CHAPTER LI¶
On the Disposition of the Ycocedron: Regular (Plates XIX–XX), Truncated (Plates XXI–XXII), and Elevated (Plates XXIII–XXIV).
We come now to the fourth regular body, the ycocedron of twenty equilateral triangular bases, and its truncated (absciso) and elevated (elevato) dependent bodies in solid and hollow perspective (Plates XIX–XXIV).
1. The Regular Ycocedron — Solid (Plate XIX) and Hollow (Plate XX)
Plate XIX (Folio 0229, Ycocedron Planum Solidu[m]) and Plate XX (Folio 0231, Ycocedron Planum Vacuum): This is the fourth regular body, whose construction in the sphere by means of our Divine Proportion we demonstrated in Chapter XXIX ([XIII.16]). It has 20 equilateral triangular bases, 12 solid angles (each contained by 5 plane angles of \(\frac{2}{3}R\)), and 30 equal edges.
2. The Truncated Icosahedron (Ycocedron Abscisum) — Solid (Plate XXI) and Hollow (Plate XXII)
Plate XXI (Folio 0233, Ycocedron Abscisum Solidu[m]) and Plate XXII (Folio 0235, Ycocedron Abscisum Vacuum): This celebrated body is derived from the regular ycocedron by dividing each of its 30 edges into three equal parts (\(\frac{1}{3}, \frac{1}{3}, \frac{1}{3}\)) and slicing off all 12 five-faced solid angles at those trisection points:
Because 5 triangular faces meet at each of the 12 vertices of the ycocedron, slicing off the 12 vertices produces 12 equilateral and equiangular pentagons (whose sides equal \(\frac{1}{3}\) of the original edge of the ycocedron).
Each of the 20 original equilateral triangular faces of the ycocedron, losing its three corners at the trisection points, is transformed into an equilateral and equiangular hexagon (of side \(\frac{1}{3}\) of the original edge)!
Thus the ycocedron absciso has 32 plane bases in all (12 regular pentagons and 20 regular hexagons), 60 solid angles (each formed by two hexagons and one pentagon), and 90 equal edges (30 remaining middle thirds of the original edges plus \(12 \times 5 = 60\) new pentagonal edges)!
3. The Elevated Icosahedron (Ycocedron Elevatum) — Solid (Plate XXIII) and Hollow (Plate XXIV)
Plate XXIII (Folio 0237, Ycocedron Elevatum Solidu[m]) and Plate XXIV (Folio 0239, Ycocedron Elevatum Vacuum): This body is formed by erecting upon each of the 20 equilateral triangular bases of the core ycocedron an equilateral triangular pyramid (whose three rising lateral edges or corausti are either equal to the edge of the ycocedron or, even more nobly, proportioned to it according to our Divine Proportion so that the twenty apices form a circumscribing duodecedron):
Externally it is bounded by \(20 \times 3 =\) 60 triangular faces (while the 20 triangular bases of the core ycocedron lie within, visible in the hollow skeletal perspective of Plate XXIV).
It has 32 vertices in all (12 interior re-entrant vertices of the core ycocedron and 20 exterior projecting cones of the twenty elevated pyramids) and 90 edges (the 30 edges of the core ycocedron plus \(20 \times 3 = 60\) rising edges).