GEOMETOR • Dürer — Underweysung der Messung (Nuremberg, 1525 / 1538)

Sann warhafftig steckt die kunst inn der natur / wer sie herauß kan reyssen / der hat sie. Überkumbstu sie / so wirdet sie dir vil fels nemen in deinem werck. Vnd durch die Geometria magstu deines wercks vil beweyssen.

—Albrecht Dürer, Vier Bücher von menschlicher Proportion (1528), Book III

GEOMETOR • durer presents a four-track bilingual Digital Scriptorium Book Codex of Albrecht Dürer’s landmark Underweysung der Messung, mit dem Zirckel un[d] Richtscheyt, in Linien Ebnen unnd gantzen Corporen (Instruction in Measurement, with Compass and Straightedge, in Lines, Planes, and Whole Solid Bodies, Nuremberg, 1525; 2nd enlarged ed., 1538; Latin ed. by Joachim Camerarius, Paris, 1532), paired with 100% exact symbolic SymPy and ``geometor.model`` constructions verifying every Euclidean proof, workshop approximation, typographic canon, planar polyhedral net, and perspective projection.

a: (0, 0) b: (1, 0) d: (\frac{1}{2}, - \frac{\sqrt{3}}{2}) c: (\frac{1}{2}, \frac{\sqrt{3}}{2}) e: (- \frac{1}{2}, - \frac{\sqrt{3}}{2}) f: (\frac{3}{2}, - \frac{\sqrt{3}}{2}) g: (\frac{1}{2}, 1 - \frac{\sqrt{3}}{2}) h: (\frac{1}{4} + \frac{\sqrt{3}}{4} + \frac{\sqrt{2} \sqrt{-2 + 3 \sqrt{3}}}{4}, - \frac{\sqrt{3}}{4} + \frac{\sqrt{2} \sqrt{-2 + 3 \sqrt{3}}}{4} + \frac{3}{4}) i: (- \frac{\sqrt{2} \sqrt{-2 + 3 \sqrt{3}}}{4} - \frac{\sqrt{3}}{4} + \frac{3}{4}, - \frac{\sqrt{3}}{4} + \frac{\sqrt{2} \sqrt{-2 + 3 \sqrt{3}}}{4} + \frac{3}{4}) k: (\frac{- \left(- \frac{\sqrt{2} \sqrt{-2 + 3 \sqrt{3}}}{4} - \frac{\sqrt{3}}{4} + \frac{3}{4}\right)^{2} + \left(- \frac{3}{4} + \frac{\sqrt{3}}{4} + \frac{\sqrt{2} \sqrt{-2 + 3 \sqrt{3}}}{4}\right)^{2} + \left(- \frac{\sqrt{2} \sqrt{-2 + 3 \sqrt{3}}}{2} - \frac{\sqrt{3}}{2} + \frac{1}{2}\right)^{2}}{2 \left(- \frac{\sqrt{2} \sqrt{-2 + 3 \sqrt{3}}}{2} - \frac{\sqrt{3}}{2} + \frac{1}{2}\right)} + \frac{1}{4} + \frac{\sqrt{3}}{4} + \frac{\sqrt{2} \sqrt{-2 + 3 \sqrt{3}}}{4}, - \frac{\sqrt{3}}{4} - \frac{\sqrt{- \left(- \left(- \frac{\sqrt{2} \sqrt{-2 + 3 \sqrt{3}}}{4} - \frac{\sqrt{3}}{4} + \frac{3}{4}\right)^{2} + \left(- \frac{3}{4} + \frac{\sqrt{3}}{4} + \frac{\sqrt{2} \sqrt{-2 + 3 \sqrt{3}}}{4}\right)^{2} + \left(- \frac{\sqrt{2} \sqrt{-2 + 3 \sqrt{3}}}{2} - \frac{\sqrt{3}}{2} + \frac{1}{2}\right)^{2}\right)^{2} + 4 \left(\left(- \frac{3}{4} + \frac{\sqrt{3}}{4} + \frac{\sqrt{2} \sqrt{-2 + 3 \sqrt{3}}}{4}\right)^{2} + \left(- \frac{\sqrt{3}}{4} + \frac{\sqrt{2} \sqrt{-2 + 3 \sqrt{3}}}{4} + \frac{3}{4}\right)^{2}\right) \left(- \frac{\sqrt{2} \sqrt{-2 + 3 \sqrt{3}}}{2} - \frac{\sqrt{3}}{2} + \frac{1}{2}\right)^{2}}}{2 \left(- \frac{\sqrt{2} \sqrt{-2 + 3 \sqrt{3}}}{2} - \frac{\sqrt{3}}{2} + \frac{1}{2}\right)} + \frac{\sqrt{2} \sqrt{-2 + 3 \sqrt{3}}}{4} + \frac{3}{4}) phi: (\frac{1}{2} + \frac{\sqrt{5}}{2}, 0) H_phi: (\frac{\sqrt{5}}{4} + \frac{3}{4}, \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{8} + \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{8}) I_phi: (\frac{1}{4} - \frac{\sqrt{5}}{4}, \frac{\sqrt{2} \sqrt{5 - \sqrt{5}}}{8} + \frac{\sqrt{10} \sqrt{5 - \sqrt{5}}}{8}) K_phi: (\frac{1}{2}, \frac{\sqrt{2 \sqrt{5} + 5}}{2}) a b d c e f g h i k phi Hphi Iphi Kphi Solid Gold: Dürer Rusty Compass (108.37°) | Dashed Blue: Euclid Q(phi) (108°)
Dürer's Single-Aperture ("Rusty Compass") Pentagon (solid gold, exact unit sides $Q = 1$, base angle $108.3661^\circ$) overlaid on Euclid's Golden-Section Pentagon over $\mathbb{Q}(\phi)$ (dashed blue, exact interior angle $108^\circ$) on shared base $ab$.

