GC-05.05 · Tran Quang Hung’s Six-Pentagon Reflection Star

Discoverer(s): Tran Quang Hung | Date: 2016 | Field: \(\mathbb{Q}(\phi, \sqrt{10 - 2\sqrt{5}})\) | Loci: 3 (Lemoine \(S = 9\))

Reflect regular pentagon ABCDE across each of its five sides to obtain five surrounding congruent pentagons. Joining the outer lateral vertices across two intervening pentagons (such as chord S1) cuts every side of ABCDE in the golden ratio, dividing AB at G and H in a 4-point golden chain.
GC-05.05 — Tran Quang Hung's Six-Pentagon Reflection Star AB: y = 0 S1: x*sqrt(5 - sqrt(5))*(-sqrt(10)/2 - sqrt(2)) + y*(3/2 + 3*sqrt(5)/2) + sqrt(10 - 2*sqrt(5))/2 = 0 S2: x*sqrt(5 - sqrt(5))*(-3*sqrt(2) - sqrt(10))/2 + y*(-sqrt(5) - 1) + sqrt(5 - sqrt(5))*(sqrt(10) + 3*sqrt(2)) = 0 Golden major AG = -1 + sqrt(5) Golden minor GB = 3 - sqrt(5) C = (1/2 + sqrt(5)/2, sqrt(2)*sqrt(5 - sqrt(5))/4 + sqrt(10)*sqrt(5 - sqrt(5))/4) D = (0, sqrt(10)*sqrt(5 - sqrt(5))/4 + 3*sqrt(2)*sqrt(5 - sqrt(5))/4) E = (-sqrt(5)/2 - 1/2, sqrt(2)*sqrt(5 - sqrt(5))/4 + sqrt(10)*sqrt(5 - sqrt(5))/4) R11 = (-sqrt(5)/2 - 1/2, -sqrt(10)*sqrt(5 - sqrt(5))/4 - sqrt(2)*sqrt(5 - sqrt(5))/4) R13 = (1/2 + sqrt(5)/2, -sqrt(10)*sqrt(5 - sqrt(5))/4 - sqrt(2)*sqrt(5 - sqrt(5))/4) R21 = (sqrt(5)/2 + 3/2, -sqrt(2)*sqrt(5 - sqrt(5))/2) R23 = (sqrt(5)/2 + 5/2, sqrt(2)*sqrt(5 - sqrt(5))/4 + sqrt(10)*sqrt(5 - sqrt(5))/4) R31 = (1 + sqrt(5), sqrt(10)*sqrt(5 - sqrt(5))/4 + 3*sqrt(2)*sqrt(5 - sqrt(5))/4) R33 = (-1/2 + sqrt(5)/2, sqrt(2)*sqrt(5 - sqrt(5)) + sqrt(10)*sqrt(5 - sqrt(5))/2) R41 = (1/2 - sqrt(5)/2, sqrt(2)*sqrt(5 - sqrt(5)) + sqrt(10)*sqrt(5 - sqrt(5))/2) R43 = (-sqrt(5) - 1, sqrt(10)*sqrt(5 - sqrt(5))/4 + 3*sqrt(2)*sqrt(5 - sqrt(5))/4) R51 = (-5/2 - sqrt(5)/2, sqrt(2)*sqrt(5 - sqrt(5))/4 + sqrt(10)*sqrt(5 - sqrt(5))/4) R53 = (-3/2 - sqrt(5)/2, -sqrt(2)*sqrt(5 - sqrt(5))/2) H = (2, 0) A = (-1, 0) B = (1, 0) G = (-2 + sqrt(5), 0) C D E R11 R13 R21 R23 R31 R33 R41 R43 R51 R53 H A B G

Given & Constructive Steps

Given: Central regular pentagon ABCDE of side \(2\) on base A \(= (-1,0)\), B \(= (1,0)\).

step Reflect ABCDE across each of its five sides AB, BC, CD, DE, EA.
step Join the outer lateral vertices in the two chiral \(\{5/2\}\) star polygons (including chord S1).
step Mark intersections G and H where the star chords cross side AB.

