Notes — Book I: Descriptive Geometry & the Eierlini Symmetry Proof

1. Birth of Two-View Descriptive Geometry (270 Years Before Monge)

Before Gaspard Monge's Géométrie descriptive (monge:1799), stonecutters and architects knew empirical plan-and-elevation templates, and Piero della Francesca applied coupled projections to human heads in De prospectiva pingendi. Yet Book I, Figures 34–37 of Dürer’s 1525 Underweysung represent the first printed mathematical application of coupled orthographic elevation and ground-plan views to determine the true metric locus of curved three-dimensional sections.

2. Exact Algebraic Resolution of Dürer’s Eierlini Paradox

Why did Dürer translate Apollonius’s ellipsis as Eierlini (“egg-line”, “darumb das sie schyer einem ey gleich ist”) and draw his 1525 woodcut Fig. 34 slightly narrower at the upper end near the cone’s apex?

Let us evaluate Dürer’s own 12-station algorithm symbolically in geometor.durer (construct_durer_conic_sections()). Take a right circular cone of base radius $R = 6$ and height $H = 12$, so at height $z \in [0, 12]$ the horizontal circular cross-section has radius

$$r(z) = \frac{12 - z}{2}.$$

Slice the cone obliquely along segment $fg$ from $F = (-4, 4)$ on the left generator to $G = (2, 8)$ on the right generator, divided into 12 equal parts $k = 0, 1, \dots, 12$:

$$x_k = -4 + \frac{k}{2}, \qquad z_k = 4 + \frac{k}{3}, \qquad r_k = 4 - \frac{k}{6}.$$

In the circular ground plan (Grundriss), the vertical projector at $x_k$ cuts the horizontal section circle $x^2 + y^2 = r_k^2$ in a chord of exact half-width $w_k$:

$$w_k^2 = r_k^2 - x_k^2 = \left(4 - \frac{k}{6}\right)^2 - \left(-4 + \frac{k}{2}\right)^2 = \left(16 - \frac{4}{3}k + \frac{k^2}{36}\right) - \left(16 - 4k + \frac{k^2}{4}\right) = \frac{2}{9}\,k\,(12 - k).$$

Hence for every station $k \in \{0, 1, \dots, 12\}$:

$$w_k = \frac{\sqrt{2\,k\,(12 - k)}}{3} = w_{12 - k}!$$

Dürer’s geometric construction rule is 100% exact and strictly symmetric under $k \mapsto 12 - k$! Because $|FG| = \sqrt{6^2 + 4^2} = 2\sqrt{13}$, the curve is a true Apollonian ellipse with semi-major axis $a = \sqrt{13}$, semi-minor axis $b = w_6 = 2\sqrt{2}$, focal distance $c = \sqrt{a^2 - b^2} = \sqrt{5}$, and eccentricity $e = \frac{\sqrt{65}}{13}$.

Why then did Dürer suspect asymmetry? In the ground plan (der grund zum kegell), the upper half of the cutting plane ($k = 7, \dots, 12$) crosses directly over the cone’s apex projection $O_{\text{plan}} = (0, 0)$ at $k = 8$ ($x_8 = 0$), crowding the upper concentric circles tightly around the center, whereas near $k = 1, 2, 3$ the arcs sweep broadly near the rim! A slight compass transfer error on the crowded upper circles immediately pinches the top of the plotted curve into an egg shape—an optical and draftsmanship trap definitively resolved by Johannes Kepler in Chapter IV of Ad Vitellionem Paralipomena (kepler:1604).

3. Exact Polar Stations of the 12-Sector Spiral over $\mathbb{Q}(\sqrt{3})$

Because $\cos(k \cdot 30^\circ), \sin(k \cdot 30^\circ) \in \left\{0, \pm\frac{1}{2}, \pm\frac{\sqrt{3}}{2}, \pm 1\right\}$ for every integer $k$, all 25 stations of Dürer’s two-turn 12-sector Archimedean spiral $P_k = \left(\frac{k}{2}\cos\frac{k\pi}{6},\, \frac{k}{2}\sin\frac{k\pi}{6}\right)$ belong entirely to the quadratic field $\mathbb{Q}(\sqrt{3})$, requiring zero floating-point evaluation in geometor.model.