CHAPTER LIII¶
On the Formation and Origin of the Body of Twenty-Six Bases: Eighteen Squares and Eight Equilateral Triangles (Plates XXXV–XXXVIII).
Beyond the immediate truncations and elevations of the five regular bodies, another most noble and useful dependent body is generated from the cube, called the Vigintisex Basium (the Body of Twenty-Six Bases, modern rhombicuboctahedron), shown both plain and elevated in Plates XXXV–XXXVIII.
1. The Twenty-Six-Base Body (Vigintisex Basium) — Solid (Plate XXXV) and Hollow (Plate XXXVI)
Plate XXXV (Folio 0261, Vigintisex Basium Planum Solidu[m]) and Plate XXXVI (Folio 0263, Vigintisex Basium Planum Vacuum): This body of 26 bases arises from the regular cube by uniform beveling (smussatura) of both its 12 edges and its 8 solid angles:
Imagine a cube whose 6 square faces are cut parallel to its 12 edges at an equal depth on every face so that:
In the middle of each of the 6 original square faces of the cube there remains an inner square (6 original squares).
Along each of the 12 bevelled edges of the cube, where two adjacent square faces met at right angles, there is formed a new square inclined equally (\(45^{\circ}\)) to both adjacent faces (12 edge squares).
At each of the 8 sliced-off corners (solid angles) of the cube, where three bevelled edges meet, there is formed an equilateral triangle whose three sides are the ends of those three edge-squares (8 corner equilateral triangles).
Thus the whole body is bounded by 26 equilateral and equiangular plane bases in all:
18 equal squares (\(6 + 12 = 18\));
8 equal equilateral triangles;
all sharing the exact same edge length!
It has 24 solid angles (each vertex surrounded by three squares and one equilateral triangle) and 48 equal edges.
Notice also its admirable octogonal cross-sections: around the body run three mutually perpendicular octogonal belts, each consisting of 8 consecutive squares (\(4\) from the original cube faces alternating with \(4\) from the bevelled edges), so that the cross-section is a regular octagon! And for this reason this body is of the greatest utility to architects, sculptors, and perspective painters (architecti, scultori, e prospectivi), since from a cubic block of marble or timber one immediately derives the regular octagon in three orthogonal planes for capitals, bases, lanterns, and cupolas.
2. The Elevated Twenty-Six-Base Body (Vigintisex Basium Elevatum) — Solid (Plate XXXVII) and Hollow (Plate XXXVIII)
Plate XXXVII (Folio 0265, Vigintisex Basium Elevatum Solidu[m]) and Plate XXXVIII (Folio 0267, Vigintisex Basium Elevatum Vacuum): Formed by erecting 18 equilateral square pyramids upon the 18 square bases of the 26-base body (\(18 \times 4 = 72\) triangular faces) and 8 equilateral triangular pyramids upon its 8 triangular bases (\(8 \times 3 = 24\) triangular faces), producing a magnificent star-like polyhedron bounded externally by \(72 + 24 =\) 96 equilateral triangular faces and 26 projecting cones!