplate their perfect symmetry, we too are reminded of the Great Architect’s hand in all things.
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