F.23 Golden Triangles — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.23 (Golden Triangles) (SeeligNotebooks).

Folio 0132 (Page 1 of 3): Golden Triangle 72-72-36 & Segment AD

[Folio 0132]

Folio 0132 (original scan 0132.jpg) from resources/seelig/23_golden-triangles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0132 (0132.jpg, Sheet 1 of 3):

Analyzes the \(72^\circ\text{–}72^\circ\text{–}36^\circ\) triangle with legs \(\phi\), base \(1\), and point \(D\) dividing \(BC\) into \(\phi-1\) and \(1\), proving \(AD = 2\sin 36^\circ \approx 1.1756\).

\[\phi = \frac{1 + \sqrt{5}}{2}, \qquad \cos 36^\circ = \frac{\phi}{2}, \qquad \cos 72^\circ = \frac{\phi - 1}{2} = \frac{1}{2\phi}, \qquad 2\sin 36^\circ = \sqrt{3 - \phi}\]

Editorial & Symbolic Notes

Archival scan 0132.jpg from Fred W. Seelig’s notebook dossier on F.23 Golden Triangles (Page 1 of 3). Companion OpenDocument worksheet(s): Golden Triangle, Polygon Table. Classical Euclidean foundations: II.11, IV.10, IV.11, VI.30.

Folio 0133 (Page 2 of 3): Golden Identity Triangles & 5th Roots of Unity (FWS 12/21/2014)

[Folio 0133]

Folio 0133 (original scan 0133.jpg) from resources/seelig/23_golden-triangles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0133 (0133.jpg, Sheet 2 of 3):

Draws nested right triangles of angles \(72^\circ, 54^\circ, 36^\circ, 18^\circ\) with altitude \(1\) and tabulates \(\operatorname{Re}\) and \(\operatorname{Im}\) of \(e^{i 2\pi n/5}\).

\[f^2 = k a^2 + k' b^2 - k k' c^2 \qquad (k + k' = 1)\]

Editorial & Symbolic Notes

Archival scan 0133.jpg from Fred W. Seelig’s notebook dossier on F.23 Golden Triangles (Page 2 of 3). Companion OpenDocument worksheet(s): Golden Triangle, Polygon Table. Classical Euclidean foundations: II.11, IV.10, IV.11, VI.30.

Folio 0134 (Page 3 of 3): Normalized Stewart’s Theorem Proof (FWS 11/18/2013)

[Folio 0134]

Folio 0134 (original scan 0134.jpg) from resources/seelig/23_golden-triangles.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0134 (0134.jpg, Sheet 3 of 3):

Proves \(f^2 = k a^2 + k' b^2 - k k' c^2\) by reducing \(X = k(1-k)^2 + k^2(1-k) = k(1-k) = kk'\).

\[\phi = \frac{1 + \sqrt{5}}{2}, \qquad \cos 36^\circ = \frac{\phi}{2}, \qquad \cos 72^\circ = \frac{\phi - 1}{2} = \frac{1}{2\phi}, \qquad 2\sin 36^\circ = \sqrt{3 - \phi}\]

Editorial & Symbolic Notes

Archival scan 0134.jpg from Fred W. Seelig’s notebook dossier on F.23 Golden Triangles (Page 3 of 3). Companion OpenDocument worksheet(s): Golden Triangle, Polygon Table. Classical Euclidean foundations: II.11, IV.10, IV.11, VI.30.