Geometry Schlaefli Triangulation3 D Triangulation Schlaefli Data

The triangulation ledger

A finite 3D triangulation is a way of filling a region of space with tetrahedra, the simplest solid shapes, meeting only along faces, edges, and vertices. In classical geometry, the Schläfli identity is a local relation: for a single tetrahedron, the sum over its six edges of the edge length times the variation of the dihedral angle at that edge equals zero. This is a differential statement, describing how the shape changes under a smooth deformation.

The framework's declaration TriangulationSchlaefliData packages this local derivative data for every tetrahedron in a finite triangulation. It is a structure that assigns to each tetrahedron a record of its six edge lengths and the derivatives of its six dihedral angles. The key theorem, global_schlaefli_of_local, proves that summing these local identities over all tetrahedra yields a global identity: the total left-hand side equals the right-hand side, which is zero. In plain terms, the local cancellations inside each tetrahedron survive the summing process, so the whole triangulation obeys the same Schläfli relation as its parts.

This is a purely combinatorial and geometric fact, established in the framework's machine-checked library of formal theorems. It does not depend on any physical assumption about space, forces, or recognition. The declaration itself is a definitional choice: it chooses how to organize the local derivative data. The theorem that follows is a proved statement, derived from the definition and the local Schläfli identity for each tetrahedron.

What this does not claim is broader significance. It does not say that all triangulations are rigid, that the identity holds for non-Euclidean geometries, or that this global cancellation has any physical consequence. It is a structural lemma about finite 3D triangulations in Euclidean space, nothing more. The framework's larger claims about space and recognition are not touched by this declaration.

/-- 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

/-- Local Schläfli derivative data on every tetrahedron of a finite
triangulation. -/
structure TriangulationSchlaefliData (K : Triangulation3D) where
  tetData : ∀ τ : Fin K.nT, TetraSchlaefliDerivativeData (K.tet τ)

What this page does not claim

The identity holds only for Euclidean tetrahedra, not for curved or non-Euclidean geometries. The declaration says nothing about rigidity, uniqueness, or physical consequences of triangulations. The global cancellation is a theorem about the sum, not a statement about any individual tetrahedron's behavior.

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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