GC-01.06 · Hofstetter’s Fixed-Aperture Rusty-Compass Division

Discoverer(s): Kurt Hofstetter | Date: 2005 | Field: \(\mathbb{Q}(\sqrt{3}, \phi)\) | Loci: 6 (Lemoine \(S = 18\))

With a rusty compass locked permanently to aperture \(r = |AB|\), circles cAB and cBA determine midpoint M via CD; a third unit circle cM centered at M meets cBA above AB at F, and line DF cuts AB at G in the golden ratio \(|AG|/|GB| = \phi\).
GC-01.06 — Hofstetter's Fixed-Aperture Rusty-Compass Division cAB: x**2 + y**2 - 1 = 0 cBA: y**2 + (x - 1)**2 - 1 = 0 cM: y**2 + (x - 1/2)**2 - 1 = 0 AB: y = 0 CD: x - 1/2 = 0 DF: x*(-sqrt(15)/4 - sqrt(3)/2) + y/4 + sqrt(15)/8 + 3*sqrt(3)/8 = 0 Golden major AG = -1/2 + sqrt(5)/2 Golden minor GB = 3/2 - sqrt(5)/2 C = (1/2, sqrt(3)/2) D = (1/2, -sqrt(3)/2) M = (1/2, 0) F = (3/4, sqrt(15)/4) A = (0, 0) B = (1, 0) G = (-1/2 + sqrt(5)/2, 0) C D M F A B G

Given & Constructive Steps

Given: Unit segment endpoints A \(= (0, 0)\) and B \(= (1, 0)\) and a fixed compass radius \(r = 1\).

step Draw line AB and unit circles cAB, cBA meeting at C and D.
step Join CD to bisect AB at M \(= (\tfrac{1}{2}, 0)\).
step With the unchanged compass opening \(r = 1\), draw circle cM centered at M, meeting cBA farthest from D at F \(= (\tfrac{3}{4}, \tfrac{\sqrt{15}}{4})\).
step Join DF to intersect AB at G \(= (1/\phi, 0)\).

Exact Symbolic Proof & Certificate

Unit circles cM \(((x - \tfrac{1}{2})^2 + y^2 = 1)\) and cBA \(((x - 1)^2 + y^2 = 1)\) intersect along their radical axis \(x = \tfrac{3}{4}\), giving F \(= (\tfrac{3}{4}, \tfrac{\sqrt{15}}{4})\)—the exact same upper intersection point achieved by radius-\(2\) circle \(E(B)\) in Hofstetter’s 2004 construction! Hence line DF meets AB at G \(= (1/\phi, 0)\), so \(|AG|/|GB| = \phi\).

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

\(|AG| = - \frac{1}{2} + \frac{\sqrt{5}}{2}\), \(|GB| = \frac{3}{2} - \frac{\sqrt{5}}{2}\)

Rational Trigonometry Quadrances

\(Q(A, G) = \frac{3}{2} - \frac{\sqrt{5}}{2}\), \(Q(G, B) = \frac{7}{2} - \frac{3 \sqrt{5}}{2}\), \(Q_{\text{long}}/Q_{\text{short}} = \frac{\sqrt{5}}{2} + \frac{3}{2} = \phi^2\)

Algebraic Number Field

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

Géométrographie Loci

2 given points · 3 lines · 3 circles · 0 polygons (elementary loci \(= 6\), Lemoine \(S = 18\))

Automated Divine Scan

geometor.divine.golden detected 1 exact collinear golden section(s) and 0 connected golden chain(s) in this model:

  • Section (A, G, B): long \(|AG|^2 = \frac{3}{2} - \frac{\sqrt{5}}{2}\) vs. short \(|GB|^2 = \frac{7}{2} - \frac{3 \sqrt{5}}{2}\)

Historical Commentary & Attribution

Poncelet–Steiner and Abū al-Wafāʾ tradition prizes constructions executed with a single fixed compass opening (compas rouillé). Kurt Hofstetter (Hofstetter 2005, Cut-the-Knot) requires only three unit circles and three lines.