Unit II · Strena seu de Nive Sexangula (1611)¶
Original GEOMETOR English translation from Johannes Kepler’s Strena seu de Nive Sexangula (Frankfurt am Main: Tampach, 1611; Frisch VII, pp. 715–730).
1. Dedication to Wackher & Snow Stars on the Prague Bridge¶
Since I am well aware, Illustrious Herr Wackher, how keenly you delight in Nothing (Nihil)—not for its bareness of price, to be sure, yet for the witty play of delightful sparrows—I readily conjecture that a New Year’s gift (Strena) will prove all the dearer to you the nearer it approaches to Nothing. Anxious in thought as to where I might find something virtually indistinguishable from Nothing, I happened just then to be crossing the bridge at Prague, mortified by my lack of a worthy gift, when by a timely chance thickened vapor condensed into snow, and scattered little flakes fell here and there upon my coat—every single one six-cornered, with feathered rays (omnes sexanguli, villosis radiis).
By Hercules! Here at last was something smaller than any drop, yet endowed with geometric figure; here was the longed-for New Year’s gift for a lover of Nothing, and one worthy for a mathematician—who neither has Nothing nor receives Nothing—to bestow, since it descends from heaven and bears a likeness to the stars! Indeed, by a pun in our German tongue, Schnee (“snow”) sounds almost like the Latin Nix (“snow” / German nichts, “nothing”). Yet why, whenever snow begins to fall before its flakes are tangled into clusters, does it always fall with six corners and six feathered rays spaced at $60^\circ$ in a single plane, rather than five or seven?
2. Bee Honeycombs & Discovery of the Two Rhombic Polyhedra¶
Let us examine familiar natural architectures where regular geometry arises from material necessity. Bees build hexagonal wax prisms (Pappus, Synagoge V) so that every wall is shared by two neighboring chambers without empty interstices. Yet inspect the bottoms (fundi) of a two-layered honeycomb! Each six-sided tube terminates not in a flat floor, but in an obtuse three-plane pyramidal keel composed of three congruent rhombi, meeting three-by-three against the staggered cells of the opposite layer.
Moved by this observation, I set my mind to inquiring via geometry whether there might exist any solid body—analogous to the five Platonic solids and Archimedes's thirteen solids, though missed by antiquity—bounded exclusively by congruent rhombi. And I discovered two:
The first, akin to the Cube and Octahedron, is enclosed by twelve congruent rhombi (Rhombic Dodecahedron, ABCD), having fourteen vertices—eight obtuse trihedral corners matching the cube and six acute four-edged corners matching the octahedron—and twenty-four edges, with long diagonal AC to short diagonal BD in exact ratio $\sqrt{2} : 1$ and dihedral angle $120^\circ$.
The second, akin to the Dodecahedron and Icosahedron, is enclosed by thirty congruent golden rhombi (Rhombic Triacontahedron), having thirty-two vertices and sixty edges, whose diagonals stand in the exact Divine Section $\phi : 1$.
Above all, the first of these two bodies shares with the Cube a most remarkable property which the second lacks: just as plane hexagons and squares cover a flat surface without leaving any gap, twelve-faced rhombic dodecahedra pack together in three-dimensional space to fill solid volume completely without a single void (impletrix loci solidi), three $120^\circ$ dihedral edges meeting seamlessly along every line ($3 \times 120^\circ = 360^\circ$)!
3. Pomegranate Seeds & the Sphere Close-Packing Conjecture¶
Turn next to the pomegranate (malum granatum), whose angular crimson kernels are packed tightly inside a tough rind. Why are those seeds faceted into rhomboidal planes? Not from any formal soul counting faces, declare I, but from material compression of round globules!
Suppose you gather equal round pellets on a single horizontal plane and press them together until they touch one another mutually: they coalesce either in quadrangular order (where four neighbors touch one central pellet at $90^\circ$) or in triangular order (as in the rosette figure above, where six pellets P0–P5 surround one central pellet O along hexagon HEX with equal center-to-center quadrances $Q = 4R^2$). Now stack such layers in three dimensions:
Stacking square layers directly atop square layers yields simple cubic order, where each sphere is touched by six neighbors ($\eta_{\text{cubic}} = \frac{\pi}{6} \approx 52.36\%$).
Seating each sphere of a square layer into the hollow pit of four spheres below—or equivalently seating each sphere of a triangular-hexagonal layer into the hollow of three spheres below—yields pyramidal close-packing, where every interior sphere is kissed simultaneously by twelve neighbors (six in its own horizontal plane, three above, and three below)!
“The fitting together will become the **tightest possible*, so that in no other arrangement whatsoever could more globules be packed into the same vessel”* (Coaptatio fiet arctissima, ut nullo praeterea ordine plures globuli in idem vas compingi queant — the Kepler Conjecture, $\eta_{\text{FCC}} = \frac{\pi}{3\sqrt{2}} \approx 74.048\%$).
And if uniform swelling or external pressure squeezes these twelve-kissed spheres until all interstitial voids vanish, each sphere deforms into an exact Rhombic Dodecahedron!
4. Pentamerous Blossoms & Self-Propagating Fibonacci Convergence¶
Why then do snowflakes exhibit sixfold crystalline rays while the blossoms of apple, pear, rose, and cinquefoil unfold in fivefold petals (ABCDE), never sixfold or sevenfold? Because crystalline vapor condenses under cold spatial geometry ($60^\circ$ planar close-packing), whereas living plants are fashioned after an archetypal formative faculty grounded in Euclid’s self-propagating Divine Proportion!
For when a line is cut in extreme and mean ratio ($y : x = x : (x + y)$), adding the whole $x + y$ to the greater part $x$ produces a new line cut once again in the very same divine ratio ([XIII.5]). Since the generative faculty of seeds cannot measure irrational radicals directly, Nature imitates this self-propagating proportion in whole numbers:
“Let the smallest terms be $1$ and $1$, which you may imagine unequal; add them, making $2$; add to this the former $1$, making $3$; add $2$ to this, making $5$; add $3$, making $8$; add $5$, making $13$; add $8$, making $21$. For always, as $5$ is to $8$, so is $8$ to $13$ very nearly, and as $8$ is to $13$, so is $13$ to $21$ very nearly. After the likeness of this **self-propagating proportion*, I hold that the seminal faculty is fashioned—and thus in the flower is displayed the authentic flag of pentamery, the pentagon!”*
Yet why do the snow star’s six rays spread flat in a single plane rather than sprouting along three orthogonal octahedral axes? Leaving chemistry and physics open at the doorway of the alchemist’s laboratory, Kepler signs off mischievously before his New Year’s snowflake melts away into Nothing!