UNIT IV NOTES — SCHOLARLY & SYMBOLIC COMMENTARY

From Fibonacci’s Sphere of Diameter Six to Piero della Francesca and Luca Pacioli’s De Divina Proportione

English Edition · Latin (1220 / Boncompagni 1862) · Summary

1. Cube-Root Extraction & the $3a(a+1)+1$ Finite Difference

Why does Leonardo divide the cubic remainder $R = N - a^3$ by $3a(a+1) + 1$? Because the exact discrete difference between consecutive integer cubes is:

$$(a + 1)^3 - a^3 = 3a^2 + 3a + 1 = 3a(a + 1) + 1.$$

Thus linear interpolation of $x \mapsto x^3$ across the unit interval $[a, a+1]$ yields the exact secant lower bound $a + \frac{R}{3a(a+1)+1} < \sqrt[3]{a^3 + R} < a + \frac{R}{3a^2}$.

2. The Egyptian–Euclidean Frustum Formula

Leonardo’s truncated-pyramid formula $V = \frac{h}{3}(B_1 + \sqrt{B_1 B_2} + B_2)$ generalizes the celebrated Moscow Mathematical Papyrus Problem $14$ ($V = \frac{h}{3}(a^2 + ab + b^2)$) from square bases to any pair of homothetic planar bases of areas $B_1, B_2$. Since linear similarity factor $\lambda = \sqrt{B_2 / B_1}$ implies $(1 - \lambda^3)/(1 - \lambda) = 1 + \lambda + \lambda^2$, multiplying by $\frac{1}{3} B_1 h$ immediately yields $\frac{h}{3}(B_1 + \sqrt{B_1 B_2} + B_2)$.

3. Direct Lineage to Piero della Francesca & Luca Pacioli

Modern scholarship (G. Arrighi, M. Clagett, E. Giusti) has established that Distinctio VI (Boncompagni Vol. II, pp. 190–202) is the direct mathematical fountainhead of Renaissance polyhedral geometry:

  • Piero della Francesca transcribed and vernacularized Leonardo’s sphere-diameter-$6$ polyhedra calculations into his Trattato d’abaco and Libellus de quinque corporibus regularibus (c. $1480$);

  • Luca Pacioli incorporated those exact numerical derivations—retaining $e_{12} = \sqrt{15} - \sqrt{3}$, $af^2 = 6 - \sqrt{7\frac{1}{5}}$, and $hf^2 = 3 + \sqrt{7\frac{1}{5}}$—into Summa de arithmetica ($1494$) and De Divina Proportione ($1509$, pacioli:divina, cross-referenced in geometor/pacioli).

In geometor.fibonacci.polyhedra_golden, polyhedra_in_sphere(6), construct_sphere_polyhedra_section(), and construct_dodecahedron_golden_chord() verify every single radical identity symbolically in SymPy:

  • sp.simplify((sp.sqrt(15) - sp.sqrt(3))**2 - (18 - sp.sqrt(180))) == 0;

  • sp.simplify((V_12 / V_20)**2 - (5 + sp.sqrt(5)) / 6) == 0, confirming Hypsicles’ Book XIV volume ratio ($V_{12} / V_{20} = e_6 / e_{20}$).