Thomas Dechand
Humanities Center
The Johns Hopkins University
Music and Mathematics in Plato's Timaeus*
I would like to begin with two quotations. The first is from Samuel Coleridge's
Biographia Literaria. As an illustration of his maxim, 'until you understand a writer's ignorance,
presume yourself ignorant of his understanding,' Coleridge writes: "I have been re-perusing with
the best energies of my mind the Timaeus of Plato. Whatever I comprehend, impresses me with a
reverential sense of the author's genius; but there is a considerable portion of the work, to which
I can attach no consistent meaning" (141). The other quote comes from the Timaeus itself, and I
found it too apposite a description of what reading the dialogue often feels like not to use: "as
you might imagine a person who is upside down and has his head leaning upon the ground and
his feet up against something in the air, and when he is in such a position, both he and the
spectator fancy that the right of either is his left, and the left right" (43e). The Timaeus is an
extraordinarily rich dialogue. It is the story of Plato's cosmology, and in it one finds not only
Plato's account of the creation of the universe, but a treatise on geometry disguised as a theory of
the structure of matter, musings on the nature of time and space, a theory of vision, a chemistry
of earth, air, fire and water, and most everything in between. Since this is obviously too much
information to try to cover in an essay, after a few general remarks, I would like to focus on a
few passages that seem particularly confusing the first time through.
Perhaps the most immediately striking feature of the dialogue, as Gregory Vlastos notes,
is that the name Plato gives his creator literally means neither ruler nor king, but "craftsman"
(Plato's Universe, 26). This is odd, since, as Vlastos tells us, "in Plato's Athens the craftsman is
*
This paper was originally prepared for English 509, University of Washington, 2001.
often a slave and as often a freeman working shoulder to shoulder with slaves in the same kind of
work" (26). The contrast is to the Greek gods, who, as portrayed by the poets, lived lives best
suited for daytime television. According to this portrait, an Athenian had to placate the gods with
constant festivals and sacrifices, while simultaneously worrying about such things as being
seduced by Zeus only to incur the jealously and wrath of Hera later on. Opposed to this image of
the gods, Plato tells us that his creator is good, and as such, can "never have any jealousy of
anything" (Timaeus, 29e). In the first of many similarities to the Biblical account of creation
found in Genesis, Plato's Demiurge desires "that all things should be as like himself as they could
be," or what comes to the same thing, "that all things should be good and nothing bad, so far as
this was attainable" (30a).
As in Genesis, Plato needs two different creation stories to cover all the ground he wants
to. The first story, describing what Plato calls "the works of intelligence," is juxtaposed with the
second, which describes "the things which come into being through necessity" (47e). The "works
of intelligence" are undertaken by the Demiurge; their telos ends in what is good. The works of
necessity, as the name suggests, describes only those events logically entailed by other events.
As Plato says: "the creation of this world is the combined work of necessity and mind. Mind, the
ruling power, persuaded necessity to bring the greater part of created things to perfection, and
thus and after this manner in the beginning, through necessity made subject to reason, this
universe was created" (47e-48a).
I will look closely at two passages, one from the works of intelligence dealing with
Plato's construction of the world-soul, the other from the works of necessity dealing with the
formation and interactions of the four elements.
Construction of the World-Soul
Plato's universe, as is to be expected, is created from a model (29b). It is based on the
form of a perfect animal (30b, 32d). As such, it is self sufficient, implying that the universe is
unique (30d-31a, 33d), and spherical, that being the most perfect shape (34a). Since it is based on
the model of an animal, Plato's universe will also need a soul (30b, 34c). Having gathered up a
sort of primordial ooze, Plato describes the Demiurge's creation of this world soul in the
following passage, which I will quote once all the way through before attempting a more
thorough line by line analysis:
And he proceeded to divide after this manner. First of all, he took away one part of the whole [1],
and then he separated a second part which was double the first [2], and then he took away a third
part which was half as much again as the second and three times as much as the first [3], and then
he took a fourth part which was twice as much as the second [4], and a fifth part which was three
times the third [9], and a sixth part which was eight times the first [8], and a seventh part which
was twenty seven times the first [27]. After this he filled up the double intervals [that is, between
1, 2, 4, 8] and the triple [that is, between 1, 3, 9, 27], cutting off yet other portions from the
mixture and placing them in the intervals, so that in each interval there were two kinds of means,
the one exceeding and exceeded by equal parts of its extremes [as, for example, 1, 4/3, 2, in which
the mean, 4/3 is one third of 1 more than 1, and one third of 2 less than 2], the other being that kind
of mean which exceeds and is exceeded by an equal number. Where there were intervals of 3/2and
of 4/3 and of 9/8, made by the connecting terms in the former intervals, he filled up all the intervals
of 4/3 with the interval of 9/8, leaving a fraction over, and the interval which this fraction expressed
was in the ratio of 256 to 243. And thus the whole mixture out of which he cut these portions was
exhausted by him. This entire compound he divided lengthwise into two parts which he joined to
one another at the center like the letter X, and bent them into a circular form, connecting them
with themselves and each other at the point opposite to their original meeting point (35b-36c).
