UNIT V — GEOMETRIC & STEREOMETRIC SUMMARY¶

Exact Symbolic Invariants of Tractatus Geometriae (geometor.pacioli.tractatus_geometriae)

1. Heron’s (13, 14, 15) Archetype (Part II, fol. 9v)¶

Coordinates: $G = (0, 0)$, $B = (14, 0)$, $A = (5, 12)$, Foot $D = (5, 0)$, Incenter $I = (6, 4)$, Circumcenter $O = \left(7, \frac{33}{8}\right)$.
Heron Product: $s = 21$, $(s - c, s - b, s - a) = (6, 7, 8)$, $\Delta = \sqrt{21 \cdot 6 \cdot 7 \cdot 8} = \sqrt{7056} = 84$.
Radii & Altitude: Cathetus $AD = 12$, Inradius $r = \Delta / s = 4$, Circumradius $R = \frac{abc}{4\Delta} = \frac{65}{8}$.

Vertex Cevian (`VI.1`): Dividing $GB$ at $K = \left(\frac{14p}{p+q}, 0\right)$ splits area $84$ into $\frac{84p}{p+q}$ and $\frac{84q}{p+q}$ (for $1:2$, $K = \left(\frac{14}{3}, 0\right)$, areas $28$ and $56$).
Parallel Cut (`VI.13` + `VI.19`): Constructing geometric mean $GU = \sqrt{14 \cdot \frac{14p}{p+q}}$ on the semicircle over $GB$ cuts off $\triangle GUV \sim \triangle GBA$ of exact rational area $\frac{84p}{p+q}$ despite radical vertex coordinates $U = (7\sqrt{2}, 0)$, $V = \left(\frac{5\sqrt{2}}{2}, 6\sqrt{2}\right)$ for $p:q = 1:1$.

Exact Stereometric Table for Sphere Diameter $D = 6$ ($R = 3$)¶
Inscribed Solid	Possanza ($e^2$)	Exact Edge ($e$)	Surface Area	Exact Volume
Tetrahedron (XIII.13)	$24$	$2\sqrt{6}$	$24\sqrt{3}$	$8\sqrt{3}$
Cube (XIII.15)	$12$	$2\sqrt{3}$	$72$	$24\sqrt{3}$
Octahedron (XIII.14)	$18$	$3\sqrt{2}$	$36\sqrt{3}$	$36$
Icosahedron (XIII.16)	$\frac{18(5 - \sqrt{5})}{5}$	$3\sqrt{\frac{10 - 2\sqrt{5}}{5}}$	$18\sqrt{3}(5 - \sqrt{5})$	$9\sqrt{10 + 2\sqrt{5}}$
Dodecahedron (XIII.17)	$18 - 6\sqrt{5}$	$\sqrt{15} - \sqrt{3}$	$18\sqrt{10(5 - \sqrt{5})}$	$30\sqrt{3} + 6\sqrt{15}$
Truncated Tetrahedron (fol. 69v)	$\frac{72}{11} = 6\frac{6}{11}$	$\frac{6\sqrt{22}}{11}$	$\frac{504\sqrt{3}}{11}$	$\frac{1656\sqrt{11}}{121}$
Cuboctahedron (fol. 69v)	$9$	$3$	$54 + 18\sqrt{3}$	$45\sqrt{2}$