Geometry Dihedral Cayley Menger Dihedral Angle3 Regular Unit

The cofactor formula at work

In solid geometry, a tetrahedron is a pyramid with four triangular faces, and a dihedral angle is the angle between two faces along a shared edge. For the regular tetrahedron, where all six edges have equal length, every dihedral angle is the same. A classical way to compute it uses the Cayley-Menger matrix, a 5 by 5 array built from the squared edge lengths. The angle's cosine is the ratio of two cofactors of that matrix: the off-diagonal cofactor for the two vertices opposite the edge, divided by the square root of the product of the two diagonal cofactors. For the regular unit tetrahedron, this formula gives cos θ = 1/3, so the angle θ is arccos(1/3), about 70.53 degrees.

This value is a standard fact of Euclidean geometry, and it appears in many contexts: the tetrahedron is the three-dimensional simplex, and its dihedral angle appears in crystallography, packing problems, and the geometry of regular polytopes. The angle is also the one that makes four regular tetrahedra fit around a shared edge in three-dimensional space, though they do not tile space without gaps.

In Recognition Science, the machine-checked library of formal theorems contains a definition that packages this classical formula. The declaration dihedralAngle3_regularUnit is a theorem stating that, for the regular unit tetrahedron, the angle computed from the Cayley-Menger cofactor formula equals the angle already defined in the framework's existing regular-tetrahedron API. The proof proceeds by first checking, for each of the six edges, that the relevant cofactors have the values 1, -3, and -3, and then substituting those values into the formula. The theorem is proved without external assumptions; the cofactor check itself is a proved theorem, not an axiom.

What this theorem does not claim is broader than what it does. It does not prove that the Cayley-Menger cofactor formula is valid for all tetrahedra; that is a separate matter, and the definition here only packages the formula for a non-degenerate tetrahedron, with the cosine assumed to lie in the interval from -1 to 1. It does not claim that the regular tetrahedron's dihedral angle is derived from the framework's physical principles; the value 1/3 is a geometric fact, independent of any recognition-cost theory. The theorem only establishes that two ways of defining the same angle agree for the regular unit case.

/-- The cofactor angle agrees with the existing regular-tetrahedron
dihedral API without external assumptions. -/
theorem dihedralAngle3_regularUnit (e : Fin 6) :
    dihedralAngle3 regularUnitTet e = regular_tet_dihedral.theta :=
  dihedralAngle3_regularUnit_of_cofactorCheck regularUnitCofactorCheck e

/-- The regular-unit cofactor check is now a theorem, not an assumption. -/
theorem regularUnitCofactorCheck : RegularUnitCofactorCheck := by
  intro e
  fin_cases e <;>
    simp [oppositeCMVertices,
      regularUnit_cofactor_34, regularUnit_cofactor_24, regularUnit_cofactor_23,
      regularUnit_cofactor_14, regularUnit_cofactor_13, regularUnit_cofactor_12,
      regularUnit_vertex_diag_cofactor]

What this page does not claim

The theorem does not prove the cofactor formula is valid for all tetrahedra. The theorem does not derive the regular tetrahedron's angle from physical principles. The theorem does not claim the framework's recognition-cost theory is needed to compute the 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/DihedralCayleyMenger.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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