The Dehn invariant of the regular tetrahedron and regular octahedron

Key references:

7 September 2026: A Geometric Conjecture for the Dark Energy Fraction [I] [2]

The Dehn invariant of a regular tetrahedron is non-zero and proportional to \(6 \cdot (s \otimes \arccos(1/3))\), while the Dehn invariant of a regular octahedron is \(-2\) times that of a tetrahedron with the same edge length, satisfying the additive relation with cubes. [1, 2, 3]

Definition of the Dehn Invariant

  • Concept: It measures whether two 3D polyhedra of equal volume can be cut into pieces and reassembled into each other (known as scissors congruence). [1]
  • Formula: Calculated over all edges \(i\) as the sum of edge lengths \(L_{i}\) tensored with dihedral angles \(\theta _{i}\) (the interior angle between faces): \(D(P) = \sum L_i \otimes \theta_i\). [1]
  • Key Property: It is additive, meaning combining shapes adds their invariants, and shapes that can form a cube (like a cube’s components) balance out to zero. [1]

Regular Tetrahedron

  • Edges: Has 6 edges of length \(s\).
  • Dihedral Angle: The interior angle between faces is \(\arccos(1/3)\) (about \(70.53^{\circ }\)).
  • Value: \(D(\text{tetrahedron}) = 6s \otimes \arccos(1/3)\).
  • Significance: Because \(\arccos(1/3)\) is an irrational multiple of \(\pi \), this invariant is non-zero, proving a regular tetrahedron cannot be cut and reassembled into a cube. [1, 2, 3]

Regular Octahedron

  • Edges: Has 12 edges of length \(s\).
  • Dihedral Angle: The interior angle between faces is \(\arccos(-1/3)\) (about \(109.47^{\circ }\)), which is supplementary to the regular tetrahedron’s angle (\(\arccos(1/3) + \arccos(-1/3) = \pi\)).
  • Relation: Twice a regular tetrahedron’s Dehn invariant plus a regular octahedron’s Dehn invariant equals zero (the invariant of a cube). [1, 2]

Preserve: