Ad Quadratum Construction and Study of the Regular Polyhedra

For 3
5
144
8
1 1
and , having respectively and from the harmonic construction, it is
15 5 15
7
and respectively on the left with the compass and
5 15
necessary to just measure out 2
swing the measurement to the right to find the proper location.
Conclusions:
The generating process for both the Pythagorean and the Just intonation scales starts with
the prime (one) and uses only 2 and 3 – under the form of the fifth 3
2
for Pythagorean
tuning and the half-cut and third-cut for Just intonation. Number 4 comes out of the
generating process through inversion of the F, in Pythagorean tuning, to bring it within
the octave 1 to 2, and through the third cut (harmonic mean) is Just intonation.
Similarly, in the adquadratum method for generating the platonic forms, only 1, 2, and 3
are used with 4 appearing only as a result of the tetrahedron construction, measuring its
height within a circumsphere of radius 3 and encompassing an insphere of radius 1. This,
in turn, gives rise to number 9 (ratio of the areas of the circumsphere to the insphere) and
27 (ratio of their volume) so that all the numbers out of which Plato compounded the
world soul, namely 1, 2, 3, 4, 8, 9, and 27 are to be found in the generation of the
tetrahedron, the simplest of the platonic forms.
The numbers 1, 2, 3, and 4 whose sum is 10 form the celebrated Tetraktys of the
Pythagoreans. Out of these simple numbers, as we have seen, the consonant ratios of the
octave (1:2), the fourth (3:4), the fifth (3:2), the double octave (1:4) and the twelfth, i.e.,
the octave plus a fifth (1:3) can be formed.
The Tetraktys, furthermore, symbolized the universal structure, starting in unity – one –
the point, moving to 2, the one-dimensional line; 3, the two dimensional triangle, and 4,
the tetrahedron, the first three dimensional form. It eventually returns through its sum to
unity, Ten. In this respect it is amusing to indulge in a bit of Pythagorean numerology by
remarking that if we sum up all the elements of the five Platonic forms as listed on
p. 130, we obtain a total of 244, the integers of which sum up to 10, i.e., again unity,
while their product comes to 32 whose product is 6, the first perfect number, since
1 2 3 1 2 6,
ha!
No wonder the Ancients were awed by such perfection. As Theon of Smyrna 42 puts it:
“Unity is the principle of all things and the most dominant
of all that is: all things emanate from it and it emanates
42 Theon of Smyrna: Mathematics Useful for Understanding Plato. Tr. By R&D Lawlor, San Diego 1979.