dihedraalikulma

Dihedraalikulma is a hypothetical geometric quantity used in speculative geometry to analyze how dihedral angles around the edges of a polyhedron contribute to its overall angular structure. The term blends the idea of dihedral angles with a root meaning corner or angle, and it is primarily encountered in thought experiments and fictional mathematical discussions rather than in standard geometry.

Definition and interpretation

For each edge e of a polyhedron, let α_e denote the dihedral angle between the two faces that meet at e. In the customary, simplified usage of dihedraalikulma, the edge-based value is defined as δ_e = α_e − π/2. This makes δ_e lie between −π/2 and π/2 for typical convex polyhedra, with δ_e = 0 precisely when the edge is right-angled (α_e = π/2). Positive δ_e indicates an edge that is more obtuse than a right angle, while negative δ_e indicates a sharper, more acute edge.

Global measures

A polyhedron’s dihedraalikulma can be summarized in several ways. The average dihedraalikulma, D(P) = (1/E) ∑_e δ_e, provides a single scalar describing the overall angular deviation from right-angled edges. A quadratic form, D_sq(P) = ∑_e δ_e^2, emphasizes larger deviations and is sometimes used for comparing angular distortion across meshes.

Examples

In a cube, α_e = π/2 for all edges, so δ_e = 0 and D(P) = 0. In a regular tetrahedron, α_e ≈ 1.231 rad, giving δ_e ≈ −0.340 rad (about −19.5°) for each edge, with a correspondingly negative average.

Notes

Dihedraalikulma remains a fictional construct, useful for illustrating how local dihedral angles contribute to global angular structure in a neutral, angle-based framework. It is not part of standard geometrical theory and appears mainly in speculative or pedagogical contexts.