Memoir on the Theory of the Articulated Octahedron

Mr. C. Stephanos posed the following question in the Interm'ediaire des Math'ematiciens: “Do there exist polyhedra with invariant facets that are susceptible to an infinite family of transformations that only alter solid angles and dihedrals?” I announced, in the same Journal, a special concave octahedron possessing the required property. Cauchy, on the other hand, has proved that there do not exist convex polyhedra that are deformable under the prescribed conditions. In this Memoir I propose to extend the above mentioned result, by resolving the problem of Mr. Stephanos in general for octahedra of triangular facets. Following Cauchy’s theorem, all the octahedra which I shall establish as deformable will be of necessity concave by virtue of the fact that they possess reentrant dihedrals or, in fact, facets that intercross, in the manner of facets of polyhedra in higher dimensional spaces.

💡 Research Summary

Raoul Bricard’s 1897 memoir, “Memoir on the Theory of the Articulated Octahedron,” addresses a question posed by C. Stephanos: whether a polyhedron can keep its facet sizes fixed while admitting an infinite family of motions that change only solid angles and dihedral angles. Bricard shows that such polyhedra exist, but only in the concave, self‑intersecting regime, thereby complementing Cauchy’s rigidity theorem for convex bodies.

The paper begins with a detailed study of a single “tetrahedral angle”: four triangular faces meeting at a vertex, with the four face angles (α, β, γ, δ) held constant. The two dihedral angles at the vertex are denoted φ and ψ; their half‑tangents t = tan(φ/2) and u = tan(ψ/2) become the only variables. By fixing a convenient orthogonal coordinate system and expressing the coordinates of the points on the edges, Bricard derives a quadratic relation between t and u:

 A t²u² + B t² + 2C tu + D u² + E = 0  (1)

where the coefficients A–E are explicit trigonometric combinations of α, β, γ, δ (e.g., A = cos γ − cos(α+β+δ)). Equation (1) is the “equation of the tetrahedral angle” and shows that for a generic set of face angles the relation is irreducible: each value of t yields two distinct values of u, which are not rational functions of t.

Bricard then asks when (1) can factor so that u becomes a rational function of t. The discriminant under the square root must be a perfect square, leading to two possibilities. The first forces one of the face angles to be 0 or π, a degenerate case. The second requires AB = 0 and DE = 0, which translates into six algebraic conditions such as A·D = 0, B·E = 0, etc. Solving these conditions yields three distinct families of tetrahedral angles:

1. General tetrahedral angles – no special relations among the four face angles; (1) remains irreducible.
2. Rhombic (or “rhomboidal”) tetrahedral angles – adjacent faces are equal or supplementary (e.g., A = 0, D = 0). Equation (1) reduces to A t²u + 2C t + D u = 0, giving a unique u for each t.
3. Opposite‑face equal or supplementary tetrahedral angles – opposite faces are equal or supplementary (e.g., A·B = 0, D·E = 0). Equation (1) reduces to either A t²u² + 2C tu + E = 0 or B t² + 2C tu + D u² = 0, which yields a one‑to‑one correspondence between t and u.

A key theorem (Section 3) states that if two tetrahedral angles undergo a continuous deformation while keeping two adjacent dihedrals equal or supplementary, then all six faces of the two tetrahedra are pairwise equal or supplementary. This “covariance of opposite dihedrals” mirrors the well‑known property of articulated quadrilaterals: a linear relation A cos φ + B cos θ + C = 0 holds between opposite dihedrals.

Having classified the elementary tetrahedral building blocks, Bricard turns to the octahedron with triangular facets (the “articulated octahedron”). By Legendre’s theorem, a generic octahedron with fixed edge lengths is rigid because the number of independent constraints equals the number of edges. Flexibility can only arise if the edge lengths satisfy special relations that reduce the effective number of constraints.

He labels the octahedron AB C D E F (Figure 2) and introduces the six tetrahedral angles at its vertices. Assuming the octahedron is deformable, all twelve dihedral angles must vary; otherwise a single fixed dihedral would rigidify the adjacent tetrahedron and, by propagation, the whole solid. The analysis focuses on the three tetrahedra at vertices A, B, C, whose half‑dihedral tangents are t = tan(∠BC/2), u = tan(∠CA/2), v = tan(∠AB/2). Their deformation equations are respectively

 A t²u² + B t² + 2C tu + D u² + E = 0  (7)
 A₀ t²v² + B₀ t² + 2C₀ tv + D₀ v² + E₀ = 0  (8)
 A₀₀ u²v² + B₀₀ u² + 2C₀₀ uv + D₀₀ v² + E₀₀ = 0  (9)

where each set of coefficients depends only on the fixed face angles of the corresponding tetrahedron, and thus ultimately on the edge lengths of the octahedron. For the octahedron to possess an infinite family of configurations, (8) and (9) must share a common root in v for infinitely many (t, u) satisfying (7). In the generic situation the two equations have a unique common root, which can be expressed rationally in terms of t and u; consequently v is not free, and the system is rigid.

Only when the constituent tetrahedra belong to the special families identified earlier does the discriminant in the expression for v become a perfect square, allowing both (8) and (9) to have two coincident roots for a continuum of (t, u). This leads to three admissible global configurations of the octahedron, now known as Bricard octahedra:

• Type I (General‑type) – all six tetrahedra are of the “opposite‑face equal or supplementary” class. The octahedron is self‑intersecting; opposite edges remain equal in length, and the structure can flex while preserving all facet sizes.
• Type II (Rhomboidal‑type) – the tetrahedra are of the rhombic class; opposite dihedrals stay equal, and a pair of opposite edges stays fixed while the other pair moves. The octahedron still self‑intersects but exhibits a different symmetry.
• Type III (Mixed‑type) – a mixture where some tetrahedra are rhombic and others are opposite‑face equal; this yields a more constrained but still flexible mechanism.

All three types are necessarily concave and involve facet intercrossing (edges or faces pass through one another) – a direct consequence of Cauchy’s theorem that convex polyhedra cannot deform under the prescribed conditions. Bricard’s construction thus resolves Stephanos’s problem in full generality for triangular‑facet octahedra.

The memoir concludes by emphasizing the geometric significance of these flexible octahedra: they provide the earliest explicit examples of non‑trivial polyhedral flexes, foreshadowing later work on flexible polyhedra (e.g., Connelly’s sphere‑eversion, Steffen’s polyhedron) and influencing modern mechanisms such as deployable structures, robotic linkages, and metamaterials that exploit controlled dihedral motion while keeping edge lengths constant.