W.19 Regular Stars, Golden Pentagons & Finite-Field Polygons

Exact Golden Quadrance Ratios Q_diag / Q_side = phi^2 and Rational Pentagons Over F_19

A regular n-star of lines has consecutive spread s satisfying S_n(s) = 0; for n = 5, a regular pentagon P_0..P_4 has side-to-diagonal spread alpha = (5 - sqrt(5))/8 = (3 - phi)/4, exterior/star spread beta = (5 + sqrt(5))/8 = (2 + phi)/4, and exact golden quadrance ratio Q_diag / Q_side = beta / alpha = phi^2 = phi + 1, existing rationally in F_p^2 whenever 5 and -1/5-dependent discriminants split (e.g. regular 5-stars in F_19).

Interactive GEOMETOR construction for W.19 (Regular Stars, Golden Pentagons & Finite-Field Polygons). Hover any geometric label in the text or figure to highlight the corresponding element.

O : Point P_0 : Point ===== c = circle(O, P_0) P_1 = golden_pentagon_vertex(O, P_0, 1) P_2 = golden_pentagon_vertex(O, P_0, 2) P_3 = golden_pentagon_vertex(O, P_0, 3) P_4 = golden_pentagon_vertex(O, P_0, 4) pent = polygon(P_0, P_1, P_2, P_3, P_4) star = polygon(P_0, P_2, P_4, P_1, P_3)

Mathematical Synthesis

In classical Euclidean geometry over \(\mathbb{Q}\), no regular pentagon can have purely rational vertex coordinates because \(S_5(s) = s(16s^2 - 20s + 5)^2 = 0\) requires \(\sqrt{5}\), and vertex placement on circle c further involves \(\sqrt{10 \pm 2\sqrt{5}}\). Yet in Rational Trigonometry (Wildberger2005), every intrinsic invariant of pentagon pent and pentagram star lives cleanly inside \(\mathbb{Q}(\phi)\) without nested radicals: the spread at P0 between two adjacent diagonals is \(\alpha = (3 - \phi)/4\), the spread between two adjacent sides is \(\beta = (2 + \phi)/4\), and the diagonal-to-side quadrance ratio is \(Q(P_0, P_2) / Q(P_0, P_1) = \phi^2 = \phi + 1\)!

Even more remarkably, over finite prime fields \(\mathbb{F}_p\) where \(5\) is a quadratic residue (\(p \equiv \pm 1 \pmod{5}\)), \(\sqrt{5}\) is an ordinary integer modulo \(p\)! In \(\mathbb{F}_{19}\), \(9^2 = 81 \equiv 5\), so the Pentagonal spread roots become \(\alpha \equiv 16\) and \(\beta \equiv 9\), producing exact 5-fold periodic spread orbits \((0, 16, 9, 9, 16, 0)\) with purely integer arithmetic!

Key Algebraic Identities

\[\frac{Q(P_0, P_2)}{Q(P_0, P_1)} = \frac{\beta}{\alpha} = \frac{5 + \sqrt{5}}{5 - \sqrt{5}} = \phi^2 = \phi + 1, \qquad \alpha + \beta = \frac{5}{4}, \quad \alpha \beta = \frac{5}{64}\]

\[\text{In } \mathbb{F}_{19}\; (9^2 \equiv 5):\quad 16 s^2 - 20 s + 5 \equiv 16(s - 16)(s - 9) \pmod{19}\]

Symbolic SymPy Verification

Invariant / Theorem Check	Exact SymPy Value	Verification Status
Unit-circle side quadrance Q(P_0, P_1)	3 - phi = (5 - sqrt(5))/2	Equals 4 * alpha
Unit-circle diagonal quadrance Q(P_0, P_2)	2 + phi = (5 + sqrt(5))/2	Equals 4 * beta
Golden quadrance ratio Q_diag / Q_side	phi^2 = phi + 1	Exact divine proportion squared
5-star spread orbits over GF(19)	[0, 16, 9, 9, 16] & [0, 9, 16, 16, 9]	9^2 == 5 (mod 19) splits S_5(s)

Cross-Repository Canon

Executable Demonstration	W.19 Regular Stars, Golden Pentagons & Finite-Field Polygons — Code (Standalone model.py & w-19.svg)
Thematic Part	Part III: Universal Triangle, Conic & Chromogeometry
Euclid’s Elements	IV.11, XIII.8, XIII.11 (EuclidElements)
GEOMETOR Elements	G.10 circle, G.12 rectilineal figures, G.25 base angles of an isosceles triangle
GEOMETOR Fundamentals	F.23 Golden Triangles, F.40 Regular Polygons and Nested Radicals
GEOMETOR Pacioli	De Divina Proportione (Pacioli1509) & geometor/divine

Constructive Dependency Graph

Depends on: W.05 Spread Polynomials S_n(s) & Pentagonal Golden Spreads, W.11 Isosceles, Equilateral & Kepler’s Golden Right Triangle, W.17 Circle Geometry: Subtended Spreads, Secant Products & Rational Points

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