KNOTT-1.4 — Regular Pentagon, Pentagram, Decagon & Paper-Knot Pentagon

The five diagonals of a regular pentagon of side 1 enclose a similar inner pentagon of side φ² and cut each diagonal in the geometric progression φ² : φ : 1 : Φ; furthermore, folding a flat uniform strip of width w into a tight overhand knot locks parallel trapezoid edges into an exact regular pentagon of side w sec 18°.

Curated Source & Catalog Provenance

Catalog ID	KNOTT-1.4
Curated Source	Dr. Ron Knott, Phi in 2D Geometry and Trigonometry (#pent)
Source Attribution	© 1996–2025 Dr. Ron Knott, University of Surrey — Explore Dr. Knott’s original exposition
Exact SymPy Validator	geometor.knott.flat_geometry.construct_pentagon_and_paper_knot() (0 sp.Float atoms)
GEOMETOR Concordance	geometor/euclid, geometor/elements, geometor/pacioli, geometor/divine

Constructive Synthesis & Exact Proof

Every regular pentagon P0P1P2P3P4 of unit edge P0P1 \(= 1\) is locked by golden triangles (KnottPhi2D, Pacioli1509):

• Pentagram Progression: All five star diagonals—P0P2, P0P3, P1P3, P1P4, and P2P4—are equal of exact length \(\Phi\). Their pairwise intersections Q0, Q1, Q2, Q3, and Q4 bound a similar inverted inner pentagon Q0Q1Q2Q3Q4 of side \(\phi^2 = 2 - \Phi\), dividing each pentagram diagonal in the four-step geometric progression \((Q_1 Q_2,\; P_0 Q_1,\; P_0 Q_2,\; P_0 P_2) = (\phi^2,\; \phi,\; 1,\; \Phi)\).

• Overhand Paper-Strip Knot: Dropping altitude P4H from vertex P4 onto the extended base line of P0P1 (parallel to opposite diagonal P2P4) at H isolates the parallel strip width \(w = \sin 72^\circ = \frac{1}{4}\sqrt{10 + 2\sqrt{5}}\). Consequently, folding a uniform paper ribbon of width \(w = 1\) into a flat overhand knot forces an exact regular pentagon of side quadrance \(Q_{\text{side}} = \sec^2 18^\circ = \frac{8}{5 + \sqrt{5}} = \frac{5 - \sqrt{5}}{5} \Phi^2\).

Machine-Verified Exact Symbolic Invariants

Invariant	Exact Symbolic Value
Q1Q2 crossbar	\(\sqrt{- \frac{\sqrt{5}}{2} - \frac{10 \sqrt{2 \sqrt{5} + 5}}{2 \sqrt{2} \sqrt{\sqrt{5} + 5} + 2 \sqrt{5} \sqrt{2 \sqrt{5} + 5} + 6 \sqrt{2 \sqrt{5} + 5}} - \frac{3 \sqrt{2 \sqrt{5} + 5}}{\sqrt{2} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{2 \sqrt{5} + 5} + 3 \sqrt{2 \sqrt{5} + 5}} - \frac{\sqrt{5} \sqrt{2 \sqrt{5} + 5}}{\sqrt{2} \sqrt{\sqrt{5} + 5} + \sqrt{5} \sqrt{2 \sqrt{5} + 5} + 3 \sqrt{2 \sqrt{5} + 5}} + \frac{2 \sqrt{5} \sqrt{2 \sqrt{5} + 5}}{2 \sqrt{2} \sqrt{\sqrt{5} + 5} + 2 \sqrt{5} \sqrt{2 \sqrt{5} + 5} + 6 \sqrt{2 \sqrt{5} + 5}} + \frac{88 \sqrt{5}}{2 \sqrt{10} \sqrt{\sqrt{5} + 5} \sqrt{2 \sqrt{5} + 5} + 6 \sqrt{2} \sqrt{\sqrt{5} + 5} \sqrt{2 \sqrt{5} + 5} + 60 \sqrt{5} + 140} + \frac{200}{2 \sqrt{10} \sqrt{\sqrt{5} + 5} \sqrt{2 \sqrt{5} + 5} + 6 \sqrt{2} \sqrt{\sqrt{5} + 5} \sqrt{2 \sqrt{5} + 5} + 60 \sqrt{5} + 140} + \frac{3}{2}}\)
P0Q1 star arm	\(\sqrt{\frac{3}{2} - \frac{\sqrt{5}}{2}}\)
P0Q2 pentagon side	\(\sqrt{\frac{88 \sqrt{5}}{2 \sqrt{10} \sqrt{\sqrt{5} + 5} \sqrt{2 \sqrt{5} + 5} + 6 \sqrt{2} \sqrt{\sqrt{5} + 5} \sqrt{2 \sqrt{5} + 5} + 60 \sqrt{5} + 140} + \frac{200}{2 \sqrt{10} \sqrt{\sqrt{5} + 5} \sqrt{2 \sqrt{5} + 5} + 6 \sqrt{2} \sqrt{\sqrt{5} + 5} \sqrt{2 \sqrt{5} + 5} + 60 \sqrt{5} + 140}}\)
P0P2 diagonal	\(\sqrt{\frac{\sqrt{5}}{2} + \frac{3}{2}}\)
strip width quadrance	\(\frac{\sqrt{5}}{8} + \frac{5}{8}\)
unit strip pentagon side quadrance	\(\frac{10 - 2 \sqrt{5}}{5}\)