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

Discoverer(s): Claudius Ptolemy, Euclid of Alexandria | Date: c. 150 CE | Field: \(\mathbb{Q}(\phi, \sqrt{10 - 2\sqrt{5}})\) | Loci: 6 (Lemoine \(S = 18\))

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-05.01 — Claudius Ptolemy's Regular Pentagon & Pentagram (Almagest I.10) cO: x**2 + y**2 - 1 = 0 cMC: y**2 + (x - 1/2)**2 - 5/4 = 0 AB: y = 0 P1P4: y - sqrt(5)/4 + 1/4 = 0 CP2: x*(sqrt(5)/4 + 5/4) - y*sqrt(10 - 2*sqrt(5))/4 + sqrt(10 - 2*sqrt(5))/4 = 0 CP3: x*(sqrt(5)/4 + 5/4) + y*sqrt(10 - 2*sqrt(5))/4 - sqrt(10 - 2*sqrt(5))/4 = 0 Golden major P1K2 = sqrt(2)*sqrt(5 - sqrt(5))/2 Golden minor K2P4 = sqrt(5 - 2*sqrt(5)) A = (-1, 0) C = (0, 1) M = (1/2, 0) E = (1/2 - sqrt(5)/2, 0) P2 = (-sqrt(2)*sqrt(5 - sqrt(5))/4, -sqrt(5)/4 - 1/4) P3 = (sqrt(2)*sqrt(5 - sqrt(5))/4, -sqrt(5)/4 - 1/4) K1 = (-3*sqrt(2)*sqrt(5 - sqrt(5))/8 + sqrt(10)*sqrt(5 - sqrt(5))/8, -1/4 + sqrt(5)/4) O = (0, 0) B = (1, 0) P1 = (-sqrt(10)*sqrt(5 - sqrt(5))/8 - sqrt(2)*sqrt(5 - sqrt(5))/8, -1/4 + sqrt(5)/4) P4 = (sqrt(2)*sqrt(5 - sqrt(5))/8 + sqrt(10)*sqrt(5 - sqrt(5))/8, -1/4 + sqrt(5)/4) K2 = (-sqrt(10)*sqrt(5 - sqrt(5))/8 + 3*sqrt(2)*sqrt(5 - sqrt(5))/8, -1/4 + sqrt(5)/4) A C M E P2 P3 K1 O B P1 P4 K2

Given & Constructive Steps

Given: Unit circle cO centered at O \(= (0,0)\) through B \(= (1,0)\), diameter AB, and perpendicular apex C \(= (0,1)\).

step Bisect radius OB at M \(= (\tfrac{1}{2}, 0)\) and swing circle cMC to meet AB at E \(= (-1/\phi, 0)\).
step Step off chord length \(|CE| = \sqrt{3 - \phi}\) around cO from C to inscribe regular pentagon CP1P2P3P4.
step Draw pentagram diagonals P1P4, CP2, and CP3, intersecting along P1P4 at K1 and K2.

Exact Symbolic Proof & Certificate

Right triangle \(\triangle MOC\) has \(|MC| = \frac{\sqrt{5}}{2}\), so \(|OE| = \frac{\sqrt{5}-1}{2} = 1/\phi\) (the regular decagon side, Euclid XIII.9–10) and \(|CE|^2 = 1 + 1/\phi^2 = 3 - \phi = 4\sin^2 36^\circ\) (the regular pentagon side). By Euclid XIII.8, every pair of crossing pentagram diagonals cuts each other in extreme and mean ratio, forming the 4-point golden chain \((P1, K1, K2, P4)\).

Machine-Verified Symbolic Certificate

Invariant

Exact Symbolic Value

Primary Golden Ratio

P1K2 \(/\) K2P4 \(= \frac{1}{2} + \frac{\sqrt{5}}{2}\) on collinear triple (P1, K2, P4)

Exact Segment Lengths

\(|P1K2| = \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{2}\), \(|K2P4| = \sqrt{5 - 2 \sqrt{5}}\)

Rational Trigonometry Quadrances

\(Q(P1, K2) = \frac{5}{2} - \frac{\sqrt{5}}{2}\), \(Q(K2, P4) = 5 - 2 \sqrt{5}\), \(Q_{\text{long}}/Q_{\text{short}} = \frac{\sqrt{5}}{2} + \frac{3}{2} = \phi^2\)

Algebraic Number Field

\(\mathbb{Q}(\phi, \sqrt{10 - 2\sqrt{5}})\)

Géométrographie Loci

2 given points · 4 lines · 2 circles · 1 polygons (elementary loci \(= 6\), Lemoine \(S = 18\))

Automated Divine Scan

geometor.divine.golden detected 8 exact collinear golden section(s) and 2 connected golden chain(s) in this model:

  • Section (A, E, O): long \(|EO|^2 = \frac{3}{2} - \frac{\sqrt{5}}{2}\) vs. short \(|AE|^2 = \frac{7}{2} - \frac{3 \sqrt{5}}{2}\)

  • Section (E, O, B): long \(|OB|^2 = 1\) vs. short \(|EO|^2 = \frac{3}{2} - \frac{\sqrt{5}}{2}\)

  • Section (P1, K1, K2): long \(|P1K1|^2 = 5 - 2 \sqrt{5}\) vs. short \(|K1K2|^2 = \frac{25}{2} - \frac{11 \sqrt{5}}{2}\)

  • Section (P1, K1, P4): long \(|K1P4|^2 = \frac{5}{2} - \frac{\sqrt{5}}{2}\) vs. short \(|P1K1|^2 = 5 - 2 \sqrt{5}\)

  • Section (P1, K2, P4): long \(|P1K2|^2 = \frac{5}{2} - \frac{\sqrt{5}}{2}\) vs. short \(|K2P4|^2 = 5 - 2 \sqrt{5}\)

  • Section (K1, K2, P4): long \(|K2P4|^2 = 5 - 2 \sqrt{5}\) vs. short \(|K1K2|^2 = \frac{25}{2} - \frac{11 \sqrt{5}}{2}\)

  • Section (P2, K1, C): long \(|P2K1|^2 = \frac{5}{2} - \frac{\sqrt{5}}{2}\) vs. short \(|K1C|^2 = 5 - 2 \sqrt{5}\)

  • Section (C, K2, P3): long \(|K2P3|^2 = \frac{5}{2} - \frac{\sqrt{5}}{2}\) vs. short \(|CK2|^2 = 5 - 2 \sqrt{5}\)

Connected Golden Chains:

  • Chain (A — E — O — B) spanning 2 overlapping golden sections (progression < <)

  • Chain (P1 — K1 — K2 — P4) spanning 2 overlapping golden sections (progression > <)

Historical Commentary & Attribution

Ptolemy introduced this streamlined circle-and-midpoint construction in Almagest I.10 to compute his chord table at \(36^\circ\) and \(72^\circ\), dramatically simplifying Euclid’s Book IV Proposition 11.