Codex & Symbolic Architecture

  1. Underweysung der Messung (Nuremberg, 1525 / Camerarius, 1532) — 4-Track Bilingual Digital Scriptorium Codex (English, German (1525), Summary, Notes) spanning Dürer’s Dedication to Willibald Pirckheimer and all Four Books (Vier Bücher):

    • Book I (unit-1-lines-spirals-conics): Point, serpentine curve (Schlangenlini), 12-sector Archimedean spirals (Schneckenlinien), spatial helices, Nicomedes’s conchoid (Muschellini), and the invention of two-view orthographic descriptive geometry for the Elipsis / Eierlini, Parabola / Brennspiegellini, and Hiperbole / Gabellini.

    • Book II (unit-2-planes-polygons-tilings): Planar tilings and pentagonal parquetries, demonstrative vs. mechanice regular polygons (3 through 16 sides), the single-aperture pentagon, the $\frac{\sqrt{3}}{2}R$ heptagon, Euclidean area quadrature, and the $\pi \approx \frac{25}{8} = 3.125$ circle quadrature.

    • Book III (unit-3-architecture-sundials-typography): Architectural columns, Nuremberg sundial gnomonics ($49^\circ$ pole), optical scaling of tower inscriptions, Roman Antiqua capitals (A–Z) inside the $10 \times 10$ square, and modular Gothic Textura minuscules.

    • Book IV (unit-4-solids-nets-perspective): The first printed planar fold-out polyhedral nets in mathematical history (all 5 Platonic + 7 Archimedean solids), the Delian Doubling of the Cube (Sporus, Plato’s mechanical square, Hero, and cannonball scaling), cube shadow sciagraphy, and the four perspective instruments.

  2. Exact Symbolic Demonstrations (demos/) — Executable Exact Symbolic Demonstrations auditing every construction in SymPy and geometor.model with zero floating-point coordinates.

  3. modules — Python Package Reference for geometor.durer.