GC-03.04 · Tran Quang Hung & Elliot McGucken’s 3×3 Grid Circle

Discoverer(s): Tran Quang Hung, Elliot McGucken | Date: 2016 | Field: \(\mathbb{Q}(\sqrt{2}, \phi)\) | Loci: 2 (Lemoine \(S = 6\))

In a \(3 \times 3\) square grid S1AS3S4 centered at O, let C be the upper-right vertex of the central grid cell, A the outer upper-right corner, and T the adjacent trisection vertex on the top edge. Circle cOT cuts diagonal segment CA at B such that \(|CB|/|BA| = \phi\).
GC-03.04 — Tran Quang Hung & Elliot McGucken's 3×3 Grid Circle cOT: x**2 + y**2 - 10 = 0 OA: -3*x + 3*y = 0 Golden major CB = -sqrt(2) + sqrt(10) Golden minor BA = -sqrt(10) + 3*sqrt(2) S1 = (-3, 3) S3 = (3, -3) S4 = (-3, -3) C1 = (-1, 1) C3 = (1, -1) C4 = (-1, -1) T = (1, 3) O = (0, 0) A = (3, 3) C = (1, 1) B = (sqrt(5), sqrt(5)) S1 S3 S4 C1 C3 C4 T O A C B

Given & Constructive Steps

Given: A \(3 \times 3\) square grid on \([-3, 3]^2\) centered at O \(= (0,0)\) with outer vertex A \(= (3,3)\).

step Identify central cell corner C \(= (1,1)\) and top boundary grid node T \(= (1,3)\).
step Draw diagonal line OA through C and A.
step Draw circle cOT centered at O through T, cutting segment CA at B.

Exact Symbolic Proof & Certificate

Circle cOT has quadrance radius \(Q(O, T) = 1^2 + 3^2 = 10\). Along diagonal \(y = x\), \(2x^2 = 10\) gives B \(= (\sqrt{5}, \sqrt{5})\). Since collinear points \(C(1,1), B(\sqrt{5},\sqrt{5}), A(3,3)\) project linearly onto the \(x\)-axis, \(|CB|/|BA| = (\sqrt{5} - 1)/(3 - \sqrt{5}) = \phi\).

Machine-Verified Symbolic Certificate

Invariant

Exact Symbolic Value

Primary Golden Ratio

CB \(/\) BA \(= \frac{1}{2} + \frac{\sqrt{5}}{2}\) on collinear triple (C, B, A)

Exact Segment Lengths

\(|CB| = - \sqrt{2} + \sqrt{10}\), \(|BA| = - \sqrt{10} + 3 \sqrt{2}\)

Rational Trigonometry Quadrances

\(Q(C, B) = 12 - 4 \sqrt{5}\), \(Q(B, A) = 28 - 12 \sqrt{5}\), \(Q_{\text{long}}/Q_{\text{short}} = \frac{\sqrt{5}}{2} + \frac{3}{2} = \phi^2\)

Algebraic Number Field

\(\mathbb{Q}(\sqrt{2}, \phi)\)

Géométrographie Loci

2 given points · 1 lines · 1 circles · 2 polygons (elementary loci \(= 2\), Lemoine \(S = 6\))

Automated Divine Scan

geometor.divine.golden detected 3 exact collinear golden section(s) and 1 connected golden chain(s) in this model:

  • Section (S4, C4, B): long \(|C4B|^2 = 4 \sqrt{5} + 12\) vs. short \(|S4C4|^2 = 8\)

  • Section (C4, C, B): long \(|C4C|^2 = 8\) vs. short \(|CB|^2 = 12 - 4 \sqrt{5}\)

  • Section (C, B, A): long \(|CB|^2 = 12 - 4 \sqrt{5}\) vs. short \(|BA|^2 = 28 - 12 \sqrt{5}\)

Connected Golden Chains:

  • Chain (C4 — C — B — A) spanning 2 overlapping golden sections (progression > >)

Historical Commentary & Attribution

Notice the structural kinship with the Greek Cross (GC-03.07 · Bùi Quang Tuân & Alexandre Borovik’s Greek Cross & Square): the four trisection nodes on the boundary of a \(3 \times 3\) grid are precisely the twelve outer corners of a five-square Greek cross.