Summary — Book II: Plane Figures, Polygons, Parquetries & Quadrature

Structural Overview (SLUB N1525 pp. 053–076, Figs. 1–36)

  1. Plane Enclosure, Angles & Quadrilateral Taxonomy (pp. 053–056, Figs. 1–8):

    • Establishes why two straight lines cannot bound a planar region (Euclid I, Def. 19), whereas lenses and crescents (Monschein) of two arcs do.

    • Distinguishes salient exterior corners (Eck) from recessed interior angles (Winckel), constructs right/acute/obtuse angles via Thales’s semicircle, and classifies triangles and the eight quadrilateral types (a–h).

  2. Epistemology of Regular Polygons: *Demonstrative* vs. *Mechanice* (pp. 057–061, Figs. 9–20):

    • Exact Euclidean Polygons (demonstrative): 6-gon and 3-gon by unchanged compass (Figs. 9–10); 4-, 8-, and 16-gons by perpendicular bisection (Figs. 13–14); 5-gon and 10-gon via Ptolemy’s golden-section arc (Fig. 15); 15-gon via Euclid IV.16 arc difference (Fig. 17).

    • Workshop Constructions (mechanice): Fixed-aperture (“rusty compass”) equilateral pentagon (Fig. 16); half-equilateral heptagon $s_7 = \frac{\sqrt{3}}{2}R$ and 14/28-gons (Figs. 11–12, 15); three-vesica (Fischblosen) nonagon (Fig. 18); hendecagon $s_{11} = frac{9}{16}R$ and tridecagon $s_{13} = \frac{1}{2}R$ (Fig. 19); iterative chord-based arc trisection (Fig. 20).

  3. Circle Packings, Gothic Tracery & Pentagonal Parquetries (pp. 061–071, Figs. 21–27):

    • Contrasts 4-neighbor square circle packing with 6-neighbor hexagonal packing (6 touch the 7th) and 12-fold Gothic tracery stars (Fig. 21).

    • Investigates periodic and locally fivefold planar tilings of 3, 4, 5, 6, 7, 8-gons, including four pentagonal parquetries (fünf rosen) with golden rhombic interstices (Fig. 24) and the Archimedean 4.8.8 octagon-square tiling (Fig. 26).

  4. Area Transformations & Circle Quadrature (pp. 071–076, Figs. 28–36):

    • Exact Euclidean squaring of triangles, rectangles, and regular $n$-gons via semicircle mean proportionals (Euclid II.14, VI.13), diagonal homothetic scaling, $2\times$, $3\times$, and $7\times$ area multiplication, generalized Pythagorean theorem (Euclid VI.31), and the $\frac{8}{10}$ diagonal circle quadrature ($\pi_{\text{Dürer}} = \frac{25}{8} = 3.125$, Fig. 34).