CHAPTER XL

How the Duodecedron Is Formed in the Ycocedron.

The sixth proposition of the fifteenth book (Campanus XV.6) demonstrates how to inscribe the duodecedron (12 pentagonal bases) inside the ycocedron (20 triangular bases).

Inscribing the Dodecahedron in the Icosahedron (Campanus XV.6)

In a given regular body of twenty equilateral triangular bases (ycocedron), to inscribe the regular body of twelve equilateral and equiangular pentagonal bases (duodecedron) so that the twenty vertices of the duodecedron stand at the twenty centers of the triangular faces of the ycocedron.

Construction and Proof ([IV.4], [XIII.16], [XIII.7])

  1. In each of the twenty equilateral triangular faces of the given ycocedron, find the center of that triangle ([IV.4]), giving twenty points in all—one at the middle of each face of the ycocedron (see the marginal diagram 0039.g3.png on Folio 0039).

  2. Around each of the twelve solid angles (vertices) of the ycocedron, exactly five equilateral triangular faces meet (Chapter XXV, Chapter XXIX). Join in order the five centers of the five triangles surrounding each vertex of the ycocedron by five straight lines.

  3. Because the five triangles meeting at a vertex of the ycocedron are mutually equal and similarly inclined around the axis passing from the center of the ycocedron through that vertex, the five straight lines joining their centers lie in one plane perpendicular to that axis and form an equilateral and equiangular pentagon.

  4. Since the ycocedron has twelve vertices, and around each vertex five triangular faces meet, joining the centers around all twelve vertices produces twelve equal regular pentagons bounded by thirty equal edges meeting three by three at the twenty face-centers of the ycocedron.

Therefore the regular duodecedron is inscribed in the ycocedron, with its twenty solid angles touching the centers of the twenty triangular bases of the ycocedron.