Compound Polyhedra

Introduction

When two polyhedra are placed with a common centre so that their faces interpenetrate, the result is a compound — a figure that is not itself a polyhedron but possesses striking symmetry and reveals deep relationships between the solids it combines. Compounds are the natural consequence of duality: dual pairs, when superimposed at their common centre, produce the most regular and symmetrical compounds possible.

When two or more polyhedra interpenetrate — sharing a common centre, their faces passing through each other — the result is called a compound polyhedron. Compounds are not polyhedra in the strict sense (their faces intersect), but they possess striking visual symmetry and deep mathematical structure. The most symmetrical of them, the regular polyhedral compounds, are as constrained and beautiful as the Platonic Solids themselves.

What is a Compound Polyhedron?

A compound polyhedron consists of two or more polyhedra placed with a common centre, typically arranged so that the compound inherits the full symmetry of its components. The components interpenetrate — their edges and faces cross — but each individual polyhedron retains its complete form within the compound.

The simplest and most famous example is the stella octangula: two tetrahedra interpenetrating to form a star-like solid. Viewed from any face-on direction, it projects as a Star of David.

Key Examples

Stella Octangula

The stella octangula (from the Latin for "eight-pointed star") is a compound of two tetrahedra, one pointing upward and one pointing downward, sharing a common centre. It was described by Kepler in his Harmonices Mundi (1619), and remains the simplest regular polyhedral compound.

The stella octangula has octahedral symmetry. Its convex hull — the smallest convex solid containing it — is a cube. The eight outer points of the stella octangula coincide with the eight vertices of this cube, while the compound's intersection (the solid region occupied by both tetrahedra simultaneously) is a regular octahedron.

This three-way relationship — tetrahedron, stella octangula, cube, octahedron — reflects the deep interconnection of the tetrahedral, cubic, and octahedral symmetry groups.

Compound of Cube and Octahedron

The cube and octahedron, being duals of each other, can be arranged as a compound with their centres coinciding. In the standard construction, the octahedron's vertices lie at the face-centres of the cube. The compound has the full cubic symmetry group and is visually striking: triangular pyramidal points protrude from each face of the cube.

Compound of Dodecahedron and Icosahedron

Similarly, the dodecahedron and icosahedron — duals of each other — form a compound where the icosahedron's vertices lie at the face-centres of the dodecahedron. This compound has full icosahedral symmetry (the largest symmetry group of any Platonic Solid pair).

The Five Regular Polyhedral Compounds

A compound is regular if both the components are regular polyhedra and the compound as a whole has full rotational symmetry. There are exactly five regular polyhedral compounds:

1. Compound of two tetrahedra (the stella octangula) — two tetrahedra, octahedral symmetry
2. Compound of five tetrahedra — five tetrahedra arranged with icosahedral symmetry; this compound is chiral (exists in left-handed and right-handed forms)
3. Compound of ten tetrahedra — superimposing both chiral versions of the five-tetrahedra compound gives a compound of ten tetrahedra with full icosahedral symmetry
4. Compound of five cubes — five cubes arranged with icosahedral symmetry; the twelve vertices of the icosahedron serve as reference points
5. Compound of five octahedra — the dual of the compound of five cubes; five octahedra with icosahedral symmetry

These five compounds are regular in the sense that they each admit a group of symmetries acting transitively on their components. The compounds with icosahedral symmetry (numbers 2–5) are particularly rich: the compound of five cubes, for instance, has 8 × 5 = 40 vertices, all lying on the vertices of a large rhombic triacontahedron.

Kepler's Mysterium Cosmographicum

In 1596, the young Johannes Kepler published Mysterium Cosmographicum ("The Cosmographic Mystery"), in which he proposed that the orbits of the six known planets were determined by the five Platonic Solids, nested one inside another:

• Saturn's sphere enclosed a cube
• Inside the cube was Jupiter's sphere
• Inside that sphere was a tetrahedron
• Mars's sphere was inside the tetrahedron
• Inside Mars's sphere was a dodecahedron
• Earth's sphere was inside the dodecahedron
• Inside Earth's sphere was an icosahedron
• Venus's sphere was inside the icosahedron
• Inside Venus's sphere was an octahedron
• Mercury's sphere was inside the octahedron

Kepler's model predicted the ratios of the planetary orbit sizes with surprising accuracy — within the precision available to him before Tycho Brahe's more accurate data revealed systematic discrepancies.

Though the model is wrong (we now know planetary distances reflect the physics of the solar nebula, not Platonic geometry), Kepler's book is historically important: it was the first attempt to find a mathematical law governing planetary distances, and it launched Kepler's career. His later discovery of the true laws of planetary motion — Kepler's three laws — grew from the same obsessive search for mathematical order in the heavens.

Star Polyhedra — the Kepler-Poinsot Solids

Extending the idea of regular polyhedra beyond convex forms, Kepler discovered two star polyhedra in 1619, and Louis Poinsot rediscovered and extended them in 1809. There are exactly four regular star polyhedra, now called the Kepler-Poinsot Solids:

1. Small Stellated Dodecahedron — twelve pentagrammic faces (five-pointed star polygons), three meeting at each vertex. It looks like a dodecahedron with a tall pentagonal pyramid on each face.

2. Great Stellated Dodecahedron — twelve pentagrammic faces, three meeting at each vertex, but with deeper interpenetration. Its convex hull is an icosahedron.

3. Great Dodecahedron — twelve pentagonal faces, five meeting at each vertex (like the dodecahedron), but the faces intersect each other. Discovered by Poinsot.

4. Great Icosahedron — twenty triangular faces, five meeting at each vertex (like the icosahedron), but again with intersecting faces.

The Kepler-Poinsot Solids satisfy a generalised form of Euler's formula: V − E + F = 2(1 − g), where g is the genus (a topological measure of how many "holes" the surface has). For these star polyhedra, g is not zero, which is why the standard formula V − E + F = 2 does not hold for them.

Connection to Sacred Geometry

The nested Platonic Solids, compound polyhedra, and star polyhedra have been interpreted throughout history as expressions of cosmic and sacred order — from Plato's Timaeus to Kepler's solar system model to the geometry of our work. The stella octangula, in particular, appears in many sacred geometry traditions as a symbol of the interpenetration of opposites: ascending and descending, spirit and matter, the six directions of space unified in a single form.

While modern science locates these structures in the symmetries of matter (crystal lattices, viral capsids, atomic orbitals) rather than in the cosmic spheres, the mathematical fact that only five regular polyhedra exist — and that they fit together in such constrained and beautiful ways — remains one of the most remarkable results in all of geometry.

Conclusion

Compound polyhedra occupy the space between isolated solids and abstract symmetry groups — they are the visible expression of what happens when dual pairs are superimposed or regular solids are nested according to their shared symmetry. The five regular polyhedral compounds, like the five Platonic Solids themselves, are uniquely determined by strict combinatorial constraints, and the Kepler-Poinsot Solids extend the concept of regularity beyond convex forms to star polyhedra that satisfy a generalised Euler formula. Together, compounds and star polyhedra complete the classical taxonomy of regular and semi-regular solids in three dimensions.

Beyond the five Platonic Solids, Archimedes discovered thirteen semi-regular polyhedra. The next chapter explores the Archimedean Solids.