The Construction Of Regular Figures

Book 2 showed how the regular plane figures fit together to form the solid figures.

The 5 solid figures (among others) in connection with the plane figures.

• They were called Cosmic by the Platonists, and to which element, and on account of what property, each of them was related.

This Chapter will treat them alone for their own sake, leading to the harmony of the heavens, and not because of the plane figures.

The rest is in the Epitome of Astronomy, Book 4.

My Mysterium Cosmographicum (The Secret of the Universe) gave a brief account of the order of the 5 figures in the universe:

• 3 are primary
• 2 are secondary.

The cube is the outermost and the largest because it is first in order of generation.

• It relates to the Whole by the very form of its generation.

Next is the tetrahedron as a part established by division of the cube.

The tetrahedron ACDF appears hidden in the cube, in such a way that any face of the tetrahedron, such as ACD, is covered by one vertex of the cube ACDB.

The cube AED appears hidden inside the dodecahedron.

Any face of the cube, such as AEU is covered by 2 vertices of the dodecahedron or by the pentahedron ABCDE which is divisible into three dissimilar tetrahedra by the 2 planes DCA and ABD.

but also itself primary, with a trilinear solid angle, like the cube. Inside the tetrahedron is the dodecahedron, 3, the last of the primaries, which of course is like a composite figure made up of parts of the cube, like a tetrahedron, that is, from irregular tetrahedra, by which the cube inside it is covered.

Next after it is the icosahedron, 4, on account of its similarity, the last of the secondaries which adopt a solid angle made up of more than three lines.

The innermost is the octahedron, 5, which is like the cube, and is the first figure among the secondaries, to which the first place among the inner figures is due, inasmuch as it can be inscribed, just as the first place among the outer figures belongs to the cube because it can be circumscribed.

However, there are two notable marriages, so to speak, of these In addition to these there is one which is, so to speak, celibate or hermaphrodite, the tetrahedron, because it is inscribed in itself, just as the feminine ones are inscribed in, and so to speak subject to, the males, and have the female tokens of their sex opposite to the masculine ones, or in other words the angles to the plane faces.

Furthermore, as the tetrahedron is an element, the entrails and, so to speak the rib of the male cube, so the octahedron, the female, is an element and a part of the tetrahedron, by another reckoning’’:thus the tetrahedron is the go-between in this marriage.

The chief difference between the couples or families consists in the following: that the relation between the cubic family is indeed expressible, for the tetrahedron is one third of the cubic solid, the octahedron half of the tetrahedric solid, and a sixth of the cube.

However, the proportion of the dodecahedric marriage is inexpressible indeed, but divine.

The conjunction of these two words warns the reader to be careful about their significance. For the word “inexpressible” here does not of itself denote any nobility, as elsewhere in theology and divine matters; but it denotes an inferior condition. For there are in geometry, as has been stated in Book I, many inexpressibles which do not thereby also participate in divine proportion. Now what divine proportion is (which should rather be called sequential) must be looked for in Book I.

For other proportions require their own four terms, and continuous proportion 3: the divine proportion also needs a particular property of its terms, in addition to that of the proportion itself, that is to say that the two lesser terms, as parts, make up the greater term, as whole.

Therefore, to the same extent as this dodecahedric marriage has lost by using an inexpressible proportion, it also gains on the other hand because the inexpressible serves the divine.

This marriage also has a solid star,^® the generation of which is from the continuation of five faces of the dodecahedron so that they all meet at a single point.

See their generation in Book 2.

Lastly, we must note the proportion of the spheres circumscribing them to those inscribed in them, as in the tetrahedron it is expressible, as 100000 to 33333 or as 3 to 1; in the cubic marriage it is inexpressible, but the radius of the inscribed circle is expressible in square, as the root of one third of the square of the diameter, that is to say as 100000 to 57735; in the dodecahedric marriage it is clearly inexpressible, as 100000 to 79465; in the star it is as 100000 to 5 2 5 7 3 , ^ 2 half the side of the icosahedron, or half the distance between the two semidiameters.