CHAPTER XLI

How the Exacedron or Cube Is Placed in the Ycocedron.

The seventh proposition of the fifteenth book (Campanus XV.7) shows how to inscribe the cube (6 square bases) inside the ycocedron (20 triangular bases) by means of the duodecedron first inscribed in the ycocedron.

Inscribing the Cube in the Icosahedron (Campanus XV.7)

In a given regular body of twenty triangular bases (ycocedron), to inscribe the cube (exacedron) so that the eight vertices of the cube stand at eight of the twenty centers of the triangular faces of the ycocedron.

Construction and Proof (via Campanus XV.6 and [XIII.17])

  1. First, in the given ycocedron inscribe the duodecedron by the method of Chapter XL (Campanus XV.6), so that the twenty vertices of the duodecedron stand at the centers of the twenty triangular faces of the ycocedron (see the marginal diagram 0040.g1.png on Folio 0040).

  2. Recall from Chapter XXX ([XIII.17]) how the duodecedron was originally constructed upon the six square faces of a cube: twelve of the twenty vertices of the duodecedron are the roof-ridge vertices, while the remaining eight vertices of the duodecedron are the eight corners of the underlying cube, and the twelve edges of that cube are twelve diagonals (corde pentagoniche) drawn across the twelve pentagonal faces of the duodecedron!

  3. Accordingly, in the twelve pentagonal faces of the duodecedron inscribed in our ycocedron, draw the twelve pentagonal chords (corde pentagoniche) connecting those eight corresponding vertices.

  4. By [XIII.17] and [XIII.8], these twelve pentagonal chords are mutually equal, meet at right angles in six square faces, and have their eight vertices among the twenty vertices of the duodecedron—which are themselves the centers of eight of the twenty triangular faces of the ycocedron!

Therefore the cube is inscribed in the ycocedron, and its edge is the pentagonal chord (corda pentagonica) of the duodecedron inscribed in the same ycocedron, so that the edge of this cube and the edge of the inscribed duodecedron stand to one another in our Divine Proportion!