[Paper Review] Memoir on the Theory of the Articulated Octahedron

Raoul Bricard|arXiv (Cornell University)|Mar 4, 2012

Mathematics and Applications1 references3 citations

TL;DR

This paper presents a comprehensive theory of deformable octahedra with invariant triangular facets, demonstrating that such polyhedra exist only as concave forms with reentrant dihedral angles or intercrossing facets. Using a tetrahedral angle deformation model based on trigonometric relations and rational parametrization via tangent half-angles, the author derives a quadratic equation governing dihedral angles, leading to three distinct types of articulated octahedra—two symmetric types and one more complex, non-intuitive type—proving their existence through geometric and algebraic analysis.

ABSTRACT

Mr. C. Stephanos posed the following question in the Intermédiaire des Mathématiciens: "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.

Motivation & Objective

• To resolve C. Stephanos' question on whether polyhedra with invariant facets can undergo infinite families of deformations altering only solid and dihedral angles.
• To extend earlier results on a special concave octahedron to a general classification of all such deformable octahedra with triangular facets.
• To establish the mathematical conditions under which articulated octahedra with fixed face shapes can flex while preserving their facet geometry.
• To demonstrate that such deformable octahedra must necessarily be concave, due to the necessity of reentrant dihedrals or intercrossing facets, in contrast to Cauchy’s rigidity theorem for convex polyhedra.

Proposed method

• Derives a fundamental 'tetrahedral angle equation' relating dihedral angles φ and ψ via trigonometric identities and coordinate geometry in a fixed face reference frame.
• Introduces rational parametrization using t = tan(φ/2) and u = tan(ψ/2) to transform the trigonometric relation into a quadratic equation in two variables.
• Analyzes the geometric multiplicity of solutions by considering symmetric configurations of the line SN relative to the plane BSM, leading to a second-degree equation in both t and u.
• Constructs articulated octahedra by pairing tetrahedral angles with equal or supplementary opposing faces, ensuring consistent facet assembly.
• Classifies the resulting deformable octahedra into three types: Type I (with an axis of symmetry), Type II (with a plane of symmetry), and Type III (more complex, non-symmetric configurations).
• Establishes equivalence between deformable octahedra and deformable oblique hexagons with fixed sides and angles, as well as systems of six hinged planes forming a closed loop.

Experimental results

Research questions

• RQ1 Can there exist polyhedra with invariant facets that undergo continuous deformation altering only solid and dihedral angles, contrary to Cauchy’s rigidity theorem for convex polyhedra?
• RQ2 What are the necessary geometric and trigonometric conditions under which a tetrahedral angle with fixed face angles can be deformed while preserving its face shapes?
• RQ3 How can the deformation of a tetrahedral angle be parameterized algebraically to yield a quadratic relation between dihedral angles?
• RQ4 What are the distinct classes of articulated octahedra with invariant triangular facets, and how do their symmetries or asymmetries affect their deformability?
• RQ5 What is the relationship between deformable octahedra and deformable hexagons or systems of six hinged planes?

Key findings

• The paper derives a quadratic equation (1) in variables t = tan(φ/2) and u = tan(ψ/2), which governs the deformation of a tetrahedral angle with fixed face angles α, β, γ, δ.
• The coefficients A, B, C, D, E in the quadratic equation are expressed in terms of trigonometric functions of the face angles, with simplified forms involving cos(γ) − cos(α ± β ± δ) expressions.
• Three distinct types of articulated octahedra with invariant facets are identified: Type I (axisymmetric), Type II (planar symmetric), and Type III (asymmetric, non-intuitive deformations).
• All such deformable octahedra must be concave, possessing reentrant dihedrals or intercrossing facets, as required by Cauchy’s rigidity theorem for convex polyhedra.
• The existence of such octahedra is equivalent to the existence of deformable oblique hexagons with fixed side lengths and angles, or to systems of six hinged planes forming a closed loop.
• The theory establishes a direct analogy with planar articulated quadrilaterals studied by Hart, Kempe, and Darboux, showing similar algebraic structures underpinning their flexibility.