Posts tagged 'field-pentagonal'

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-05.02 · Dao Thanh Oai’s Regular Pentagon Edge Perpendicular

In regular pentagon ABCDE, erect line EF perpendicular to side DE at vertex E. Line EF intersects opposite non-adjacent side AB internally at F such that \(|FB|/|AF| = |AB|/|FB| = \phi\).

GC-05.05 · Tran Quang Hung’s Six-Pentagon Reflection Star

Reflect regular pentagon ABCDE across each of its five sides to obtain five surrounding congruent pentagons. Joining the outer lateral vertices across two intervening pentagons (such as chord S1) cuts every side of ABCDE in the golden ratio, dividing AB at G and H in a 4-point golden chain.