So you can see that Coleridge was not exaggerating. To try and better understand what Plato is
up to here, it is necessary to parse this passage one bit at a time.
After separating out the numbers 1, 2, 4, and 8, as well as 1, 3, 9, and 27, Plato's
Demiurge fills up the spaces between each of these numbers by inserting two kinds of means. I
will illustrate how Plato's Demiurges divides and fills the line by looking only at the numbers
between 1 and 2. One kind of mean, that which "exceeds and is exceeded by an equal number" is
more commonly known as the arithmetic mean. It is obtained by adding two numbers and then
dividing their sum by two and given by the formula (a + b)/2. The arithmetic mean of 1 and 2 is 3/2.
3
1
/2
2
The other kind of mean, what Plato describes as "exceeding and exceeded by equal parts of its
extremes" is better known as the harmonic mean. It is given by the formula 2/(1/a + 1/b). The
harmonic mean of 1 and 2 is 4/3.
4
1
/3
2
Placing these two means in the same diagram, Plato notes there are intervals of 4/3, 3/2, and 9/8,
the latter obtained by dividing 3/2 by 4/3 (36a-b).
4
1
3
/3
/2
2
(x 9/8)
Plato then fills up the entire segment with such 9/8 intervals, noting that this does not quite work
out because it leaves remainder 256/243 intervals (36b). Plato's construction, reading off the
numbers between 1 and 2 from left to right, thus gives us the sequence 1, 9/8, 81/64, 4/3, 3/2, 27/16,
243
/128, and 2. Each term in this sequence is obtained by multiplying the prior term with either the
interval 9/8 or the interval 256/243.
9
1
9
(x /8)
81
/8
9
(x /8)
4
/64
/3
256
(x /243)
3
9
(x /8)
27
/2
243
/16
9
(x /8)
9
(x /8)
/128
2
(x 256/243)
Having thus filled up all the intervals between 1, 2, 3, 4, 8, 9, and 27, Plato's Demiurge
stops dividing, telling us only that the "whole mixture out of which he cut these portions was
exhausted" (36b). And without telling us what, if anything, these numbers mean, Plato is off and
running in the very next sentence, cutting this strip into two and tying it back together so as to
create two circles which he uses to recreate the motions of the fixed stars in relation to the
motion of the planets, sun, and moon about the earth.
So what in the world is going on here? What significance does Plato see in the arithmetic
and harmonic means, and why does he stop dividing the intervals at 9/8 and 256/243? Having
sought in vain for a mathematical interpretation of this sequence, I eventually happened to hit
upon a musical interpretation.
If one considers the ratios as the relative frequencies of various notes, or as was more
likely in Plato's time, the relative lengths of string plucked to produce pleasing sounds, a scheme
quickly emerges for making sense of Plato's construction of the world-soul. One can verify the
following ratios: an octave corresponds to a 2/1 ratio; a perfect fifth to 3/2; a fourth to 4/3; a tone to
9
/8; and a near half-tone to 256/243. Looking back at the division between 1 and 2 of the world-
soul, one notices that Plato has constructed a musical scale! This scale is nothing but Plato's
version of a sort of 'Do Re Me Fa So La Ti Do.'
9
1
/8
9
(x /8)
C
D
81
4
/64
/3
9
256
(x /8)
(x /243)
E
F
3
27
/2
9
/16
9
(x /8)
/128
2
256
(x /8)
(x /243)
B
C
9
(x /8)
G
243
A
Had Plato built a piano, it would sound somewhat along the lines of what one would hear, if,
starting with middle C, Plato played only the white keys until he hit another C. This scale is
known as the Pythagorean scale, and it is Pythagoras who is credited with first discovering the
rations governing the octave, fourth, fifth, and tone.
This interpretation may also help explain the nature of the stuff that Plato's Demiurge
used to create the world soul. This mixture, as Gregory Vlastos notes, "contains no physical
matter, and none of the properties of physical matter except one: it can move" (Plato's Universe,
31). Its movement is special both because soul is the only material capable of moving itself, and
also because its motion is confined to revolving in a circle (31). Plato describes circular motion
as "most appropriate to mind and intelligence" (Timaeus, 34a). The musical ascent of one octave,
since it begins and ends on the same note, could be thought of as describing a circle.
