CHAPTER XLII¶
How the Tetracedron Is Formed in the Ycocedron.
The eighth proposition of the fifteenth book (Campanus XV.8) shows how to inscribe the tetracedron (4 triangular bases) inside the ycocedron (20 triangular bases) by means of the cube inscribed in the ycocedron.
Inscribing the Tetrahedron in the Icosahedron (Campanus XV.8)
In a given regular body of twenty triangular bases (ycocedron), to inscribe the regular body of four triangular bases (tetracedron) so that the four vertices of the tetracedron stand at four of the twenty centers of the triangular faces of the ycocedron.
Construction and Proof (via Campanus XV.7 and XV.2)
First, in the given ycocedron inscribe the duodecedron (Chapter XL, Campanus XV.6), and within that duodecedron inscribe the cube by drawing twelve pentagonal chords across its twelve faces (Chapter XLI, Campanus XV.7), so that the eight vertices of the cube stand at eight centers of the triangular faces of the ycocedron (see the marginal diagram
0040.g2.pngon Folio 0040).Next, in this inscribed cube, inscribe the tetracedron by the method of Chapter XXXVI (Campanus XV.2), drawing the six face diagonals across the six square faces of the cube to connect four alternating corners of the cube.
Because the eight corners of the cube stand at eight face-centers of the ycocedron, the four corners of the tetracedron likewise stand at four centers of the triangular faces of the ycocedron, and all six edges of the tetracedron are mutually equal (\(s_{\text{tetra}} = s_{\text{cube}}\sqrt{2}\)).
Therefore the regular tetracedron is inscribed in the ycocedron.