CHAPTER LIII — Critical Commentary & Notes

On the Formation and Origin of the Body of Twenty-Six Bases: Eighteen Squares and Eight Equilateral Triangles (Plates XXXV–XXXVIII).

Historical & Mathematical Commentary

• Paganini 1509 Plates Referenced:

  • Plate XXXV (0261.rst): Vigintisex Basium Planum Solidum (Rhombicuboctahedron, Archimedean solid \((3.4^3)\) of 18 squares and 8 equilateral triangles, solid).

  • Plate XXXVI (0263.rst): Vigintisex Basium Planum Vacuum (Rhombicuboctahedron, skeletal).

  • Plate XXXVII (0265.rst): Vigintisex Basium Elevatum Solidum (Cumulated rhombicuboctahedron, 96 triangular faces, solid).

  • Plate XXXVIII (0267.rst): Vigintisex Basium Elevatum Vacuum (Cumulated rhombicuboctahedron, skeletal).

• Jacopo de’ Barbari’s 1495 Portrait of Pacioli: The glass Vigintisex Basium half-filled with water hanging from a thin cord in Jacopo de’ Barbari’s famous 1495 Capodimonte portrait of Fra Luca Pacioli (and the wooden dodecahedron resting on Pacioli's Summa) is this exact 26-base rhombicuboctahedron!

Constantin Winterberg’s Commentary (1889 Edition)

• Winterberg Footnotes 1–2 (Cap. LIII, p. 243): (1) Compare Chapter LXXI. (2) Namely, the 8 edges of the regular octagon lying in the same plane—a construction that can also be found by purely elementary geometry without the ‘profound science’ presupposed by the author. (Winterberg Fussnoten 1–2 (Cap. LIII, S. 243): 1) cfr. Cap. LXXI. 2) scil. welche in derselben Ebene liegen, was übrigens auch ohne die vom Autor vorausgesetzten „tiefen Kenntnisse“ auf rein elementarem Wege zu finden ist.)

Philological and Folio Apparatus

• 1509 Paganini Witness Folios: 0046, 0047 (f. 15v–f. 16r)

• Winterberg 1889 Italian Source: winterberg/01_divina_proportione_italian/053_cap_LIII.txt

• Winterberg 1889 German Source: winterberg/03_divina_proportione_german/053_cap_LIII.txt

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