GC-01.01 · Euclid’s Classical Golden Section (Elements II.11 & VI.30)

Discoverer(s): Euclid of Alexandria | Date: c. 300 BCE | Field: \(\mathbb{Q}(\phi)\) | Loci: 3 (Lemoine \(S = 9\))

Given a straight segment AB, erect the square ABCD, bisect side AD at M, and transfer MB along ray DA beyond A to E. Swinging AE onto AB at G cuts AB in extreme and mean ratio, so that \(|AB|/|AG| = |AG|/|GB| = \phi\).
GC-01.01 — Euclid's Classical Golden Section (Elements II.11 & VI.30) cMB: x**2 + (y - 1)**2 - 5 = 0 cAE: x**2 + y**2 - (-1 + sqrt(5))**2 = 0 AD: x = 0 Golden major AG = -1 + sqrt(5) Golden minor GB = 3 - sqrt(5) D = (0, 2) C = (2, 2) M = (0, 1) E = (0, 1 - sqrt(5)) F = (-1 + sqrt(5), 1 - sqrt(5)) A = (0, 0) B = (2, 0) G = (-1 + sqrt(5), 0) D C M E F A B G

Given & Constructive Steps

Given: Endpoints A at \((0, 0)\) and B at \((2, 0)\) defining a base segment AB of length \(2\).

step Erect square ABCD on base AB with D at \((0, 2)\) and C at \((2, 2)\).
step Bisect vertical side AD at midpoint M \(= (0, 1)\) and join diagonal-radius MB of length \(\sqrt{5}\).
step Draw compass circle cMB centered at M through B, cutting line AD extended below A at E \(= (0, 1 - \sqrt{5})\).
step Draw circle cAE centered at A through E, cutting AB internally at G \(= (\sqrt{5} - 1, 0)\) and completing the golden square AGFE.

Exact Symbolic Proof & Certificate

In right triangle \(\triangle MAB\), \(Q(M, A) = 1\) and \(Q(A, B) = 4\), so quadrance \(Q(M, B) = 5\) and \(|ME| = |MB| = \sqrt{5}\). Subtracting \(|MA| = 1\) gives \(|AG| = |AE| = \sqrt{5} - 1 = 2/\phi\), while the remainder on AB has length \(|GB| = 2 - (\sqrt{5} - 1) = 3 - \sqrt{5} = 2/\phi^2\). Hence \(Q(A, G) = 6 - 2\sqrt{5}\) and \(Q(G, B) = 14 - 6\sqrt{5}\), satisfying \(Q(A, G) = (\phi + 1)\,Q(G, B)\) and \(|AG|/|GB| = \phi\).

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

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

Géométrographie Loci

2 given points · 1 lines · 2 circles · 2 polygons (elementary loci \(= 3\), Lemoine \(S = 9\))

Automated Divine Scan

geometor.divine.golden detected 2 exact collinear golden section(s) and 0 connected golden chain(s) in this model:

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

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

Historical Commentary & Attribution

In Book II Proposition 11 of the Elements, Euclid formulates this construction purely in the language of geometric algebra—requiring that rectangle \(AB \cdot GB\) equal square AGFE—and returns to it in Book VI Proposition 30 using theory of proportions (Euclid). In Alexander Bogomolny’s survey (Cut-the-Knot), this construction anchors all subsequent minimal-step refinements.