GC-03.07 · Bùi Quang Tuân & Alexandre Borovik’s Greek Cross & Square

Discoverer(s): Bùi Quang Tuân, Alexandre Borovik | Date: 2015 | Field: \(\mathbb{Q}(\phi)\) | Loci: 4 (Lemoine \(S = 12\))

Inscribe a symmetrical five-square Greek cross CROSS and a concentric axis-aligned square S1S2S3S4 in the same circle cO. Along any inner arm line AC of the cross, the square edge S1S2 cuts AC at B such that \(|BC|/|AB| = \phi\), while \(|BD|/|CD| = \phi\).
GC-03.07 — Bùi Quang Tuân & Alexandre Borovik's Greek Cross & Square cO: x**2 + y**2 - 10 = 0 AC: x - 1 = 0 S1S2: y - sqrt(5) = 0 S4S3: y + sqrt(5) = 0 Golden major BC = -1 + sqrt(5) Golden minor AB = 3 - sqrt(5) X1 = (-1, 3) X2 = (3, 1) X3 = (3, -1) D = (1, -1) X4 = (1, -3) X5 = (-1, -3) X6 = (-1, -1) X7 = (-3, -1) X8 = (-3, 1) X9 = (-1, 1) S1 = (-sqrt(5), sqrt(5)) S2 = (sqrt(5), sqrt(5)) S3 = (sqrt(5), -sqrt(5)) S4 = (-sqrt(5), -sqrt(5)) E = (1, -sqrt(5)) O = (0, 0) A = (1, 3) C = (1, 1) B = (1, sqrt(5)) X1 X2 X3 D X4 X5 X6 X7 X8 X9 S1 S2 S3 S4 E O A C B

Given & Constructive Steps

Given: Circle cO centered at O \(= (0,0)\) through cross corner A \(= (1,3)\) (radius \(\sqrt{10}\)).

step Inscribe the five-square Greek cross CROSS of square side \(2\), with right upper arm edge passing through A \(= (1,3)\), C \(= (1,1)\), and D \(= (1,-1)\).
step Inscribe concentric square S1S2S3S4 of side \(2\sqrt{5}\) with vertices \((\pm\sqrt{5}, \pm\sqrt{5})\) on cO.
step Mark B \(= (1, \sqrt{5})\) and E \(= (1, -\sqrt{5})\) where horizontal square edges S1S2 and S4S3 meet line AC.

Exact Symbolic Proof & Certificate

Each cross arm has half-width \(1\) and outer reach \(3\), so \(R^2 = 1^2 + 3^2 = 10\). An axis-aligned square in the same circle satisfies \(x^2 + y^2 = 10\) with \(|x| = |y| = \sqrt{5}\). On vertical line \(x = 1\), points \(A(1,3), B(1,\sqrt{5}), C(1,1), D(1,-1)\) have \(|BC| = \sqrt{5} - 1 = 2/\phi\), \(|AB| = 3 - \sqrt{5} = 2/\phi^2\), and \(|BD| = \sqrt{5} + 1 = 2\phi\), giving both \(|BC|/|AB| = \phi\) and \(|BD|/|CD| = \phi\).

Machine-Verified Symbolic Certificate

Invariant

Exact Symbolic Value

Primary Golden Ratio

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

Exact Segment Lengths

\(|BC| = -1 + \sqrt{5}\), \(|AB| = 3 - \sqrt{5}\)

Rational Trigonometry Quadrances

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

Algebraic Number Field

\(\mathbb{Q}(\phi)\)

Géométrographie Loci

2 given points · 3 lines · 1 circles · 2 polygons (elementary loci \(= 4\), Lemoine \(S = 12\))

Automated Divine Scan

geometor.divine.golden detected 10 exact collinear golden section(s) and 3 connected golden chain(s) in this model:

  • Section (X4, E, D): long \(|ED|^2 = 6 - 2 \sqrt{5}\) vs. short \(|X4E|^2 = 14 - 6 \sqrt{5}\)

  • Section (X4, D, B): long \(|DB|^2 = 2 \sqrt{5} + 6\) vs. short \(|X4D|^2 = 4\)

  • Section (E, D, C): long \(|DC|^2 = 4\) vs. short \(|ED|^2 = 6 - 2 \sqrt{5}\)

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

  • Section (D, C, B): long \(|DC|^2 = 4\) vs. short \(|CB|^2 = 6 - 2 \sqrt{5}\)

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

  • Section (S4, X6, C): long \(|X6C|^2 = 8\) vs. short \(|S4X6|^2 = 12 - 4 \sqrt{5}\)

  • Section (X6, C, S2): long \(|X6C|^2 = 8\) vs. short \(|CS2|^2 = 12 - 4 \sqrt{5}\)

  • Section (S1, X9, D): long \(|X9D|^2 = 8\) vs. short \(|S1X9|^2 = 12 - 4 \sqrt{5}\)

  • Section (X9, D, S3): long \(|X9D|^2 = 8\) vs. short \(|DS3|^2 = 12 - 4 \sqrt{5}\)

Connected Golden Chains:

  • Chain (X4 — E — D — C — B — A) spanning 4 overlapping golden sections (progression < < > >)

  • Chain (S4 — X6 — C — S2) spanning 2 overlapping golden sections (progression < >)

  • Chain (S1 — X9 — D — S3) spanning 2 overlapping golden sections (progression < >)

Historical Commentary & Attribution

Alexandre Borovik noted that the overlap between the vertical arm of width \(2\) and segment BD of height \(2\phi\) forms an exact golden rectangle.