W.10 Law V: Triple Spread, Triple Cross & Quadruple Spread Formulas

Cubic Polynomial Harmony Replacing theta_1 + theta_2 + theta_3 = pi

For any three coplanar non-null lines l_1, l_2, l_3 with pairwise spreads s_1, s_2, s_3 and crosses c_i = 1 - s_i, the spreads satisfy the cubic Triple Spread Formula (s_1 + s_2 + s_3)^2 = 2(s_1^2 + s_2^2 + s_3^2) + 4*s_1*s_2*s_3 and the crosses satisfy (c_1 + c_2 + c_3 - 1)^2 = 4*c_1*c_2*c_3; for four lines, the Quadruple Spread polynomial R(s_12, s_23, s_34, s_14) = 0 is identically invariant under s_ij <-> 1 - s_ij.

Interactive GEOMETOR construction for W.10 (Law V: Triple Spread, Triple Cross & Quadruple Spread Formulas). Hover any geometric label in the text or figure to highlight the corresponding element.

A_1 : Point A_2 : Point A_3 : Point ===== l_1 = line(A_2, A_3) l_2 = line(A_1, A_3) l_3 = line(A_1, A_2) t = polygon(A_1, A_2, A_3)

Mathematical Synthesis

Just as the Triple Quad Formula replaces \(d_1 + d_2 = d_3\) by a quadratic relation with no square roots, the Triple Spread Formula replaces Euclid’s angle-sum theorem \(\theta_1 + \theta_2 + \theta_3 = \pi\) with a symmetric cubic equation in the three spreads \((s_1, s_2, s_3)\) of lines l1, l2, l3 (Wildberger2005). Notice the structural miracle: it differs from \(\mathcal{A}(s_1, s_2, s_3) = 0\) by the single cubic term \(4 s_1 s_2 s_3\)!

Substituting \(s_i = 1 - c_i\) produces the Triple Cross Formula \((c_1 + c_2 + c_3 - 1)^2 = 4 c_1 c_2 c_3\). Extending from three to four lines yields the Quadruple Spread Formula \(R(s_{12}, s_{23}, s_{34}, s_{14}) = 0\), which possesses an unexpected self-duality: replacing every spread \(s_{ij}\) with its cross \(c_{ij} = 1 - s_{ij}\) leaves the polynomial \(R\) identically unchanged!

Key Algebraic Identities

\[(s_1 + s_2 + s_3)^2 = 2(s_1^2 + s_2^2 + s_3^2) + 4 s_1 s_2 s_3 \iff (s_3 - (s_1 + s_2 - 2 s_1 s_2))^2 = 4 s_1 s_2 (1 - s_1)(1 - s_2)\]

\[(c_1 + c_2 + c_3 - 1)^2 = 4 c_1 c_2 c_3, \qquad R(a, b, c, d) \equiv R(1 - a, 1 - b, 1 - c, 1 - d)\]

Symbolic SymPy Verification

Invariant / Theorem Check	Exact SymPy Value	Verification Status
Spreads (s_1, s_2, s_3)	(9/13, 81/130, 9/10)	Exact rationals
Triple Spread Formula S(s_1, s_2, s_3)	0	Verified == 0
Triple Cross Formula C(c_1, c_2, c_3)	0	Verified == 0
Quadruple Spread Formula R(s_12,s_23,s_34,s_14)	0	Verified == 0 & self-dual under s <-> 1-s

Cross-Repository Canon

Executable Demonstration	W.10 Law V: Triple Spread, Triple Cross & Quadruple Spread Formulas — Code (Standalone model.py & w-10.svg)
Thematic Part	Part II: The Five Fundamental Laws
Euclid’s Elements	I.32 (Angle sum of a triangle, EuclidElements)
GEOMETOR Elements	G.3 extremities of a segment, G.5 surface
GEOMETOR Fundamentals	F.35 Obtuse and Acute Triangles, F.53 Geometric Half-Angle Trigonometry
3D Algebraic Surface	Defines the ellipson surface in [0, 1]^3 (W.28)

Constructive Dependency Graph

Depends on: W.04 Spread, Cross, Twist & Line Separation Ratios, W.08 Law III: The Spread Law & Extended Circumquadrance Law, W.09 Law IV: The Cross Law & Quadrea-Spread Identity

Required for: W.11 Isosceles, Equilateral & Kepler’s Golden Right Triangle, W.12 Vertex Bisectors, Square Vertices & The Four Incenters, W.23 Chromogeometry & Universal Symmetric Bilinear Forms, W.24 Exact Triangle Solvers & Surveying Triangulation, W.25 Snellius–Pothenot Resection & Hansen’s Surveying Problem, W.28 Rational Polar Coordinates, Classical Curves & The Ellipson