The Music of Pythagoras (21 page)

• Next, a section “three times the third.”

• Next, a section “eight times the first.”

• Last, a section “twenty-seven times the first.”

Counting the squares in each line gives 1, 2, 3, 4, 9, 8, 27. The first four of those numbers are the numbers in the
tetractus
and the Pythagorean musical ratios, 2:1, 3:2, 4:3, but it is an interesting challenge to discern a meaningful pattern in the rest of the numbers, and how they could work in a creation scheme. The answer is that if you square each of the first two numbers 2 and 3 (1 not being a “number”), you get 4 and 9. Cube the same two numbers, 2 and 3, and you get 8 and 27. For a Pythagorean it was significant that each pair was an even and an odd number. Plato stopped with the cubes because in the creation of three-dimensional, solid physical reality, only three dimensions are needed.

Next, according to the account Plato put in Timaeus’ mouth, the creator divided his material into smaller parts, filling in harmonic and arithmetical means between those numbers and connecting the “world soul” with a diatonic scale in music.
*
Plato used a scale developed by Philolaus, not the one developed by Archytas.

Astronomy in his
Timaeus
was also worked out in numbers, with the “world soul” cut into two strips bent around to form an “X” at one point, making an inner and outer ring. Two such rings really exist in astronomy, the celestial equator and the ecliptic. The celestial equator is on the plane of Earth’s equator and anchors the sphere of the fixed stars that do not change their positions in the sky relative to one another and the celestial equator. This was the ring Timaeus called “the Same.” It stays the same and never changes. The ecliptic is the circular path that the Sun appears to follow in its daily round, with the planets appearing to orbit in a band centered on it. This ring was Timaeus’s “the Different,”
for it changes He called the planets “instruments by which Time can be measured.”
*
The creator cut the Different into seven narrower strips to accommodate Sun, Moon, and five planets, with the radiuses of their orbits proportional to the numbers 1, 2, 3, 4, 8, 9, and 27. Both rings—Same and Different—were in constant motion, which, Plato thought, nothing but a living soul could be, unless something else pushed it. The rings moved in opposite directions, the Same east to west, the Different west to east, and the seven strips of the Different moved at different speeds, corresponding to the speeds of the Sun, the Moon and the planets.

Plato had Timaeus explain that the movement humans see in the sky is the result of this combination: The daily rotation of the Same with the sphere of fixed stars carries everything around with it, east to west, including Sun, Moon, and planets. But the Sun, Moon, and planets—the seven bodies of the Different—have in addition their own contrary west-to-east motion against that background. They “swim upstream,” so
to speak, against the current of the Same, at varying speeds, and sometimes back up. This, says Timaeus, is because they are souls, and souls exercise independent choices and power of movement. It is believed to be one of the Pythagorean triumphs, showing up in Philolaus’ fragments, in more detail in Archytas’ work, and then in Plato, to have explained heavenly motion correctly as a combination of opposite movements.

Geometry, Plato had Timaeus explain, had a detailed role in creation when primordial disorder was sorted into four elements—earth, fire, air, and water—and the creator introduced four geometric figures—cube, pyramid or tetrahedron, octahedron, and icosahedron. These “Pythagorean” or “Platonic” solids are four of the five possible solids in which all the edges are the same length and all the faces are the same shape.
*
Each element—earth, fire, air, and water—was made up of tiny pieces in one of those shapes, too small to be visible to the eye.

Plato had Timaeus continue: The four elements and four solids were not the alphabet of the universe. The solids were constructed of something even more basic, two types of right triangles. Plato, through Timaeus, admitted there was room for argument about which triangles were most basic, but he thought he was correct to choose the isosceles triangle and scalene triangle. Both are right triangles.

The isosceles triangle is made by cutting a square into equal halves on the diagonal. Obviously, two isosceles triangles make a square, and squares make up cubes (one of the solids).

In a scalene triangle, the diagonal is twice as long as the shortest side. Two scalene triangles set back to back create an equilateral triangle—none other than the Pythagorean
tetractus
. The faces of the tetrahedron, octahedron, and icosahedron are equilateral triangles.

Here is Plato’s explanation.

Cube: Fasten together the edges of six squares (each made by pairing two isosceles triangles). The result is a
cube
, the only regular solid that uses the isosceles triangle or square for its construction.

Pyramid or tetrahedron: Fasten together the edges of four equilateral triangles (each made by pairing two right scalene triangles). The result is a
pyramid
or
tetrahedron
.

Octahedron: Fasten together the edges of eight equilateral triangles. The result is an
octahedron
.

Icosahedron: Fasten together the edges of twenty equilateral triangles. The result is an
icosahedron
.

The Pythagoreans and Plato knew the dodecahedron, the only regular solid made of pentagons (12 of them), but Plato did not use it in his scheme.

Beyond those five—cube, pyramid, octahedron, icosahedron, and dodecahedron—there are no other regular solids (polyhedrons). Try to fasten together
any other number
of
any regular figure
(polygon). You get no fit. No wonder the Pythagoreans, Plato, and later Kepler thought these solids were mysterious.

Timaeus explains to Socrates and the other characters in the dialogue that earth is made up of microscopic cubes, fire of tetrahedrons, air of octahedrons, water of icosahedrons. The pairings were based on how easily movable each solid was, how sharp, how penetrating, and on considerations of what qualities it would give an element to be made up of tiny pieces in this shape.

Timaeus pairs the fifth regular solid, the dodecahedron, with “the whole spherical heaven,” and in his
Phaedo
, Plato associated it with the spherical Earth, in spite of the fact that in his time most of the Greek world, except for the scattered Pythagorean communities, still assumed the Earth was flat. The dodecahedron comes close to actually being a
sphere. In fact, the earliest mention of a dodecahedron was in sports, with twelve pentagonal pieces of cloth sewn together and the result in-flated to create a ball. Each of the five solids fits into a sphere with each of its points touching the inner surface of the sphere, and a sphere can be fitted into each of the solids so as to touch the center of each surface, which makes sense of Philolaus’ enigmatic (and controversial) fragment: “The bodies in the sphere are five: fire, water, earth, and air, and fifthly the hull of the sphere.”

Though the triangles making up the solids in Plato’s scheme may have been the basic “alphabet” of creation, he thought they were not the fundamentals or
archai
. In the dialogue
Philebus
, Socrates says knowledge of the principles of
unlimited
and
limiting
is “a gift of the gods to human beings, tossed down from the gods by some Prometheus together with the most brilliant fire. And the ancients, our superiors who dwelt nearer to the gods, have passed this word on to us.”
³
Plato’s contemporaries and generations of later readers thought that by “some Prometheus,” he meant Pythagoras, and that “the ancients, our superiors who dwelt nearer to the gods,” were the Pythagoreans, which contributed substantially to the image of Pythagoras as a channel for superhuman knowledge and wisdom. If Plato meant that, he short-changed Anaximander, who had talked of “unlimited” and “limiting” earlier.

According to Plato, one thing that “some Prometheus” tossed down concerning the unlimited and the limiting was that “all things that are said to be are always derived from One and from Many, having Limit and Unlimitedness inherent in their nature.”
⁴
He explained this in unpublished lectures at his Academy that Aristotle reported firsthand.