Unit II · Historical & Mathematical Commentary¶
1. Literary Paradox (Nix / Nihil) and Morphological Method¶
Composed at Imperial Prague in late 1610 as a playful humanist gift for Privy Councillor J. M. Wackher von Wackenfels, Strena stands in the Renaissance tradition of paradoxical encomia (Praise of Nothing). Yet within twenty-four quarto pages Kepler inaugurated three modern disciplines simultaneously:
Geometric Crystallography & Lattice Theory (later developed by Steno, Hooke, Huygens, Haüy, and Bravais);
Discrete & Computational Geometry (sphere packing, kissing numbers, and 3D Voronoi/Dirichlet tessellations, Thompson 1942);
Mathematical Phyllotaxis (linking Fibonacci parastichy numbers to the golden section, developed across
geometor/phyllotaxis).
2. Equivalence of Kepler’s Square-Pyramidal and Hexagonal Packings¶
A remarkable geometric subtlety appears in Strena §§11–12: Kepler describes two ways of stacking horizontal sphere layers so that each interior sphere touches twelve neighbors:
Start with a square grid in the horizontal $(x, y)$ plane and seat each sphere of the next layer into the fourfold hollow of four touching spheres below (cannonball pyramid).
Start with a triangular-hexagonal grid (fig03) and seat each sphere of the next layer into a threefold hollow ($A \to B \to C \to A\dots$ or $A \to B \to A\dots$).
In fact, the square-based cannonball pyramid is the face-centered cubic (FCC) lattice viewed along its [001] fourfold axis, whose [111] diagonal planes are hexagonal layers in ABCABC stacking! Both ABCABC (FCC, Voronoi cell = Rhombic Dodecahedron) and ABAB (HCP, Voronoi cell = Trapezo-Rhombic Dodecahedron) achieve the exact same packing fraction $\frac{\pi}{3\sqrt{2}}$. After standing open for 387 years, Kepler’s optimality claim was proved by Thomas C. Hales in 1998 (published 2005) and formally verified by machine in the Flyspeck theorem-proving project (2014).
3. The Maraldi Angle of the Honeycomb Rhombus¶
In geometor.kepler.strena_packing.analyze_rhombic_dodecahedron, each face of the Rhombic Dodecahedron has half-diagonals $\sqrt{2}$ and $1$ and edge length $\sqrt{3}$. By the law of cosines on the rhombus triangles:
$$\cos\alpha_{\text{obtuse}} = \frac{3 + 3 - 8}{2 \cdot 3} = -\frac{1}{3} \implies \alpha_{\text{obtuse}} = \arccos\!\left(-\frac{1}{3}\right) \approx 109^\circ 28' 16'',$$
the celebrated Maraldi tetrahedral angle (rediscovered experimentally in beehives by Giacomo Filippo Maraldi in 1712), minimizing wax surface area for a given hexagonal cell volume!