GC-01 · Classical & Minimal-Step Compass Constructions¶
Euclid II.11, Lemoine–Hofstetter 5-step records, Mascheroni 5-circle compass-only, and rusty compass
From Euclid’s Elements Book II Proposition 11 to early twenty-first-century Forum Geometricorum papers by Kurt Hofstetter and Bùi Quang Tuân, geometers have asked how economically a straight segment can be divided in extreme and mean ratio. This opening family catalogs the classical Euclidean gnomon, the Heron–Ptolemy right-triangle hypotenuse transfer, four distinct 5-step minimal Euclidean and rusty-compass divisions, and Hofstetter’s 5-circle Mascheroni compass-only construction.
Cataloged Constructions (7)¶
GC-01.01 · Euclid’s Classical Golden Section (Elements II.11 & VI.30)
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-01.02 · Right-Triangle Hypotenuse Transfer (1 : 2 : √5)
On segment AB of length \(2\), erect perpendicular leg BC of length \(1 = \tfrac{1}{2}|AB|\). Circle cCB cuts hypotenuse AC at D (and ray AC at E), and circle cAD transfers AD onto AB at G, dividing AB in the golden ratio \(|AG|/|GB| = \phi\).
GC-01.03 · Lemoine–Reusch–Hofstetter 5-Step Division
Given points A and B on line AB, let unit vesica circles cAB and cBA meet at C and D. Circle cCA cuts cAB at E and chord CD at F; then circle cEF cuts AB internally at G and externally at G1 so that \(|AG|/|GB| = |G_1A|/|AB| = \phi\).
GC-01.04 · Hofstetter’s 2004 Five-Step Secant Division
Given points A and B, draw line AB and unit circles cAB, cBA meeting at C and D, with cAB meeting ray BA beyond A at E. Circle cEB cuts cBA above AB at F, and secant DF divides AB at G in the golden ratio \(|AG|/|GB| = \phi\) in exactly five Euclidean loci.
GC-01.05 · Mascheroni Five-Circle Compass-Only Construction
Using a compass alone and zero straightedge lines, five circles constructed from given points A and B generate four collinear points Y, C, D, X such that \(|CX|/|CD| = |CD|/|DX| = \phi\).
GC-01.06 · Hofstetter’s Fixed-Aperture Rusty-Compass Division
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-01.07 · Bùi Quang Tuân’s Five-Step Three-Circle Secant
On line L through O, draw circle cO meeting L at A and B, circle cBA meeting L at C, and circle cCO meeting cO above L at M and cBA below L at N. Then chord MN cuts cO at P with \(|MN|/|MP| = |MP|/|PN| = \phi\).