UNIT IV SUMMARY — DISTINCTIO V–VI: SOLIDS & THE FIVE REGULAR POLYHEDRA¶
Structural Synopsis of Boncompagni Vol. II, pp. 148–202 (fol. 92v–130v)
English Edition · Latin (1220 / Boncompagni 1862) · GEOMETOR Notes
Overview¶
Units V and VI ascend from planar fields to three-dimensional bodies: first building the arithmetic of cube roots (Distinctio V), then establishing the mensuration of prisms, pyramids, frustums, and spheres (Distinctio VI, Part 1), and culminating in the exact quadratic and biquadratic radical invariants of all five regular polyhedra inscribed in a sphere of diameter $d = 6$ (Distinctio VI, Part 2).
Exact Invariants of the Five Platonic Solids in Sphere $d = 6$ ($R = 3$)¶
Regular Solid |
Edge Squared $e^2$ |
Exact Edge $e$ |
Exact Solid Volume $V$ |
|---|---|---|---|
Tetrahedron ($4$ faces) |
$\frac{2}{3} d^2 = 24$ |
$2\sqrt{6}$ |
$8\sqrt{3} = \sqrt{192}$ |
Cube ($6$ faces) |
$\frac{1}{3} d^2 = 12$ |
$2\sqrt{3}$ |
$24\sqrt{3} = \sqrt{1{,}728}$ |
Octahedron ($8$ faces) |
$\frac{1}{2} d^2 = 18$ |
$3\sqrt{2}$ |
$36$ (exact integer!) |
Dodecahedron ($12$ faces) |
$18 - 6\sqrt{5} = 18 - \sqrt{180}$ |
$\sqrt{15} - \sqrt{3} = \frac{2\sqrt{3}}{\phi}$ |
$30\sqrt{3} + 6\sqrt{15}$ |
Icosahedron ($20$ faces) |
$18 - \frac{18}{5}\sqrt{5} = 18 - \sqrt{64\frac{4}{5}}$ |
$\frac{3\sqrt{10 - 2\sqrt{5}}}{\sqrt{5}}$ |
$18\sqrt{10 + 2\sqrt{5}}$ |