CHAPTER XLV

How the Octocedron Is Placed in the Duodecedron.

Returning now to the duodecedron, the eleventh proposition of the fifteenth book (Campanus XV.11) shows how to inscribe the octocedron (8 triangular bases) inside the duodecedron (12 pentagonal bases)—a construction of singular elegance because the six vertices of the octocedron do not touch the faces or vertices of the duodecedron, but rather the midpoints of six of its thirty edges!

Inscribing the Octahedron in the Dodecahedron (Campanus XV.11)

In a given regular body of twelve pentagonal bases (duodecedron), to inscribe the regular body of eight equilateral triangular bases (octocedron) so that the six solid angles of the octocedron stand upon the midpoints of six opposite edges of the duodecedron.

Construction and Proof (Campanus XV.11, via [XIII.17] and [XIII.14])

  1. Recall from Chapter XXX ([XIII.17]) how the duodecedron was built by erecting six hipped roofs over the six square faces of an inscribed cube: over each of the six square faces of that underlying cube there stands one ridge edge (corausto or roof-crest, such as \(vx\) in 0035.g1.png) of the duodecedron, parallel to the median of that square face of the cube and lying directly above its center!

  2. Thus among the thirty edges of the duodecedron, there is a set of six privileged edges—three pairs of opposite, parallel roof-ridge edges—whose six midpoints lie directly on the three mutually perpendicular axes passing through the center of the duodecedron (and of the underlying cube) and perpendicular to the six faces of that cube (see the marginal diagram 0041.g2.png on Folio 0041).

  3. Mark the midpoints \(f, g, h, k, l, m\) of these six opposite edges of the duodecedron. The three straight lines joining the three opposite pairs of midpoints pass straight through the center of the duodecedron, are all equal in length (each being equal to the diameter of the sphere that touches those six edge-midpoints), and intersect one another at right angles at the center!

  4. Join the six midpoints \(f, g, h, k, l, m\) to one another by twelve straight lines. By the exact same proof as in Chapter XXVIII ([XIII.14]), because the three axes are mutually equal and bisect one another at right angles at the center, all twelve connecting edges are mutually equal (each having its square equal to half the square on the axis joining two opposite midpoints), and all eight faces are equilateral triangles!

Therefore the regular octocedron is inscribed in the duodecedron, with its six vertices standing at the midpoints of six opposite edges of the duodecedron.