CHAPTER LVII¶
How All Five Regular Bodies Are Placed in the Sphere, How Stonemasons May Carve Them from a Spherical Stone, and the Anecdote of Count Girolamo Riario and Melozzo da Forlì in Rome.
In this sphere, most lofty Duke, all five regular bodies are imagined and carved in the following manner—an explanation that will also show how a stonemason (lapicida or scarpellino) can fashion any of the five regular bodies out of a spherical stone cannonball (vna pietra de bombarda / balotta):
1. Carving the Five Regular Solids from a Spherical Cannonball
The Tetracedron (4 Points, 6 Edges, 4 Triangular Faces): If upon the spherical surface (spoglia or veste) of the sphere one marks 4 points equidistant from one another in every direction and joins them by 6 straight lines passing inside the sphere, the regular tetracedron is formed within it. And if a stonemason takes a spherical stone cannonball (pietra de bombarda) marked with those 4 equidistant points and chisels away (stempiasse ouer sfaciasse) the four spherical caps with his iron tools along the flat planes passing through those 6 straight lines, leaving the 4 points intact, he will have reduced the spherical stone to a naked tetracedron.
The Exacedron or Cube (8 Points, 12 Edges, 6 Square Faces): Similarly, if 8 equidistant points are marked upon the spherical surface and joined by 12 straight lines, there is placed in the sphere the second regular body, called the exacedron or cube—the figure of that diabolical instrument called the gaming die (el diabolico instrumento dicto taxillo)! And a stonemason chiseling flat along those six planes will reduce the round cannonball to cubic form.
The Octocedron (6 Points, 12 Edges, 8 Triangular Faces): If 6 equidistant points are marked upon the sphere and joined by 12 straight lines, the third regular body, the octocedron of 8 triangular bases, is formed; and the stonemason chiseling along those planes will make from the cannonball the body of 8 equilateral triangles.
The Ycocedron (12 Points, 30 Edges, 20 Triangular Faces): If 12 equidistant points are marked upon the sphere and joined by 30 straight lines, the fourth regular body, the ycocedron of 20 triangular bases, is formed in the sphere and carved from the stone.
The Duodecedron (20 Points, 30 Edges, 12 Pentagonal Faces): Finally, if 20 equidistant points are marked upon the sphere and joined by 30 straight lines, the fifth and noblest regular body, the duodecedron of 12 pentagonal bases, is formed in the sphere and carved from the round stone.
And in every one of these five bodies, when it is placed in the sphere, if one of its solid angles touches the spherical surface, all its solid angles necessarily touch the sphere at the same time; nor is it ever possible for one vertex to touch without all the others touching.
2. How to Expose Boastful Artisans, and the Anecdote of Count Girolamo Riario and Melozzo da Forlì in Rome
By means of this infallible science (Chapter XXV), Your Ducal Highness may at times amuse yourself—as we ourselves have often done—by exposing the ignorance of boastful stonemasons (lapicidi) and rash braggarts (temerarii milantatori) who promise “Rome and Toma, seas and mountains” (Roma e toma, maria e montes), neither knowing geometry nor caring to learn it, against the moral precept:
Ne pudeat quae nescieris te velle doceri (“Be not ashamed to wish to be taught what you do not know”).
Simply command them to carve from such a stone a capital, base, or frieze block bounded by equal faces, equal sides, and equal solid angles, but which is none of the five regular forms—for instance:
A regular body of 4 equal faces that are not triangles; or
A regular body of 6 equal faces that are not squares (say triangles or pentagons); or
A regular body of 8 or 20 equal faces that are not triangles; or
A regular body of 12 equal faces that are not pentagons!
All of which are absolutely impossible in nature, as we proved demonstratively in Chapter XXV.
Such ignorant artisans are like:
The carpenter (carpentieri) who, when asked what he would do if he had no plane (pialla), replied that he would plane a block of wood into a plane using another plane!
Or the other joiner (marangone) who declared that his large try-square (squadra) was “too large” to test a small square, imagining that right angles vary in size according to the length of their legs (contra [I.Post.4]: All right angles are equal to one another)!
Or the simpleton who, placing two equal little rods in the form of a Tau (\(\mathbf{T}\)) before his eyes, judged first the vertical stem and then the horizontal crossbar to be the longer!
Just such an incident actually occurred in Rome at the time of the building of the palace of Count Girolamo Riario (Conte Girolamo, nephew of Pope Sixtus IV, today the Palazzo Altemps) of blessed memory. While we were conversing in the Count’s presence and inspecting the building works along with many worthy men of diverse faculties in his retinue—among them the renowned painter Melozzo da Forlì (Melozzo da Furlì)—in order to give pleasure to speculation, Melozzo and I urged the Count to commission a certain capital in one of those impossible forms, without revealing the geometric impossibility to the Count, saying only that it would be a most worthy piece.
The Count agreed, summoned his chief master stonemason—and at that time the Roman stonemasons were famous—and asked him whether he knew how to carve it. The mason replied that this was a trifling matter (piccola facenda) and that he had made such pieces many times before! Whereupon the Count began to doubt whether the form was really as rare and worthy as we had praised it to be. Melozzo and I stood firm, openly declaring that the mason would never make it because of the geometric impossibility proved above.
Recalling the stonemason, the Count asked him again if he would truly make it. Laughing scornfully, the mason offered to wager upon it (al sì e al non sempre fia pronto lo impegnare). The Count asked: “If you fail to make it, what are you willing to forfeit?” The mason cleverly replied: “My Lord, whatever more Your Most Illustrious Lordship pleases beyond what I stand to earn!” They agreed on a term of twenty days, though the mason boasted he needed only four.
In the end, after spoiling many fine blocks of marble and cutting one bungled piece after another, he could never make the faces and angles meet! The Count graciously held him liable only for the cost of the ruined marble, and the mason was left thoroughly shamed (scornato). Yet he never rested until he learned who had proposed the problem; and discovering that it was the Friar (el frate, namely myself!), he bore me no small grudge until he found me and said: “Messer, Messer, I will never forgive you the injury you did me unless you teach me the reason and the way of these bodies!” I gladly offered him whatever skill I possessed, and during several days in Rome I was not ungenerous to him, opening to him the secrets of these and other geometric matters pertaining to his art; and that courteous master insisted on having a fine friar’s cloak (vna degna cappa) made for me in his name!
Thus Your Highness sees how great lords can cure presumptuous artisans of their boasting—just as King Hiero of Syracuse did with the poet Simonides, as Cicero recounts in De natura deorum (I.22.60): when Hiero asked Simonides what God was, Simonides rashly asked for a single day to state the exact definition; after one day he asked for two, then four, and after many extensions finally confessed that the more he pondered it, the less he understood, remaining confounded in his initial rashness.