GC-03.08 · John Arioni’s 30°-Rotated Unit Square & Diagonal Circle

Discoverer(s): John Arioni | Date: 2016 | Field: \(\mathbb{Q}(\sqrt{3}, \phi)\) | Loci: 2 (Lemoine \(S = 6\))

Rotate unit square DGHJ clockwise by \(30^\circ\) around vertex D to obtain congruent square DCBA. Circle cAC centered at A through opposite vertex C intersects line DG at E and F such that \(|DG|/|GE| = |DE|/|DG| = \phi\).
GC-03.08 — John Arioni's 30°-Rotated Unit Square & Diagonal Circle cAC: (x - 1/2)**2 + (y - sqrt(3)/2)**2 - (1/2 + sqrt(3)/2)**2 - (1/2 - sqrt(3)/2)**2 = 0 DG: y = 0 Golden major DG = 1 Golden minor GE = -1/2 + sqrt(5)/2 H = (1, 1) J = (0, 1) A = (1/2, sqrt(3)/2) B = (1/2 + sqrt(3)/2, -1/2 + sqrt(3)/2) C = (sqrt(3)/2, -1/2) F = (1/2 - sqrt(5)/2, 0) D = (0, 0) G = (1, 0) E = (1/2 + sqrt(5)/2, 0) H J A B C F D G E

Given & Constructive Steps

Given: Unit square DGHJ on base D \(= (0,0)\), G \(= (1,0)\).

step Rotate DGHJ clockwise by \(30^\circ\) about D so A \(= (\tfrac{1}{2}, \tfrac{\sqrt{3}}{2})\) and C \(= (\tfrac{\sqrt{3}}{2}, -\tfrac{1}{2})\).
step Draw diagonal AC (of length \(\sqrt{2}\)) of the rotated unit square DCBA.
step Draw circle cAC centered at A through C, cutting base line DG at E and F.

Exact Symbolic Proof & Certificate

Center A has altitude \(y_A = \frac{\sqrt{3}}{2}\) above line DG (\(y = 0\)) and abscissa \(x_A = \frac{1}{2}\), while radius \(|AC| = \sqrt{2}\). Setting \(y = 0\) in \((x - \tfrac{1}{2})^2 + (\tfrac{\sqrt{3}}{2})^2 = 2\) gives \((x - \tfrac{1}{2})^2 = \frac{5}{4}\), hence \(x_E = \frac{1 + \sqrt{5}}{2} = \phi\) and \(x_F = -1/\phi\), forming the 4-point golden chain \((F, D, G, E)\).

Machine-Verified Symbolic Certificate

Invariant

Exact Symbolic Value

Primary Golden Ratio

DG \(/\) GE \(= \frac{1}{2} + \frac{\sqrt{5}}{2}\) on collinear triple (D, G, E)

Exact Segment Lengths

\(|DG| = 1\), \(|GE| = - \frac{1}{2} + \frac{\sqrt{5}}{2}\)

Rational Trigonometry Quadrances

\(Q(D, G) = 1\), \(Q(G, E) = \frac{3}{2} - \frac{\sqrt{5}}{2}\), \(Q_{\text{long}}/Q_{\text{short}} = \frac{\sqrt{5}}{2} + \frac{3}{2} = \phi^2\)

Algebraic Number Field

\(\mathbb{Q}(\sqrt{3}, \phi)\)

Géométrographie Loci

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

Automated Divine Scan

geometor.divine.golden detected 2 exact collinear golden section(s) and 1 connected golden chain(s) in this model:

  • Section (F, D, G): long \(|DG|^2 = 1\) vs. short \(|FD|^2 = \frac{3}{2} - \frac{\sqrt{5}}{2}\)

  • Section (D, G, E): long \(|DG|^2 = 1\) vs. short \(|GE|^2 = \frac{3}{2} - \frac{\sqrt{5}}{2}\)

Connected Golden Chains:

  • Chain (F — D — G — E) spanning 2 overlapping golden sections (progression < >)

Historical Commentary & Attribution

Rotating by \(30^\circ\) converts the square’s diagonal quadrance \(2\) minus the equilateral altitude quadrance \(3/4\) directly into \(5/4\).