CHAPTER LII¶
On the Disposition of the Duodecedron: Regular (Plates XXV–XXVI), Truncated (Plates XXVII–XXVIII), Elevated (Plates XXIX–XXX), and the Elevated Truncated Bodies (Plates XXXI–XXXIV).
Now we describe the fifth and noblest regular body, the duodecedron of twelve regular pentagonal bases, together with its truncated (absciso) and elevated (elevato) dependents, as well as the further bodies formed by elevating the truncated bodies (abscisi elevati) (Plates XXV–XXXIV).
1. The Regular Duodecedron — Solid (Plate XXV) and Hollow (Plate XXVI)
Plate XXV (Folio 0241, Duodecedron Planum Solidu[m]) and Plate XXVI (Folio 0243, Duodecedron Planum Vacuum): This is the fifth regular body, whose construction in the sphere by means of our Divine Proportion we demonstrated in Chapter XXX ([XIII.17]). It has 12 equilateral and equiangular pentagonal bases, 20 solid angles (each contained by 3 pentagonal angles of \(1\frac{1}{5}R\)), and 30 equal edges.
2. The Truncated Dodecahedron (Duodecedron Abscisum / Icosidodecahedron) — Solid (Plate XXVII) and Hollow (Plate XXVIII)
Plate XXVII (Folio 0245, Duodecedron Abscisum Solidu[m]) and Plate XXVIII (Folio 0247, Duodecedron Abscisum Vacuum): This dependent body is formed from the regular duodecedron by bisecting all 30 edges at their midpoints and slicing off all 20 three-faced solid angles by plane cuts passing through those midpoints:
Because 3 pentagonal faces meet at each of the 20 vertices of the duodecedron, slicing off the 20 vertices produces 20 equilateral triangles (whose sides equal half the pentagonal chord of the original duodecedron, so that the original edge and the new edge stand in the Divine Proportion!).
Each of the 12 original pentagonal faces, having its five corners cut off at the midpoints of its five sides, is reduced to an inscribed regular pentagon sharing the exact same edges as the 20 new equilateral triangles!
Thus the duodecedron absciso (the icosidodecahedron) has 32 plane bases in all (12 regular pentagons and 20 equilateral triangles), 30 solid angles (at the 30 midpoints of the original edges, each vertex surrounded by two pentagons and two triangles), and 60 equal edges.
3. The Elevated Dodecahedron (Duodecedron Elevatum) — Solid (Plate XXIX) and Hollow (Plate XXX)
Plate XXIX (Folio 0249, Duodecedron Elevatum Solidu[m]) and Plate XXX (Folio 0251, Duodecedron Elevatum Vacuum): This body is formed by erecting upon each of the 12 regular pentagonal bases of the core duodecedron an equilateral pentagonal pyramid (each bounded by 5 equilateral triangles meeting at an outer apex or cone—recall that 5 equilateral triangles have angle sum \(5 \times \frac{2}{3}R = 3\frac{1}{3}R < 4R\), so they form a genuine solid pyramid over a regular pentagon!):
Externally it is bounded by \(12 \times 5 =\) 60 equilateral triangular faces (while the 12 pentagonal bases of the core duodecedron lie within, visible in the hollow skeletal perspective of Plate XXX).
It has 32 vertices in all (20 interior re-entrant vertices of the core duodecedron and 12 exterior projecting cones of the twelve pentagonal pyramids) and 90 equal edges (the 30 edges of the core duodecedron plus \(12 \times 5 = 60\) rising edges).
4. The Elevated Truncated Cube (Exacedron Abscisum Elevatum) and Elevated Truncated Dodecahedron (Duodecedron Abscisum Elevatum) — Plates XXXI–XXXIV
By the same art, one can also elevate the truncated bodies (abscisi elevati) by erecting pyramids upon all the faces of a truncated body:
Plate XXXI (Folio 0253, Exacedron Abscisum Elevatum Solidu[m]) and Plate XXXII (Folio 0255, Exacedron Abscisum Elevatum Vacuum): Formed by taking the 14-base exacedron absciso (the cuboctahedron of 6 squares and 8 equilateral triangles, Plates IX–X) and erecting 6 square pyramids upon its 6 square bases (\(6 \times 4 = 24\) triangles) and 8 triangular pyramids upon its 8 triangular bases (\(8 \times 3 = 24\) triangles), yielding a star-like body bounded externally by \(24 + 24 =\) 48 triangular faces and 14 projecting cones!
Plate XXXIII (Folio 0257, Duodecedron Abscisum Elevatum Solidu[m]) and Plate XXXIV (Folio 0259, Duodecedron Abscisum Elevatum Vacuum): Formed by taking the 32-base duodecedron absciso (the icosidodecahedron of 12 pentagons and 20 equilateral triangles, Plates XXVII–XXVIII) and erecting 12 pentagonal pyramids upon its 12 pentagonal bases (\(12 \times 5 = 60\) triangles) and 20 triangular pyramids upon its 20 triangular bases (\(20 \times 3 = 60\) triangles), yielding a magnificent star-like body bounded externally by \(60 + 60 =\) 120 triangular faces and 32 projecting cones!