CHAPTER XXXII — Critical Commentary & Notes

On the Proportion of the Said Regular Bodies to One Another, Especially that of the Ycocedron to the Duodecedron.

Historical & Mathematical Commentary

• Hypsicles’ Book XIV: Book XIV of the Elements was written by Hypsicles of Alexandria (2nd century BCE), who cites the lost treatise On the Comparison of the Five Figures by Aristaeus the Elder and the treatise on the dodecahedron and icosahedron by Apollonius of Perga.

• Mathematical Identity: For a sphere of diameter \(D\):

  • The circumradius \(\rho\) of a face of the dodecahedron and of a face of the icosahedron is identical: \(\rho^{2} = \frac{5 - \sqrt{5}}{30} D^{2}\)? Wait: \(a_{20}^{2} = 3\rho^{2}\) and \(a_{20}^{2} = \frac{5-\sqrt{5}}{10}D^{2}\), so \(\rho^{2} = \frac{5-\sqrt{5}}{30}D^{2} = \frac{3-\sqrt{5}}{6}\cdot\dots\) Wait: for the dodecahedron, \(a_{12} = \frac{\sqrt{5}-1}{2}\frac{D}{\sqrt{3}}\), so \(a_{12}^{2} = \frac{6-2\sqrt{5}}{12}D^{2} = \frac{3-\sqrt{5}}{6}D^{2}\). The circumradius \(\rho_{12}\) of a regular pentagon of side \(a_{12}\) satisfies \(a_{12}^{2} = \frac{5-\sqrt{5}}{2}\rho_{12}^{2}\), so \(\rho_{12}^{2} = \frac{2}{5-\sqrt{5}}\frac{3-\sqrt{5}}{6}D^{2} = \frac{(5+\sqrt{5})(3-\sqrt{5})}{60}D^{2} = \frac{10 - 2\sqrt{5}}{60}D^{2} = \frac{5-\sqrt{5}}{30}D^{2} = \rho_{20}^{2}\)!

  • Because both solids have the exact same face circumradius \(\rho\) and therefore the exact same insphere radius \(r_{\text{in}} = \sqrt{\frac{D^{2}}{4} - \rho^{2}} = D\sqrt{\frac{5+2\sqrt{5}}{60}}\), the ratio of their volumes equals the ratio of their surface areas: \(\frac{V_{20}}{V_{12}} = \frac{A_{20}}{A_{12}} = \frac{s_{\text{cube}}}{s_{20}} = \sqrt{\frac{1/3}{(5-\sqrt{5})/10}} = \sqrt{\frac{5+\sqrt{5}}{6}} \approx 1.09818\).

Philological and Folio Apparatus

• 1509 Paganini Witness Folios: 0040 (f. 12v–f. 12v)

• Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/032_cap_XXXII.txt

• Winterberg 1889 German Source: winterberg/03_divina_proportione_german/032_cap_XXXII.txt

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