CHAPTER XLV — Critical Commentary & Notes

How the Octocedron Is Placed in the Duodecedron.

Historical & Mathematical Commentary

• Geometric Insight of Campanus XV.11: In Euclid’s construction of the dodecahedron (Euclid XIII.17), 6 of the 30 edges of the dodecahedron are the roof ridges \(vx\) centered over the 6 faces of the inscribed cube. The midpoints of those 6 edges lie on the 3 coordinate axes of the cube at distance \(\frac{1}{2}s_{\text{cube}} + \text{lesser} = \frac{\varphi}{2}s_{\text{cube}} = \frac{\varphi^{2}}{2}a_{12}\) from the origin. Because the 3 coordinate axes are orthogonal and equal in length, those 6 edge-midpoints are the exact vertices of a regular octahedron of edge \(a_{8} = \frac{\varphi^{2}}{\sqrt{2}}a_{12}\)!

Philological and Folio Apparatus

• 1509 Paganini Witness Folios: 0043 (f. 14r–f. 14r)

• Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/045_cap_XLV.txt

• Winterberg 1889 German Source: winterberg/03_divina_proportione_german/045_cap_XLV.txt

• Marginal Woodcut Diagrams: 0 figure(s) extracted from the 1509 Venice witness.