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.

Curated Source & Catalog Provenance

Catalog ID	KNOTT-3.2
Curated Source	Dr. Ron Knott, The Golden Geometry of Solids or Phi in 3 Dimensions (#cubeinDod)
Source Attribution	© 1996–2025 Dr. Ron Knott, University of Surrey — Explore Dr. Knott’s original exposition
Exact SymPy Validator	geometor.knott.solid_geometry.construct_cube_in_dodecahedron_model() (0 sp.Float atoms)
GEOMETOR Concordance	geometor/euclid, geometor/elements, geometor/pacioli

Constructive Synthesis & Exact Proof

In Elements XIII.17 (EuclidElements), Euclid constructs the regular dodecahedron by erecting roof-like pentagonal hipped wedges onto the six faces of a cube; Dr. Ron Knott (KnottPhi3D) highlights the resulting symmetric compound of five cubes inscribed inside a single dodecahedron.

• Primary Cube & Roof Ridges: Starting from primary cube vertices C1 through C8 at \((\pm 1, \pm 1, \pm 1)\) with front face C1C2C3C4 and cube edges C1C2, C2C3, and C1C5 of length \(2\), adjoining the twelve golden ridge vertices D1 – D12 at cyclic permutations of \((0, \pm\phi, \pm\Phi)\) produces thirty dodecahedral edges—such as ridge D1D2 and rafters C1D1, C1D5—each of exact 3D quadrance \(4\phi^2\) (length \(2\phi\)).

• Partition of the 60 Face Diagonals: Because every pentagonal face of side \(2\phi\) has five diagonals of length \(\Phi \cdot (2\phi) = 2\), the \(12 \times 5 = 60\) face diagonals partition cleanly into 5 congruent inscribed cubes of volume \(8\), with every one of the 20 dodecahedral vertices belonging to exactly \(2\) cubes and volume ratio \(V_{\text{dod}} / V_{\text{cube}} = 1 + \Phi/2\).

Machine-Verified Exact Symbolic Invariants

Invariant	Exact Symbolic Value
cube count	\(5\)
cubes per vertex	\(2\)
cube edge	\(2\)
dodecahedron edge	\(-1 + \sqrt{5}\)
face diagonal to edge ratio	\(\frac{1 + \sqrt{5}}{2}\)
cube volume	\(8\)
volume ratio dodecahedron to cube	\(\frac{\sqrt{5}}{4} + \frac{5}{4}\)