Posts tagged 'euclid'

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-05.01 · Claudius Ptolemy’s Regular Pentagon & Pentagram (Almagest I.10)

In unit circle cO with perpendicular radii OB and OC, let M bisect OB. Circle cMC cuts AB at E with \(|OB|/|EO| = \phi\) and chord \(|CE|\) equal to the side of inscribed regular pentagon CP1P2P3P4, whose pentagram diagonals P1P4, CP2, CP3 cut one another in golden chains \(|P1K2|/|K2P4| = \phi\).

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 \(|AG|/|GB| = \phi\).