Unit II: Capillarity, Cellular Foams, the Bee’s Cell, and Spicular Skeletons — Summary

Analytical Synopsis — 1917 Cambridge First Edition — Chapters V–X (pp. 201–492)

Overview of Unit II (Chapters V–X)

Unit II traces the geometry of surface tension, minimal area partitions, and biomineralized skeletons across six closely linked chapters of the 1917 First Edition (pp. 201–492):

1. Chapter V — The Forms of Cells (pp. 201–276): Applies the Laplace–Young capillary law $\Delta P = T(1/R_1 + 1/R_2)$ and Joseph Plateau's six axisymmetric surfaces of constant mean curvature—plane, sphere, cylinder, catenoid, unduloid, and nodoid (plateau:1873)—to solitary protists (Lagena, Amoeba, ciliates, trypanosomes, and red blood corpuscles).

2. Chapter VI — A Note on Adsorption (pp. 277–292): Explains Gibbs–Thomson interfacial concentration at cell membranes, paving the way for localized pellicles, cell walls, and mineral spicule deposition along liquid edges.

3. Chapter VII — The Forms of Tissues, or Cell-Aggregates (pp. 293–345): Establishes Plateau's laws for 2D and 3D cellular froths:

   • Planar triple junctions meet at $120^\circ$ (Fermat–Torricelli point X(13)), yielding hexagonal epithelia.

   • The Bee’s Cell (pp. 327–336): Traces the two-century mathematical drama from Pappus and Kepler (kepler:1611) through Maraldi (maraldi:1712), Réaumur, König, Maclaurin (maclaurin:1743), and Boscovich (boscovich:1760), demonstrating that the three-rhombus pyramidal cap of the honeycomb with Maraldi angle $\arccos(-1/3) = 109^\circ 28' 16''$ arises automatically from mutual compression and soap-film tension rather than geometric calculation by the bee.

   • Parenchyma & Kelvin’s Foam (pp. 336–345): Connects Plateau’s wire-cube soap film (Fig. 134) to Lord Kelvin's space-filling 14-hedral tetrakaidecahedron (kelvin:1887).

4. Chapter VIII — The Forms of Tissues, continued (pp. 346–410): Derives Sachs’s law of orthogonal cell-plate intersection and Errera’s minimal-area cleavage partitions in dividing embryos, cambium, and cork.

5. Chapter IX — On Concretions, Spicules, and Spicular Skeletons (pp. 411–487): Shows how sponge triradiate ($120^\circ$) and tetraradiate ($109^\circ 28'$) spicules crystallize within intercellular Plateau channels, and reinterprets Ernst Haeckel's Challenger Radiolaria (haeckel:1887) via spherical foam geometry and Euler’s polyhedral theorem.

6. Chapter X — A Parenthetic Note on Geodetics (pp. 488–492): Explains helical thickenings in plant vessels and Spirogyra chloroplasts as geodesics minimizing elastic bending energy on cylindrical walls.