KNOTT-3.1 — Three Orthogonal Golden Rectangles → Regular Icosahedron & Dodecahedron

Three mutually perpendicular 2 × 2Φ golden rectangles have 12 corners (0, ±1, ±Φ) forming a regular icosahedron of edge 2, volume 20Φ²/3, and rational dihedral spread 4/9, while (±1, ±1, ±1) ∪ cyc(0, ±φ, ±Φ) forms a regular dodecahedron of edge 2φ, volume 8 + 4Φ, and dihedral spread 4/5.

Curated Source & Catalog Provenance

Catalog ID	KNOTT-3.1
Curated Source	Dr. Ron Knott, The Golden Geometry of Solids or Phi in 3 Dimensions (#coords)
Source Attribution	© 1996–2025 Dr. Ron Knott, University of Surrey — Explore Dr. Knott’s original exposition
Exact SymPy Validator	geometor.knott.solid_geometry.construct_orthogonal_golden_rectangles_solids() (0 sp.Float atoms)
GEOMETOR Concordance	geometor/pacioli, geometor/euclid, geometor/wildberger

Constructive Synthesis & Exact Proof

Moving from the plane into three-space (KnottPhi3D), Luca Pacioli (Pacioli1509) and H. S. M. Coxeter (Coxeter1973) showed that three congruent \(2 \times 2\Phi\) golden rectangles slotted symmetrically along the coordinate planes interlock to span the twelve vertices of a regular icosahedron.

• Three Orthogonal Golden Frames: In our exact rational cabinet projection, golden rectangle X1X2X3X4 in \(x = 0\) (\((0, \pm 1, \pm\Phi)\)), Y1Y2Y3Y4 in \(y = 0\) (\((\pm\Phi, 0, \pm 1)\)), and Z1Z2Z3Z4 in \(z = 0\) (\((\pm 1, \pm\Phi, 0)\)) contribute six short edges of length \(2\)—such as X1X2, Y1Y2, and Z1Z2—together with twenty-four inter-frame edges such as X1Y1, X1Z1, and Y1Z1 whose 3D quadrance is identically \(1^2 + (\Phi - 1)^2 + \Phi^2 = 1 + \phi^2 + \Phi^2 = 4\).

• Exact Volumes & Rational Dihedral Spreads: Summing all twenty triangular pyramids from the origin gives exact icosahedral volume \(\frac{20}{3}\Phi^2 = \frac{10(3+\sqrt{5})}{3}\), while dualizing via \((\pm 1, \pm 1, \pm 1) \cup \operatorname{cyc}(0, \pm\phi, \pm\Phi)\) yields the regular dodecahedron of edge \(2\phi\) and volume \(8 + 4\Phi\). Both solids exhibit pure rational dihedral spreads (Wildberger2005): \(s_{\text{ico}} = 4/9\) and \(s_{\text{dod}} = 4/5\).

Machine-Verified Exact Symbolic Invariants

Invariant	Exact Symbolic Value
icosahedron edge	\(2\)
icosahedron volume	\(\frac{10 \sqrt{5}}{3} + 10\)
icosahedron dihedral spread	\(\frac{4}{9}\)
dodecahedron edge	\(-1 + \sqrt{5}\)
dodecahedron volume	\(2 \sqrt{5} + 10\)
dodecahedron dihedral spread	\(- \frac{144}{720 - 320 \sqrt{5}} + 1 + \frac{64 \sqrt{5}}{720 - 320 \sqrt{5}}\)