KNOTT-3.3 — Acute & Obtuse Golden Rhombohedra (3D Quasicrystal / Ammann Tiles)

Triples of five-fold icosahedral vectors of quadrance Φ + 2 span acute (prolate) and obtuse (oblate) golden rhombohedra bounded by six congruent golden rhombi of diagonal ratio Φ : 1, with exact volumes 2Φ² and 2Φ in ratio Φ : 1 and 10 + 10 assembly into a rhombic triacontahedron of volume 20Φ³.

Curated Source & Catalog Provenance

Catalog ID	KNOTT-3.3
Curated Source	Dr. Ron Knott, The Golden Geometry of Solids or Phi in 3 Dimensions (#quasicrystalsPhi)
Source Attribution	© 1996–2025 Dr. Ron Knott, University of Surrey — Explore Dr. Knott’s original exposition
Exact SymPy Validator	geometor.knott.solid_geometry.construct_golden_rhombohedra_model() (0 sp.Float atoms)
GEOMETOR Concordance	geometor/pacioli, geometor/divine

Constructive Synthesis & Exact Proof

Just as Roger Penrose’s thick and thin golden rhombs tile the plane with five-fold symmetry, three-dimensional icosahedral quasicrystals (Ammann–Kramer–Neri tilings, cataloged by Dr. Ron Knott on KnottPhi3D) are built from two golden rhombohedra bounded by six congruent golden rhombi.

• Acute (Prolate) vs. Obtuse (Oblate) Hexahedra: Spanning vectors \((0, 1, \Phi)\), \((1, \Phi, 0)\), and \((\Phi, 0, 1)\) from A0 through A1, A2, A3 to apex A7 generate the acute rhombohedron (with golden rhomb face A0A1A4A2 and edges A0A1, A0A2, A0A3, and A4A7). Reversing two signs from B0 through B1, B2, B3 to B7 generates the obtuse rhombohedron (front face B0B1B4B2 and edges B0B1, B4B7).

• Golden Volume Ratio & Triacontahedron: Every edge has 3D quadrance \(1 + \Phi^2 = \Phi + 2\), every rhombic face has diagonals \(2\Phi\) and \(2\) (ratio \(\Phi : 1\)), and exact 3D determinant evaluation proves \(V_{\text{acute}} = 2\Phi^2\) and \(V_{\text{obtuse}} = 2\Phi\) (ratio \(\Phi : 1\)). Ten acute and ten obtuse tiles assemble Johannes Kepler’s 30-faced rhombic triacontahedron of exact volume \(10(2\Phi^2 + 2\Phi) = 20\Phi^3\).

Machine-Verified Exact Symbolic Invariants

Invariant	Exact Symbolic Value
edge quadrance	\(\frac{\sqrt{5}}{2} + \frac{5}{2}\)
face diagonal ratio	\(\frac{1}{2} + \frac{\sqrt{5}}{2}\)
volume acute	\(\sqrt{5} + 3\)
volume obtuse	\(1 + \sqrt{5}\)
volume ratio acute to obtuse	\(\frac{\sqrt{5}}{1 + \sqrt{5}} + \frac{3}{1 + \sqrt{5}}\)
triacontahedron volume	\(40 + 20 \sqrt{5}\)