W.27 3D Planes, De Gua’s Quadrea Law & Platonic Dihedral Spreads

Miraculous Cancellation of sqrt(5): Purely Rational Face Spreads 8/9, 1, 8/9, 4/9, 4/5

In 3D space over F, quadrance is Q(A_1, A_2) = dx^2 + dy^2 + dz^2, a right-corner tetrahedron satisfies De Gua’s Quadrea Theorem A_base = A_xy + A_yz + A_zx, and the dihedral (face) spreads of ALL FIVE regular Platonic solids—Tetrahedron (8/9), Cube (1), Octahedron (8/9), Icosahedron (4/9), and Dodecahedron (4/5)—are exact rational numbers in Q!

Interactive GEOMETOR construction for W.27 (3D Planes, De Gua’s Quadrea Law & Platonic Dihedral Spreads). Hover any geometric label in the text or figure to highlight the corresponding element.

V_0 : Point V_1 : Point V_2 : Point V_3 : Point ===== t_base = polygon(V_1, V_2, V_3) M_edge = midpoint(V_1, V_2) s_ap1 = segment(V_0, M_edge) s_ap2 = segment(V_3, M_edge)

Mathematical Synthesis

Extending quadrance to three dimensions (\(\Delta x^2 + \Delta y^2 + \Delta z^2\)) defines the dihedral spread \(S(\Pi_1, \Pi_2)\) between two planes via the spread of their normal vectors—equivalently, the spread between face apothems sap1 and sap2 meeting at edge midpoint Medge on base tbase (V0, V1, V2, V3). For a right-corner tetrahedron, Pythagoras’ theorem lifts directly to De Gua’s Quadrea Law: the slant face quadrea equals the sum of the three orthogonal face quadreas!

The crowning 3D revelation of Wildberger2005 (Chapter 26) concerns the five Platonic solids celebrated in Book XIII of Euclid’s Elements and Luca Pacioli’s De Divina Proportione (Pacioli1509). Although the icosahedron and dodecahedron require the pentagon spreads \(\alpha, \beta = (5 \mp \sqrt{5})/8\) solving \(16 s^2 - 20 s + 5 = 0\) (W.19), in the face-spread formulas \(S_{\text{ico}} = \frac{4\beta}{3\alpha}(1 - \frac{\beta}{3\alpha}) = \frac{4}{9}\) and \(S_{\text{dod}} = \frac{4\alpha - 1}{4\alpha^2} = \frac{4}{5}\), every factor of \(\sqrt{5}\) cancels identically, leaving five pure fractions in \(\mathbb{Q}\): \(8/9, 1, 8/9, 4/9, 4/5\)!

Key Algebraic Identities

\[16\alpha^2 - 20\alpha + 5 = 0 \implies S_{\text{ico}} = \frac{4\beta}{3\alpha}\!\left(1 - \frac{\beta}{3\alpha}\right) = \frac{4}{9}, \qquad S_{\text{dod}} = \frac{4\alpha - 1}{4\alpha^2} = \frac{4}{5}\]

\[\big(S_{\text{tet}},\; S_{\text{cube}},\; S_{\text{oct}},\; S_{\text{ico}},\; S_{\text{dod}}\big) = \left(\frac{8}{9},\; 1,\; \frac{8}{9},\; \frac{4}{9},\; \frac{4}{5}\right) \in \mathbb{Q}^5\]

Symbolic SymPy Verification

Invariant / Theorem Check	Exact SymPy Value	Verification Status
Regular Tetrahedron & Octahedron face spreads	S_tet = 8/9, S_oct = 8/9	Exact rational spreads
Cube orthogonal face spread	S_cube = 1	Mutually perpendicular planes
Regular Icosahedron face spread (vertices [0,+/-1,+/-phi])	S_ico = 4/9	sqrt(5) cancels completely!
Regular Dodecahedron face spread	S_dod = 4/5	sqrt(5) cancels completely!
De Gua's 3D Right-Corner Quadrea Theorem	A_slant = 2016 == 2016	3D generalization of Pythagoras

Cross-Repository Canon

Executable Demonstration	W.27 3D Planes, De Gua’s Quadrea Law & Platonic Dihedral Spreads — Code (Standalone model.py & w-27.svg)
Thematic Part	Part IV: Practical Solvers, 3D Platonic Geometry & Polar Curves
Euclid’s Elements	XI.4, XIII.13, XIII.14, XIII.15, XIII.16, XIII.17, XIII.18 (EuclidElements)
GEOMETOR Pacioli	De Divina Proportione Platonic polyhedra plates (Pacioli1509)
GEOMETOR Fundamentals	F.18 Equilateral Triangle, F.23 Golden Triangles
SymPy Proof	Verified from exact 3D vertex coordinates in prove_platonic_dihedral_spreads()

Constructive Dependency Graph

Depends on: W.05 Spread Polynomials S_n(s) & Pentagonal Golden Spreads, W.06 Law I: Pythagoras’ Theorem & Right Triangle Invariants, W.07 Law II: Triple Quad Formula, Archimedes’ Function & Quadrea, W.11 Isosceles, Equilateral & Kepler’s Golden Right Triangle, W.19 Regular Stars, Golden Pentagons & Finite-Field Polygons