Geometry Dihedral Cofactor Formula Geometric Dihedral Cos Edge0 Eq Cofactor Rati

The geometric cosine

A tetrahedron, the simplest three-dimensional solid, has four triangular faces. Two faces meet along an edge, and the angle between those faces is called a dihedral angle. Its cosine can be computed directly from the geometry: build a vector perpendicular to each face, using the cross product of two edge vectors, then take the normalized inner product of those two normals. This is the geometric definition, and it works for any tetrahedron whose vertices are points in ordinary three-dimensional space.

The same cosine can also be computed from a purely algebraic object: the matrix of squared edge lengths, called the Gram matrix. From that matrix one forms cofactors, and the ratio of a specific cofactor to a square root of a product of two others gives a number. The theorem in question, proved in the machine-checked library (a collection of formal theorems verified by computer), states that for edge 0 of any tetrahedron, the geometric cosine equals this algebraic cofactor ratio, provided a certain square-root identity holds. That identity, stated as a hypothesis, says the square root of the product of two cofactors equals four times the square root of the geometric denominator. The theorem is conditional: it does not assert the identity holds for all tetrahedra, only that if it does, the two cosines agree.

The proof proceeds by expanding both sides in terms of the Gram matrix entries. The geometric numerator for edge 0 becomes a combination of Gram entries, and the denominator becomes a product of two face-normal squared lengths. The cofactor ratio, when the square-root condition is met, simplifies to exactly the same expression. The library also proves the unconditional version for edge 0, using a separate argument that the square-root identity always holds. That unconditional result, geometricDihedralCos_edge0_eq_cmCofactorRatio, is the one that appears in the final Berger cofactor formula, which identifies the geometric cosine with the Cayley-Menger cofactor ratio for all six edges.

What this does not claim: the conditional theorem does not prove the square-root identity for every tetrahedron, nor does it cover edges other than edge 0. It does not assert anything about the sign of the square root, only equality of the real-valued cosines. It does not claim the geometric and algebraic definitions agree for degenerate tetrahedra, where faces collapse and the denominator could vanish. The unconditional version, proved separately, does establish the edge-0 equality for all realized tetrahedra, and the full Berger formula extends it to all edges, but that is a distinct theorem with its own proof.

/-- Edge `0`: Berger's cofactor formula reduced to the remaining square-root
scaling/positivity fact. -/
theorem geometricDihedralCos_edge0_eq_cofactorRatio_of_sqrt
    (T : RealizedTet)
    (hsqrt :
      Real.sqrt (cmCofactor3 (sqEdgeOfPoints T) 3 3 *
        cmCofactor3 (sqEdgeOfPoints T) 4 4)
        = 4 * Real.sqrt (geometricDihedralDenomSq T 0)) :
    geometricDihedralCos T 0 = dihedralCos3Sq (sqEdgeOfPoints T) 0 := by
  unfold geometricDihedralCos dihedralCos3Sq dihedralDenom3
  change geometricDihedralNumerator T 0 / Real.sqrt (geometricDihedralDenomSq T 0) =
    cmCofactor3 (sqEdgeOfPoints T) 3 4 /
      Real.sqrt (cmCofactor3 (sqEdgeOfPoints T) 3 3 *
        cmCofactor3 (sqEdgeOfPoints T) 4 4)
  rw [cmCofactor3_edge0_eq_four_geometricNumerator, hsqrt]
  field_simp

/-- Edge `0`: Berger's cofactor formula is fully proved. -/
theorem geometricDihedralCos_edge0_eq_cmCofactorRatio (T : RealizedTet) :
    geometricDihedralCos T 0 = dihedralCos3Sq (sqEdgeOfPoints T) 0 :=
  geometricDihedralCos_edge0_eq_cofactorRatio_of_sqrt T
    (cmCofactor3_edge0_sqrt_diag_product T)

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

What this page does not claim

The conditional theorem does not prove the square-root identity for all tetrahedra. The theorem does not cover edges other than edge 0. The equality is not asserted for degenerate tetrahedra where the denominator could vanish.

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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