Exact Symbolic Proof & Certificate

By fivefold dihedral symmetry \(D_5\), the two chiral \(\{5/2\}\) star polygons on the reflected lateral vertices meet side AB at mirror-symmetric points G and H. Using exact reflection across unit-quadrance-scaled sides, chord S1 meets \(y = 0\) at \(x = \pm(\sqrt{5} - 2)\), partitioning \([-1, 1]\) into segments of lengths \(2/\phi^2, 2/\phi^3, 2/\phi^2\) and proving \(|AG|/|GB| = \phi^{\pm 1}\).

Machine-Verified Symbolic Certificate

Invariant

Exact Symbolic Value

Primary Golden Ratio

AG \(/\) GB \(= \frac{1}{2} + \frac{\sqrt{5}}{2}\) on collinear triple (A, G, B)

Exact Segment Lengths

\(|AG| = -1 + \sqrt{5}\), \(|GB| = 3 - \sqrt{5}\)

Rational Trigonometry Quadrances

\(Q(A, G) = 6 - 2 \sqrt{5}\), \(Q(G, B) = 14 - 6 \sqrt{5}\), \(Q_{\text{long}}/Q_{\text{short}} = \frac{\sqrt{5}}{2} + \frac{3}{2} = \phi^2\)

Algebraic Number Field

\(\mathbb{Q}(\phi, \sqrt{10 - 2\sqrt{5}})\)

Géométrographie Loci

2 given points · 3 lines · 0 circles · 6 polygons (elementary loci \(= 3\), Lemoine \(S = 9\))

Automated Divine Scan

geometor.divine.golden detected 13 exact collinear golden section(s) and 5 connected golden chain(s) in this model:

  • Section (A, G, B): long \(|AG|^2 = 6 - 2 \sqrt{5}\) vs. short \(|GB|^2 = 14 - 6 \sqrt{5}\)

  • Section (R11, G, R31): long \(|GR31|^2 = 2 \sqrt{5} + 14\) vs. short \(|R11G|^2 = 16 - 4 \sqrt{5}\)

  • Section (R53, A, C): long \(|AC|^2 = 2 \sqrt{5} + 6\) vs. short \(|R53A|^2 = 4\)

  • Section (A, C, R31): long \(|AC|^2 = 2 \sqrt{5} + 6\) vs. short \(|CR31|^2 = 4\)

  • Section (R11, A, D): long \(|AD|^2 = 2 \sqrt{5} + 6\) vs. short \(|R11A|^2 = 4\)

  • Section (A, D, R33): long \(|AD|^2 = 2 \sqrt{5} + 6\) vs. short \(|DR33|^2 = 4\)

  • Section (R41, D, B): long \(|DB|^2 = 2 \sqrt{5} + 6\) vs. short \(|R41D|^2 = 4\)

  • Section (D, B, R13): long \(|DB|^2 = 2 \sqrt{5} + 6\) vs. short \(|BR13|^2 = 4\)

  • Section (R43, E, B): long \(|EB|^2 = 2 \sqrt{5} + 6\) vs. short \(|R43E|^2 = 4\)

  • Section (E, B, R21): long \(|EB|^2 = 2 \sqrt{5} + 6\) vs. short \(|BR21|^2 = 4\)

  • Section (R51, E, C): long \(|EC|^2 = 2 \sqrt{5} + 6\) vs. short \(|R51E|^2 = 4\)

  • Section (E, C, R23): long \(|EC|^2 = 2 \sqrt{5} + 6\) vs. short \(|CR23|^2 = 4\)

  • Section (R51, G, R21): long \(|R51G|^2 = 2 \sqrt{5} + 14\) vs. short \(|GR21|^2 = 16 - 4 \sqrt{5}\)

Connected Golden Chains:

  • Chain (R53 — A — C — R31) spanning 2 overlapping golden sections (progression < >)

  • Chain (R11 — A — D — R33) spanning 2 overlapping golden sections (progression < >)

  • Chain (R41 — D — B — R13) spanning 2 overlapping golden sections (progression < >)

  • Chain (R43 — E — B — R21) spanning 2 overlapping golden sections (progression < >)

  • Chain (R51 — E — C — R23) spanning 2 overlapping golden sections (progression < >)

Historical Commentary & Attribution

Remarkably, at each golden division point G and H on a side of ABCDE, two distinct star chords concur directly on the pentagon edge.