CHAPTER XLIII¶
How the Ycocedron Is Formed in the Duodecedron.
Now we come to the ninth proposition of the fifteenth book (Campanus XV.9), where we begin inscribing all four of the other regular bodies inside the fifth and noblest body, the duodecedron (12 pentagonal bases)—which, as Plato says in the Timaeus, is the receptacle and lodging (receptaculo e albergo) of all the others—starting with the ycocedron (20 triangular bases).
Inscribing the Icosahedron in the Dodecahedron (Campanus XV.9)
In a given regular body of twelve pentagonal bases (duodecedron), to inscribe the regular body of twenty equilateral triangular bases (ycocedron) so that the twelve vertices of the ycocedron stand at the twelve centers of the pentagonal faces of the duodecedron.
Construction and Proof ([IV.14], [XIII.17], [XI.10])
In each of the twelve regular pentagonal faces of the given duodecedron, find the center of that pentagon by the fourteenth of the fourth (bisecting two adjacent angles of the pentagon; the intersection of the bisectors is the center of the pentagon), giving twelve face-centers in all (see the marginal diagram
0040.g3.pngon Folio 0040).Around each of the twenty solid angles (vertices) of the duodecedron, exactly three regular pentagonal faces meet (Chapter XXV). Join the three centers of the three pentagons meeting at each vertex of the duodecedron by three straight lines.
Because the three pentagonal faces meeting at any vertex of the duodecedron are equal and equally inclined to one another, the three straight lines connecting their centers are mutually equal and form an equilateral triangle opposite that vertex.
Carrying out this construction around all twenty vertices of the duodecedron produces twenty equal equilateral triangles bounded by thirty equal edges meeting five by five at the twelve centers of the pentagonal faces of the duodecedron.
Therefore the regular ycocedron is inscribed in the duodecedron, with its twelve solid angles touching the centers of the twelve pentagonal bases of the duodecedron.