The Platonic solids and the tetractys
Five perfect solids, two dual pairs and one solid that is dual to itself, assigned to the start figures of the tetractys.
The five Platonic solids. 'Dual' shows the figure formed by their face centres.
There are exactly five solids whose faces are identical regular polygons and at whose vertices the same number of faces meet: tetrahedron, cube (hexahedron), octahedron, dodecahedron and icosahedron. Join the face centres of such a solid and you get its dual:
- The tetrahedron is dual to itself.
- Octahedron and cube are dual to each other.
- Icosahedron and dodecahedron are dual to each other.
All of them can be built from tetrahedra: the octahedron as the intersection of two tetrahedra, the cube as their hull, the icosahedron from five tetrahedra (bulging outwards), the dodecahedron from five tetrahedra (inwards). Conversely, each solid can be described by several of the others. A dodecahedron, for example, by five cubes or five octahedra.
Assignment to the start figures
The solids can be assigned to the four start figures via the figure you see when looking at the solid along an axis of symmetry:
The four start figures: triangle, hexagram, pentagram and double pentagram.
| Solid | View along symmetry axis | Start figure |
|---|---|---|
| Tetrahedron | triangle, onto a face | triangle (3) |
| Octahedron and cube | sixfold, along the space diagonal | hexagram (6 : 2) |
| Icosahedron and dodecahedron | fivefold and tenfold | pentagram (5 : 2) and double pentagram (10 : 4) |
That the primes 3 and 5 double, triangle to six-pointed star, five-pointed star to ten-pointed star, is on this reading a consequence of even and odd numbers (see Number theory). Drawn upright and inverted, the start figures show the two dual pairs, 2 × 2 = 4, a tetractys. The tetrahedron, the only self-dual solid, thus stands above the other four in a fractal hierarchy.
The star polyhedra
It is said that the five Platonic solids are, apart from the sphere, the only perfectly symmetrical solids. That is only partly true. The star polyhedra belong with them: the four Kepler–Poinsot polyhedra and the stella octangula. They correspond to the star polygons among the start figures, just as the Platonic solids correspond to the polygons.
Open questions
Still to be investigated:
- the solid-angle sums of the Platonic solids, measured in full spheres, compared with the full circles of the start figures,
- the view of the simplexes as two-dimensional projections of higher-dimensional tetrahedra, and of the Platonic solids as a “distorted image” of our three-dimensional experience.