Geometry Freudenthal Cube Triangulation Edge In Tet Vertices

Edge and tetrahedron bookkeeping

The standard Freudenthal decomposition cuts one unit cube into six tetrahedra by drawing the body diagonal from vertex 0 to vertex 7 and taking the six monotone paths between them. Each tetrahedron has six edges: three unit cube edges, two face diagonals, and one body diagonal. Across all six tetrahedra these edges are not all distinct; the triangulation uses only 19 unique edges in total. The declaration edgeInTet_vertices proves a bookkeeping fact about that arrangement: whenever a global edge belongs to a tetrahedron, the two endpoints of that edge, as listed in the tetrahedron's vertex list, are exactly the two endpoints of the global edge, in either order.

This is a statement about incidence data, not about geometry. It says that the map from global edges to local edge slots, and the map from local slots back to vertices, are consistent with each other. The proof is a finite check over all 19 edges, 6 tetrahedra, and 6 slots, done by a machine-checked library of formal theorems. The result is a theorem, not a definition: it derives the endpoint agreement from the earlier definitions of the edge and tetrahedron vertex lists.

In Recognition Science, this declaration is part of a larger framework that models physical structure from a discrete ledger of recognition events. Here the ledger is the incidence bookkeeping of the cube triangulation. The declaration does not claim that the Freudenthal decomposition is unique, that it is the only way to triangulate a cube, or that it has any physical meaning by itself. It only establishes the internal consistency of one particular combinatorial description.

The practical consequence is that any later use of the triangulation, for example in a discrete model of space or in a computation of volumes, can rely on the fact that the edge and vertex data agree. This is the kind of low-level guarantee that makes larger proofs possible without rechecking every case by hand.

theorem edgeInTet_vertices
    (e : Fin 19) (τ f : Fin 6) (h : edgeInTet e τ = some f) :
      let ev := edgeVerts e
      let tv := ReggeRigorousFoundation.edgeVertices f
      (tetVerts τ tv.1 = ev.1 ∧ tetVerts τ tv.2 = ev.2) ∨
        (tetVerts τ tv.1 = ev.2 ∧ tetVerts τ tv.2 = ev.1) := by
  fin_cases e <;> fin_cases τ <;> fin_cases f <;>
    simp [edgeInTet, edgeVerts, tetVerts, ReggeRigorousFoundation.edgeVertices] at h ⊢

What this page does not claim

The Freudenthal decomposition is the only way to triangulate a cube into six tetrahedra. The edgeInTet_vertices theorem assigns any physical meaning to the cube triangulation. The declaration proves that the six tetrahedra have no overlapping interiors.

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

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