Geometry Schlaefli Triangulation3 D Global Schlaefli Rhs

The vanishing global term

The Schläfli identity is a classical relation in geometry: for a family of polyhedra or simplices that deform smoothly, a particular combination of edge-length variations and dihedral angles (the angles between faces) cancels out. In three dimensions, the identity says that for each tetrahedron, the sum over its six edges of the edge length times the derivative of the opposite dihedral angle equals zero. This is a local statement, one tetrahedron at a time.

The framework's machine-checked library of formal theorems extends this local statement to a finite triangulation, a mesh of tetrahedra glued face-to-face. The declaration globalSchlaefliRHS defines the right-hand side of the global identity as the number 0. That is the entire content: after summing the local tetrahedral identities over every tetrahedron in the mesh, the total left-hand side, which is the sum over all tetrahedra and all six edges of the same edge-length-times-angle-derivative product, equals zero. The theorem global_schlaefli_of_local proves this equality by applying the local identity to each tetrahedron and then adding up the results.

What this establishes is a clean global analogue of the classical local fact. The sum over the whole triangulation vanishes, not because of any special geometry of the mesh, but because each individual tetrahedron already contributes zero. The proof is a direct summation; it introduces no new geometric content beyond the local identity.

The declaration does not claim that any particular triangulation has zero total curvature, nor that the dihedral angles themselves are constant. It does not assert that the identity holds for curved spaces or for non-Euclidean geometries; the framework's statement is specifically for finite 3D triangulations with Euclidean tetrahedra. It also does not say that the left-hand side is zero for each edge index separately; the equality holds for the summed quantity, one edge index at a time, after the full double sum.

/-- The global Schläfli right-hand side: the Euclidean angle-variation term
vanishes. -/
def globalSchlaefliRHS (K : Triangulation3D)
    (_D : TriangulationSchlaefliData K) (_e' : Fin 6) : ℝ :=
  0

/-- Summing local tetrahedral Schläfli identities gives the global finite
triangulation identity. -/
theorem global_schlaefli_of_local
    (K : Triangulation3D) (D : TriangulationSchlaefliData K) (e' : Fin 6) :
    globalSchlaefliLHS K D e' = globalSchlaefliRHS K D e' := by
  unfold globalSchlaefliLHS globalSchlaefliRHS
  have hlocal : ∀ τ : Fin K.nT,
      (∑ e : Fin 6,
        Real.sqrt ((K.tet τ).sqEdge e) * (D.tetData τ).dihedralDeriv e e')
        = 0 := by
    intro τ
    exact (D.tetData τ).schlaefli e'
  simp_rw [hlocal]
  simp

/-- Summing local tetrahedral Schläfli identities gives the global finite
triangulation identity. -/
theorem global_schlaefli_of_local
    (K : Triangulation3D) (D : TriangulationSchlaefliData K) (e' : Fin 6) :
    globalSchlaefliLHS K D e' = globalSchlaefliRHS K D e' := by
  unfold globalSchlaefliLHS globalSchlaefliRHS
  have hlocal : ∀ τ : Fin K.nT,
      (∑ e : Fin 6,
        Real.sqrt ((K.tet τ).sqEdge e) * (D.tetData τ).dihedralDeriv e e')
        = 0 := by
    intro τ
    exact (D.tetData τ).schlaefli e'
  simp_rw [hlocal]
  simp

What this page does not claim

The identity does not hold for curved or non-Euclidean geometries. The vanishing sum does not imply any individual dihedral angle is constant. The declaration does not assert the identity for each edge separately without the full double sum.

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/SchlaefliTriangulation3D.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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