Geometry Dihedral Cofactor Formula Cm Cofactor3 Edge2 Diag Product Eq Sixteen De

The product formula

A tetrahedron has four triangular faces meeting along six edges. At each edge, the two adjacent faces form a dihedral angle, the angle between the planes of those faces. One way to compute its cosine uses the normals to the two faces: take their dot product and divide by the product of their lengths. That is the geometric side of the story, expressed directly from the coordinates of the four vertices.

The algebraic side comes from the Cayley-Menger matrix, built from the squared edge lengths of the tetrahedron. For each edge, the cofactor of that matrix, after removing the row and column of the two opposite vertices, gives a number. The declaration cmCofactor3_edge2_diag_product_eq_sixteen_denomSq proves a precise identity: for the edge indexed by 5 (the edge between vertices 1 and 2), the product of the two cofactors corresponding to the opposite vertices 1 and 2 equals 16 times the square of the denominator used in the geometric cosine formula. In symbols, cmCofactor3 (sqEdgeOfPoints T) 1 1 * cmCofactor3 (sqEdgeOfPoints T) 2 2 = 16 * geometricDihedralDenomSq T 5.

This identity is one of six similar edge-wise bridges. Together they support the Berger cofactor formula, which states that the geometric dihedral cosine equals the ratio of the Cayley-Menger cofactor to the square root of the product of the two diagonal cofactors. The declaration here supplies the square-root relation needed for that equality in the edge-5 case. It is a formal theorem in the machine-checked library: it is derived from the definitions of face normals, dot products, and the Cayley-Menger cofactors, with no additional assumptions beyond the tetrahedron's coordinates.

The theorem does not claim that the product equals sixteen times the denominator itself; it relates to the squared denominator. It also does not assert that the dihedral cosine equals the cofactor ratio for all edges; that is a separate theorem, bergerCofactorFormula3, which covers all six edges. Nor does it say anything about the physical meaning of the cofactors or the dihedral angle; it is a purely algebraic-geometric identity within the framework's geometry module.

theorem cmCofactor3_edge5_diag_product_eq_sixteen_denomSq (T : RealizedTet) :
    cmCofactor3 (sqEdgeOfPoints T) 1 1 *
      cmCofactor3 (sqEdgeOfPoints T) 2 2 =
        16 * geometricDihedralDenomSq T 5 := by
  rw [cmCofactor3_edge5_left_diag_eq_neg_four_normalSq,
    cmCofactor3_edge5_right_diag_eq_neg_four_normalSq]
  unfold geometricDihedralDenomSq
  simp [edgeVertices3, adjacentFaceOppositeVertices, ReggeRigorousFoundation.edgeVertices]
  ring

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

What this page does not claim

The theorem does not claim that the product equals sixteen times the denominator itself, only the squared denominator. It does not claim that the dihedral cosine equals the cofactor ratio for a specific edge; that is the separate Berger cofactor formula. It does not assign physical meaning to the cofactors or the dihedral angle.

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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