CHAPTER XLIV — Critical Commentary & Notes¶
How the Ycocedron Is Placed in the Cube.
Historical & Mathematical Commentary¶
Two Ways to Inscribe the Icosahedron in the Cube:
Edge-on-Face Inscription (Golden Rectangles): An icosahedron of edge \(a_{20}\) has 12 vertices that form three mutually perpendicular golden rectangles of dimensions \(a_{20} \times \varphi\,a_{20}\). Placing these three golden rectangles inside a cube of side \(s_{\text{cube}} = \varphi\,a_{20}\) puts 6 edges of the icosahedron (and all 12 of its vertices!) directly on the 6 square faces of the cube, with the cube edge and icosahedron edge in the Divine Proportion (\(s_{\text{cube}} : a_{20} = \varphi : 1\)).
Campanus XV.10 Inscription: Campanus routes the construction through the octahedron inscribed in the cube (XV.3) and the circles of Euclid XIII.16.
Philological and Folio Apparatus¶
1509 Paganini Witness Folios: 0042 (f. 13v–f. 13v)
Winterberg 1889 Italian Source:
winterberg/01_divina_proportione_italian/044_cap_XLIV.txtWinterberg 1889 German Source:
winterberg/03_divina_proportione_german/044_cap_XLIV.txtMarginal Woodcut Diagrams: 0 figure(s) extracted from the 1509 Venice witness.