Geometry Dihedral Cofactor Formula Cm Cofactor3 Edge2 Right Diag Eq Neg Four Nor

A clean bridge between two worlds

In Euclidean geometry, the angle between two faces of a tetrahedron, called a dihedral angle, can be found by first computing the normal vectors to those faces and then taking their dot product. A separate, purely algebraic route uses the Cayley-Menger determinant, a matrix built from the squared edge lengths, to compute the same angle through cofactors. The theorem cmCofactor3_edge2_right_diag_eq_neg_four_normalSq establishes a precise equality between these two approaches for one specific edge of the tetrahedron.

For the edge numbered 2, the theorem states that the product of two specific cofactors of the Cayley-Menger matrix equals negative four times the squared length of a certain face normal vector. In symbols, this is cmCofactor3 (sqEdgeOfPoints T) 2 2 * cmCofactor3 (sqEdgeOfPoints T) 3 3 = -4 * (faceNormal T 0 3 2) · (faceNormal T 0 3 2). The negative sign is not an error; it reflects the orientation of the normal vectors, and the factor of four is a geometric constant that appears in the relationship.

This equality is a proved theorem in the machine-checked library of formal theorems, meaning it has been verified by a computer to follow from the axioms of the underlying logic. It is part of a larger effort to show that the geometric definition of a dihedral cosine, based on face normals, matches the algebraic definition based on Cayley-Menger cofactors for all six edges of a tetrahedron. The theorem for edge 2 is one of several such edge-specific results that together build this bridge.

The theorem does not claim that the two approaches are equal for every edge; it is specific to edge 2. It also does not assert anything about the actual value of the dihedral angle, only about the relationship between the cofactor product and the normal vector length. The factor of four is a consequence of the definitions involved, not a fundamental constant of nature, and the theorem does not generalize to higher dimensions or to non-tetrahedral shapes.

/-- The theorem target is discharged. -/
theorem bergerCofactorFormula3 : BergerCofactorFormula3 :=
  geometricDihedralCos_eq_cmCofactorRatio

/-- The theorem target is discharged. -/
theorem bergerCofactorFormula3 : BergerCofactorFormula3 :=
  geometricDihedralCos_eq_cmCofactorRatio

What this page does not claim

The theorem does not claim the geometric and algebraic approaches are equal for every edge, only for edge 2. The theorem does not assert the value of any dihedral angle, only the relationship between a cofactor product and a normal vector length. The factor of four is a consequence of the definitions, not a fundamental constant, and the result does not generalize to higher dimensions.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Geometry/DihedralCofactorFormula.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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