Posts tagged 'john-arioni'

GC-06.06 — John Arioni's Golden Secant in the Yin-Yang Circle cO: x**2 + y**2 - 1 = 0 cA: y**2 + (x + 1/2)**2 - 1/4 = 0 cC: y**2 + (x - 1/2)**2 - 1/4 = 0 BA: x - y/2 + 1/2 = 0 Golden major HI = 1 Golden minor BH = -1/2 + sqrt(5)/2 PL = (-1, 0) PR = (1, 0) A = (-1/2, 0) C = (1/2, 0) O = (0, 0) B = (0, 1) H = (-1/2 + sqrt(5)/10, sqrt(5)/5) I = (-1/2 - sqrt(5)/10, -sqrt(5)/5) PL PR A C O B H I

GC-06.06 · John Arioni’s Golden Secant in the Yin-Yang Circle

Inscribe twin tangent circles cA and cC of radius \(\tfrac{1}{2}\) along horizontal diameter PLPR of unit circle cO, and let B be the apex of perpendicular radius OB. Secant BA cuts cA at H and I such that \(|HI|/|BH| = |BI|/|HI| = \phi\).
GC-03.08 — John Arioni's 30°-Rotated Unit Square & Diagonal Circle cAC: (x - 1/2)**2 + (y - sqrt(3)/2)**2 - (1/2 + sqrt(3)/2)**2 - (1/2 - sqrt(3)/2)**2 = 0 DG: y = 0 Golden major DG = 1 Golden minor GE = -1/2 + sqrt(5)/2 H = (1, 1) J = (0, 1) A = (1/2, sqrt(3)/2) B = (1/2 + sqrt(3)/2, -1/2 + sqrt(3)/2) C = (sqrt(3)/2, -1/2) F = (1/2 - sqrt(5)/2, 0) D = (0, 0) G = (1, 0) E = (1/2 + sqrt(5)/2, 0) H J A B C F D G E

GC-03.08 · John Arioni’s 30°-Rotated Unit Square & Diagonal Circle

Rotate unit square DGHJ clockwise by \(30^\circ\) around vertex D to obtain congruent square DCBA. Circle cAC centered at A through opposite vertex C intersects line DG at E and F such that \(|DG|/|GE| = |DE|/|DG| = \phi\).