KNOTT-3 — 3D Golden Geometry of Solids

Three-dimensional golden coordinates spanning the regular icosahedron and dodecahedron from three mutually orthogonal golden rectangles, Euclid XIII.17’s inscribed cube and five-cube compound, and the acute and obtuse golden rhombohedra of icosahedral quasicrystals.

Curated from Dr. Ron Knott’s University of Surrey archive (© 1996–2025 Dr. Ron Knott, University of Surrey).

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.

KNOTT-3.2 — Inscribed Cube & Compound of Five Cubes in a Dodecahedron

Inside a regular dodecahedron of edge 2φ, the 60 face diagonals of length 2 = Φ · (2φ) partition into five congruent inscribed cubes of edge 2 and volume 8, with each of the 20 dodecahedral vertices belonging to exactly two cubes.

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.

Collection Concordance

Catalog ID	GEOMETOR Catalog Entry	Curated Source Anchor
KNOTT-3.1	Three Orthogonal Golden Rectangles → Regular Icosahedron & Dodecahedron	The Golden Geometry of Solids or Phi in 3 Dimensions (#coords)
KNOTT-3.2	Inscribed Cube & Compound of Five Cubes in a Dodecahedron	The Golden Geometry of Solids or Phi in 3 Dimensions (#cubeinDod)
KNOTT-3.3	Acute & Obtuse Golden Rhombohedra (3D Quasicrystal / Ammann Tiles)	The Golden Geometry of Solids or Phi in 3 Dimensions (#quasicrystalsPhi)