CHAPTER XLVI

How the Exacedron and the Tetracedron Are Placed in the Duodecedron, and Why Only Twelve Inscriptions Are Possible.

Now we complete the twelfth proposition of the fifteenth book (Campanus XV.12 and its corollary), showing how both the cube (exacedron) and the tetracedron are inscribed in the duodecedron, and explaining why only twelve of the twenty theoretical inscriptions can be regularly formed.

Inscribing the Cube and the Tetrahedron in the Dodecahedron (Campanus XV.12)

In a given regular body of twelve pentagonal bases (duodecedron), to inscribe the cube (exacedron) and the pyramid of four triangular bases (tetracedron) so that their vertices stand at vertices of the duodecedron.

Construction and Proof ([XIII.17], [XIII.8], Campanus XV.2)

  1. Inscribing the Cube in the Duodecedron: As was already demonstrated in Chapter XXX ([XIII.17]) and used in Chapter XLI, across each of the twelve pentagonal faces of the given duodecedron draw one pentagonal chord (corda pentagonica, subtending one pentagonal angle) so that the twelve chords meet three by three at eight of the twenty solid angles (vertices) of the duodecedron (see the marginal diagram 0041.g3.png on Folio 0041).

    • By [XIII.8] and [XIII.17], all twelve pentagonal chords are equal, and they form six equal square faces bounding a cube whose eight vertices are eight of the twenty vertices of the duodecedron!

    • Furthermore, by [XIII.8], if the pentagonal chord (which is the edge of this inscribed cube) is divided according to our Divine Proportion, its greater segment is the edge of the duodecedron!

  2. Inscribing the Tetracedron in the Duodecedron: Having inscribed the cube in the duodecedron so that its eight vertices are eight vertices of the duodecedron, inscribe the tetracedron in that cube by the method of Chapter XXXVI (Campanus XV.2), joining four alternating corners of the cube by the six face diagonals of the cube’s six squares.

    • Because the eight vertices of the cube are vertices of the duodecedron, the four vertices of the tetracedron are likewise four of the twenty vertices of the duodecedron (and its six edges are six interior diagonals of the duodecedron)!

Why Only Twelve Inscriptions Are Regularly Assigned, and the Mystery of the Holy Cross in the Octocedron

Let Your Ducal Highness now consider why, out of the \(5 \times 4 = 20\) combinations that one might imagine by pairing each of the five regular bodies with the other four, our author and Campanus assign only these twelve:

  • The Duodecedron Receives All Four: As the figure of the fifth essence and of the universal heaven (il cielo) which embraces all things created, the duodecedron alone receives all four of the other regular bodies within itself: the ycocedron at its 12 face-centers (Cap. XLIII), the octocedron at 6 of its 30 edge-midpoints (Cap. XLV), the cube at 8 of its 20 vertices (Cap. XLVI), and the tetracedron at 4 of its 20 vertices (Cap. XLVI).

  • Why the Ycocedron Receives Three, Denying Only the Octocedron: In the ycocedron (20 triangular faces, 12 vertices, 30 edges), we inscribed the duodecedron (Cap. XL), the cube (Cap. XLI), and the tetracedron (Cap. XLII), because 20 face-centers accommodate the 20 vertices of the duodecedron, 8 of those 20 accommodate the cube, and 4 of those 8 accommodate the tetracedron. Why then does the ycocedron refuse to receive the octocedron?

    • Because the octocedron has six vertices at the ends of three axes that intersect one another at right angles (ad angoli recti in croce), forming the glorious sign of the Holy Cross (el glorioso segno dela sancta Croce)! Neither the 20 face-centers nor the 12 vertices of the ycocedron contain six points at the extremities of three mutually perpendicular diameters; and just as water (represented by the ycocedron) yields to the Cross, or as the devil flees the sign of the Cross, so the ycocedron cannot enclose the octocedron at its face-centers or vertices! (Though if one wished to use the midpoints of six opposite edges among the 30 edges of the ycocedron, just as we did in the duodecedron in Chapter XLV, three orthogonal axes do pass through six edge-midpoints of the ycocedron.)

  • Why the Other Eight Combinations Fail at Face-Centers or Vertices: In the tetracedron (4 faces, 4 vertices, 6 edges), only the octocedron (6 vertices) can be inscribed using the 6 edge-midpoints (Cap. XXXV), for neither the cube (8 vertices), the ycocedron (12 vertices), nor the duodecedron (20 vertices) can fit symmetrically on 4 faces, 4 vertices, or 6 edges. Similarly, the octocedron (8 faces, 6 vertices, 12 edges) receives the cube (8 vertices at its 8 face-centers, Cap. XXXVIII) and the tetracedron (4 vertices at 4 of those 8 face-centers, Cap. XXXIX), but its 8 faces, 6 vertices, and 12 edges cannot symmetrically seat the 12 vertices of the ycocedron or the 20 vertices of the duodecedron.