Tetrahedron

The tetrahedron has four triangular faces, four vertices and six edges, with three faces meeting at each vertex. It is the smallest number of flat faces that can enclose a volume: three planes cannot close a space, so no simpler polyhedron exists. The counts satisfy Euler's formula for convex polyhedra, V, E + F = 2, which here gives 4, 6 + 4 = 2.

It is the only one of the five regular solids that is self-dual. Take the centre of each of its four faces, join the neighbouring centres, and the figure that appears is another tetrahedron, smaller and inverted. The other four pair off with each other instead: the cube and the octahedron are duals, and so are the dodecahedron and the icosahedron.

That there are exactly five convex regular solids and no more is not a tradition or a claim. It is a theorem, and the argument is short enough to give in full. At least three faces must meet at every vertex, and the angles gathered there must add to less than 360 degrees or the surface will not fold into a corner. Equilateral triangles have 60 degree angles, so three, four or five of them fit: that gives the tetrahedron, the octahedron and the icosahedron. Squares have 90 degree angles, so only three fit: the cube. Regular pentagons have 108 degree angles, so only three fit: the dodecahedron. Regular hexagons have 120 degree angles, and three of those already lie flat and tile the plane. Nothing with more sides can be used at all. Five arrangements, and the list closes.

Euclid sets this out at the end of the Elements, in Book XIII, which constructs the five figures and then argues that no other is possible. A scholium attached to that book credits the cube, the pyramid and the dodecahedron to the Pythagoreans, and the octahedron and the icosahedron to Theaetetus, who is generally taken to have written the first systematic treatment of all five. Plato, whose name the solids carry, did not discover them.

In the Timaeus, written in the fourth century BC, Plato has his speaker build the physical world out of geometry and hand four of the five regular solids to the four elements. The tetrahedron goes to fire.

The reasoning is given, and it is worth reading as reasoning rather than as decoration. Plato argues that the elements differ in mobility and in sharpness, that fire is the most mobile and the most cutting of them, and that among the solids the tetrahedron has the fewest faces, the sharpest angles and the least bulk. The most piercing element therefore receives the most piercing shape. He is not reporting an observation. He is arguing that like should answer to like.

Underneath the assignment is a second structure that usually gets dropped. Plato does not treat the solids as fundamental. He builds their faces from two elementary right triangles, the half-equilateral and the half-square, and it is the triangles that are basic. That has a consequence. Fire, air and water are all made from the half-equilateral, so they share a building block and can be taken apart and reassembled into one another. Earth is made from the half-square, and Plato is explicit that it cannot turn into any of the rest. The tetrahedron being fire is a detail of the scheme. The triangles are the theory.

None of this is a statement about matter, and Plato claimed no experimental support for it. It is a statement about what one philosopher in Athens thought a rational cosmos ought to look like. Nearly every later association of the tetrahedron with fire, will, energy and action descends from this passage, whether or not the writer knows it. Where a modern source presents that link as ancient universal knowledge about the element, what it is actually reporting is a few pages of a single dialogue.

Two tetrahedra can be set point for point in opposite orientations so that they interpenetrate, their eight vertices marking the corners of a cube and an octahedron occupying the space where they overlap. The figure was illustrated in Luca Pacioli's De divina proportione of 1509, with woodcuts attributed to Leonardo, and Kepler later named it the stella octangula. As mathematics it is a compound of two solids rather than a sixth regular solid, and it does not extend Euclid's list.

In modern esoteric writing this compound is called the Merkaba and described as a light-body vehicle activated by breath and rotation. The word is worth separating from the shape. Merkavah in Hebrew means chariot, and Merkavah mysticism is a documented early Jewish tradition, roughly the first millennium AD, concerned with visionary ascent to the throne-chariot of Ezekiel's opening chapter. That tradition has nothing to say about two interlocking tetrahedra. The geometric identification comes from the sacred-geometry literature of the late twentieth century, principally the Flower of Life workshops and books of Drunvalo Melchizedek in the 1990s. It is a modern teaching using an old name.

The chemistry, by contrast, needs no qualification. A carbon atom bonded to four others places them at the corners of a regular tetrahedron, because four things repelling one another equally around a centre settle into exactly that arrangement. The angle between any two bonds is about 109.47 degrees, the angle whose cosine is minus one third. Methane has this shape, and so does every carbon atom in diamond, whose hardness comes from an unbroken tetrahedral network. Silicon and oxygen build the silicate minerals from tetrahedral units in the same way, which makes the arrangement one of the commonest in the Earth's crust.

The tetrahedron is also the rigid one. A frame of four rods hinged at the corners cannot deform without a rod bending, while a cube of hinged rods folds over. Trusses and space frames exploit this. It is a genuine property of the shape and needs no dressing up.