W.05 Spread Polynomials S_n(s) & Pentagonal Golden Spreads

Universal Chebyshev Analogues, Semigroup Composition, and Exact Golden Roots

The integer-coefficient spread polynomials S_0(s) = 0, S_1(s) = s, S_n(s) = 2(1 - 2s)S_{n-1}(s) - S_{n-2}(s) + 2s satisfy the semigroup composition law S_n(S_m(s)) = S_{nm}(s) over every commutative ring, and the fivefold polynomial S_5(s) = s(16s^2 - 20s + 5)^2 vanishes precisely at the Pentagonal Golden Spreads s = (5 -/+ sqrt(5))/8 = (3 - phi)/4 and (2 + phi)/4.

Interactive GEOMETOR construction for W.05 (Spread Polynomials S_n(s) & Pentagonal Golden Spreads). Hover any geometric label in the text or figure to highlight the corresponding element.

O : Point V_0 : Point V_1 : Point ===== l_0 = line(O, V_0) l_1 = line(O, V_1) l_2 = reflect_line(l_0, l_1) l_3 = reflect_line(l_1, l_2) l_4 = reflect_line(l_2, l_3)

Mathematical Synthesis

Reflecting a line l0 successively across l1 to form a fan of equally spread lines \(\ell_0, \ell_1, \ell_2, \dots\) with \(s(\ell_k, \ell_{k+1}) = s\) produces \(s(\ell_0, \ell_n) = S_n(s)\), where \(S_n(s) \in \mathbb{Z}[s]\) is the \(n\)-th spread polynomial (Wildberger2005). Over \(\mathbb{R}\), \(S_n(\sin^2\theta) = \sin^2(n\theta)\) rescales Pafnuty Chebyshev’s polynomials, while \(S_2(s) = 4s(1-s)\) is the maximal logistic map and \(S_n(S_m(s)) = S_{nm}(s)\) forms a commutative polynomial semigroup!

Where Rational Trigonometry meets GEOMETOR’s Divine Proportion (\(\phi = \frac{1+\sqrt{5}}{2}\)) is in \(S_5(s) = s(16s^2 - 20s + 5)^2 = 0\): its non-zero roots are the two Pentagonal spread constants \(\alpha = \frac{5 - \sqrt{5}}{8} = \frac{3 - \phi}{4}\) and \(\beta = \frac{5 + \sqrt{5}}{8} = \frac{2 + \phi}{4}\), whose ratio is \(\beta / \alpha = \phi^2 = \phi + 1\)! Over \(\mathbb{F}_{19}\) (where \(9^2 \equiv 5\)), these roots reduce to exact residues \(s \equiv 9, 16 \pmod{19}\).

Key Algebraic Identities

\[S_0(s) = 0, \quad S_1(s) = s, \quad S_2(s) = 4s(1-s), \quad S_3(s) = s(3 - 4s)^2, \quad S_n(S_m(s)) = S_{nm}(s)\]

\[S_5(s) = s(16s^2 - 20s + 5)^2 = 0 \iff s \in \left\{0,\; \alpha = \frac{5 - \sqrt{5}}{8} = \frac{3 - \phi}{4},\; \beta = \frac{5 + \sqrt{5}}{8} = \frac{2 + \phi}{4}\right\}\]

Symbolic SymPy Verification

Invariant / Theorem Check	Exact SymPy Value	Verification Status
Reflection fan spreads (s = 1/5)	S_2=16/25, S_3=121/125, S_4=576/625	Exact match s(l_0, l_n) == S_n(1/5)
Fivefold polynomial S_5(s)	s*(16*s**2 - 20*s + 5)**2	s*(16*s^2 - 20*s + 5)^2
Golden root alpha = (3 - phi)/4	(5 - sqrt(5))/8	S_5(alpha) = 0
Golden root beta = (2 + phi)/4	(5 + sqrt(5))/8	beta/alpha = phi^2, S_5(beta) = 0
Pentagonal orbit in GF(19) at s = 16	[0, 16, 9, 9, 16]	Exact period-5 integer cycle

Cross-Repository Canon

Executable Demonstration	W.05 Spread Polynomials S_n(s) & Pentagonal Golden Spreads — Code (Standalone model.py & w-05.svg)
Thematic Part	Part I: Preliminaries & Foundational Invariants
Euclid’s Elements	IV.11, XIII.8 (EuclidElements)
GEOMETOR Elements	G.12 rectilineal figures, G.15 construct a circle (Golden section & pentagonal symmetry)
GEOMETOR Pacioli	De Divina Proportione Cap. VII–XXIII (Pacioli1509)
Sister Codices	geometor/divine (golden ratio \(\phi\) algebraic field Q(phi))

Constructive Dependency Graph

Depends on: W.02 Lineations, Central Rotations & Reflections in Lines, W.04 Spread, Cross, Twist & Line Separation Ratios

Required for: W.11 Isosceles, Equilateral & Kepler’s Golden Right Triangle, W.17 Circle Geometry: Subtended Spreads, Secant Products & Rational Points, W.19 Regular Stars, Golden Pentagons & Finite-Field Polygons, W.26 Algebraic Physics: Projectile Parabolas, Snell’s Law & Relativity, W.27 3D Planes, De Gua’s Quadrea Law & Platonic Dihedral Spreads