F.45 Squares, Subdivided Lattices, and the Japanese Theorem — Folios

Archival manuscript scans and diplomatic transcriptions from seelig’s notebooks for F.45 (Squares, Subdivided Lattices, and the Japanese Theorem) (SeeligNotebooks).

Folio 0287 (Page 1 of 6): Compass Construction of Inscribed Square & Equilateral Triangle (001)

[Folio 0287]

Folio 0287 (original scan 0287.jpg) from resources/seelig/45_squares.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0287 (0287.jpg, Sheet 1 of 6):

Inscribes a square of side \(\sqrt{2}R\) and an equilateral triangle of side \(\sqrt{3}R\) inside a circle of radius \(R\).

\[s_{\square} = \sqrt{2}\,R, \qquad s_{\triangle} = \sqrt{3}\,R, \qquad OM_A + OM_B + OM_C = R + r\]

Editorial & Symbolic Notes

Archival scan 0287.jpg from Fred W. Seelig’s notebook dossier on F.45 Squares, Subdivided Lattices, and the Japanese Theorem (Page 1 of 6). Companion OpenDocument worksheet(s): Polygon Table, General Reference. Classical Euclidean foundations: IV.6, IV.7.

Folio 0288 (Page 2 of 6): Line Intersection N Inside a Square — Setup (002)

[Folio 0288]

Folio 0288 (original scan 0288.jpg) from resources/seelig/45_squares.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0288 (0288.jpg, Sheet 2 of 6):

Sets up a square of side \(s\) on Cartesian axes and writes the two-intercept equations of intersecting cevians meeting at \(N = (X_N, Y_N)\).

\[X_N = \frac{s \cdot u v}{u + v - u v}, \qquad Y_N = \frac{s \cdot u (1 - v)}{u + v - u v}\]

Editorial & Symbolic Notes

Archival scan 0288.jpg from Fred W. Seelig’s notebook dossier on F.45 Squares, Subdivided Lattices, and the Japanese Theorem (Page 2 of 6). Companion OpenDocument worksheet(s): Polygon Table, General Reference. Classical Euclidean foundations: IV.6, IV.7.

Folio 0289 (Page 3 of 6): Line Intersection N Inside a Square — Solution (003)

[Folio 0289]

Folio 0289 (original scan 0289.jpg) from resources/seelig/45_squares.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0289 (0289.jpg, Sheet 3 of 6):

Solves the \(2\times 2\) linear system for \((X_N, Y_N)\) and computes the areas of the four resulting sub-regions.

\[s_{\square} = \sqrt{2}\,R, \qquad s_{\triangle} = \sqrt{3}\,R, \qquad OM_A + OM_B + OM_C = R + r\]

Editorial & Symbolic Notes

Archival scan 0289.jpg from Fred W. Seelig’s notebook dossier on F.45 Squares, Subdivided Lattices, and the Japanese Theorem (Page 3 of 6). Companion OpenDocument worksheet(s): Polygon Table, General Reference. Classical Euclidean foundations: IV.6, IV.7.

Folio 0290 (Page 4 of 6): Subdivided Square & Circle Packing Sketch (004)

[Folio 0290]

Folio 0290 (original scan 0290.jpg) from resources/seelig/45_squares.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0290 (0290.jpg, Sheet 4 of 6):

Sketches nested squares and inscribed circles scaling by powers of \(1/\sqrt{2}\).

\[X_N = \frac{s \cdot u v}{u + v - u v}, \qquad Y_N = \frac{s \cdot u (1 - v)}{u + v - u v}\]

Editorial & Symbolic Notes

Archival scan 0290.jpg from Fred W. Seelig’s notebook dossier on F.45 Squares, Subdivided Lattices, and the Japanese Theorem (Page 4 of 6). Companion OpenDocument worksheet(s): Polygon Table, General Reference. Classical Euclidean foundations: IV.6, IV.7.

Folio 0291 (Page 5 of 6): Subdivided Equilateral Triangles of Sides 6 and 1 (005)

[Folio 0291]

Folio 0291 (original scan 0291.jpg) from resources/seelig/45_squares.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0291 (0291.jpg, Sheet 5 of 6):

Partitions an equilateral triangle of side 6 into 36 unit equilateral triangles and computes sub-triangle altitudes and centroids.

\[s_{\square} = \sqrt{2}\,R, \qquad s_{\triangle} = \sqrt{3}\,R, \qquad OM_A + OM_B + OM_C = R + r\]

Editorial & Symbolic Notes

Archival scan 0291.jpg from Fred W. Seelig’s notebook dossier on F.45 Squares, Subdivided Lattices, and the Japanese Theorem (Page 5 of 6). Companion OpenDocument worksheet(s): Polygon Table, General Reference. Classical Euclidean foundations: IV.6, IV.7.

Folio 0292 (Page 6 of 6): Carnot’s Theorem & the Japanese Sangaku Theorem (006)

[Folio 0292]

Folio 0292 (original scan 0292.jpg) from resources/seelig/45_squares.

Diplomatic Mathematical & Geometric Transcription

Diplomatic Transcription — Folio 0292 (0292.jpg, Sheet 6 of 6):

Recalls \(OM_A + OM_B + OM_C = R + r\) and explains how summing over a cyclic polygon triangulation proves the invariance of the inradii sum.

\[X_N = \frac{s \cdot u v}{u + v - u v}, \qquad Y_N = \frac{s \cdot u (1 - v)}{u + v - u v}\]

Editorial & Symbolic Notes

Archival scan 0292.jpg from Fred W. Seelig’s notebook dossier on F.45 Squares, Subdivided Lattices, and the Japanese Theorem (Page 6 of 6). Companion OpenDocument worksheet(s): Polygon Table, General Reference. Classical Euclidean foundations: IV.6, IV.7.