CHAPTER XXXVI — Critical Commentary & Notes¶
How the Tetracedron Is Placed in the Cube.
Historical & Mathematical Commentary¶
Johannes Kepler’s *:term:`Stella Octangula <octahedron-elevatum>`* and the Volume Ratio: Slicing away the four right-angled corner pyramids (\(abcf\), \(acdh\), \(aefh\), \(cfgh\)) from the cube of side \(s\) leaves the regular tetrahedron \(afch\) of edge \(s\sqrt{2}\). Since each corner pyramid has three mutually perpendicular legs of length \(s\) and therefore volume \(\frac{1}{6}s^{3}\), the four corner pyramids have total volume \(4 \times \frac{1}{6}s^{3} = \frac{2}{3}s^{3}\), leaving one-third of the cube’s volume (\(\frac{1}{3}s^{3}\)) for the inscribed tetrahedron!
Philological and Folio Apparatus¶
1509 Paganini Witness Folios: 0041 (f. 13r–f. 13r)
Winterberg 1889 Italian Source:
winterberg/01_divina_proportione_italian/036_cap_XXXVI.txtWinterberg 1889 German Source:
winterberg/03_divina_proportione_german/036_cap_XXXVI.txtMarginal Woodcut Diagrams: 0 figure(s) extracted from the 1509 Venice witness.