CHAPTER XLIV

How the Ycocedron Is Placed in the Cube.

Before completing the remaining inscriptions in the duodecedron, Campanus places as the tenth proposition of the fifteenth book (Campanus XV.10) the inscription of the ycocedron (20 triangular bases) inside the cube (6 square bases).

Inscribing the Icosahedron in the Cube (Campanus XV.10)

In a given cube (exacedron), to inscribe the regular body of twenty equilateral triangular bases (ycocedron) so that the twelve vertices of the ycocedron stand in pairs along six median lines on the six square faces of the cube (or at the twelve centers via the inscribed octocedron/duodecedron).

Construction and Proof (Campanus XV.10, via XV.3 and [XIII.16])

  1. Campanus demonstrates this inscription by first inscribing the octocedron in the given cube by the method of Chapter XXXVII (Campanus XV.3), so that the six vertices of the octocedron stand at the six centers of the square faces of the cube, and the three axes of the octocedron join the opposite face-centers of the cube (see the marginal diagram 0041.g1.png on Folio 0041).

  2. Alternatively—and even more manifestly from the Divine Proportion—on the six square faces of the cube, draw six median lines through the six face-centers in three mutually perpendicular directions (so that on each pair of opposite square faces the medians are parallel, and on each adjacent face the median is perpendicular to the shared edge, just as we did in Chapter XXX when erecting the roofs of the duodecedron).

  3. Divide the half-median of each square face according to our Divine Proportion, and mark on each of the six face-medians two points equidistant from the face-center by half the lesser (or greater) segment so that six edges of the ycocedron lie flat on the six square faces of the cube, and all twelve vertices of the ycocedron touch the six square faces of the cube!

  4. In Campanus’s construction (XV.10), the twelve vertices of the ycocedron lie on the circles inscribed in the faces (or along the twelve edges of the inscribed octocedron divided in extreme and mean ratio), forming twenty equilateral triangles whose twelve vertices touch the interior of the cube.

Thus the regular ycocedron is inscribed in the cube.