Posts tagged 'johannes-kepler'

GC-04.02 — Johannes Kepler's Geometric-Progression Right Triangle (1597) cOB: y**2 + (x - 3/4 - sqrt(5)/4)**2 - (sqrt(5)/4 + 3/4)**2 = 0 AB: y = 0 Golden major AH = 1/2 + sqrt(5)/2 Golden minor HB = 1 O = (sqrt(5)/4 + 3/4, 0) C = (1/2 + sqrt(5)/2, sqrt(1/2 + sqrt(5)/2)) A = (0, 0) H = (1/2 + sqrt(5)/2, 0) B = (sqrt(5)/2 + 3/2, 0) O C A H B

GC-04.02 · Johannes Kepler’s Geometric-Progression Right Triangle (1597)

Divide hypotenuse AB at H in the golden ratio (\(|AH|/|HB| = \phi\)), and erect perpendicular HC to semicircle cOB on diameter AB. Right triangle ABC has side lengths in geometric progression \(1 : \sqrt{\phi} : \phi\) and side quadrances in progression \(1 : \phi : \phi^2\).