As mentioned earlier, Plato uses this scale to construct the motions of the fixed stars,
planets, sun, and moon around the earth. An interesting application of this idea, one I have had
absolutely no luck with, would be to apply this scale to Plato's description of the motion of the
heavenly bodies.
Plato's Particle Physics
In a celebrated line from his television series Cosmos, astronomer Carl Sagan said, "If
you want to make an apple pie from scratch, you must first invent the universe." Plato's version,
at least according to the Timaeus, would likely read: "If you wish to make goat cheese from
scratch, you must first invent the triangle." Remembering that Plato marked the entrance to his
academy with the words: "Let no one who is destitute of geometry pass beyond these walls," let
us proceed to examine what may be termed 'Plato's Particle Physics.'
Plato chooses the two triangles he believes have "the most beautiful forms" as the
fundamental particles of his universe (54a). The first is "that triangle which has its hypotenuse
twice the lesser side," or a 30-60-90 degree triangle (54d). The second is an isosceles triangle,
which has angles equal to 45, 45, and 90 degrees (55b).
The 30-60-90 degree triangle (left) and the isosceles triangle (right) constitute Plato's fundamental particles.
The four elements in Plato's universe, earth, air, fire, and water, are made by combinations of
these two triangles. Plato envisions this system on a similar part to whole relation as he uses for
letters and words in the Cratylus.
Three of the four elements, those made from 30-60-90 degree triangles, have equilateral
triangles as their faces. Plato describes the construction of a face as follows: "When two such
triangles are joined at their diagonal, and this is repeated three times, and the triangles rest their
diagonals and shorter sides on the same point as a center, one equilateral triangle is produced
from six of these" (54d-e). The following diagram shows such a face.
The first object Plato makes with these equilateral triangles is a tetrahedron. In his words:
"four equilateral triangles, if put together, make out of every three plane angles one solid angle.
And out of the combination of these four angles arises the first solid" (54e-55a).
The second object constructed in this manner is a octahedron: "the second species of solid is
formed out of the same triangles, which unite as eight equilateral triangles and form one solid
angle out of four plane angles, and out of six such angles the second body is completed" (55a).
The third object is an icosahedron: "the third body is made up of one hundred and twenty
triangular elements, forming twelve solid angles, each of them included in five plane equilateral
triangles, having altogether twenty bases, each of which is an equilateral triangle" (55a-b).
The object constructed from the isosceles triangle, however, has squares, not equilateral
triangles, as its faces. Plato describes the creation of the square face: "the isosceles triangle
produced the fourth elementary figure, which is compounded of four such triangles, joining their
right angles in a center, and forming one equilateral quadrangle" (55b-c).
Six of these square faces are then used to construct a cube (55c).
Earth, being the most stable and plastic of the elements, is associated with the cube, since
the cube has the most stable basis of all the polyhedrons (55e). The tetrahedron, the smallest and
most movable shape, is assigned to fire, and the icosahedron, the largest and least movable, is
assigned to water (56a). Air, by default, gets the octahedron (56a).
These shapes are all known in geometry as regular polyhedra. A regular polyhedron is
simply an object whose faces are made up of polygons that have the same size and shape. In the
tetrahedron (fire), octahedron (air), and icosahedron (water), the faces are made up of equilateral
triangles. The faces of the cube (earth) are made up of squares. A well-known theorem in
geometry shows that only five such shapes exist. The fifth shape is called a dodecahedron and its
faces are made up of pentagons.
This theorem makes its first appearance in Euclid's Elements, written approximately fifty
years after Plato's time. However, I believe that Plato was aware of the theorem. If so, the
existence of five shapes for only four elements would seem to put a serious dent in his theory. So
what does Plato do about this problem? In the style of Socrates' advice in the Phaedo of throwing
away evidence which does not agree with the theory judged the soundest (100a), he does
nothing. He ignores the difficulty entirely, commenting only that "there was yet a fifth
combination which God used in the delineation of the universe with figures of animals," (55c)
never to so much as hint at the existence of the dodecahedron again.
Having described the constituents of earth, air, fire, and water, Plato needs to give an
account of their interactions. He needs to explain what happens when, for instance, water boils.
Plato gives the following examples: "water, when divided by fire or by air, on re-forming, may
become one part fire and two parts air, and a single volume of air divided becomes two parts
fire" (56d-e). As recurs throughout this dialogue, one reads this passage and wonders where the
numbers come from, what they mean, and what Plato had in mind when choosing them.
Plato pictures the elements interacting by crashing into one another. (57b). In one of these
collisions, some of the elements may be cut into smaller pieces which then recombine to form
other elements (57a-b). As Plato says: "When one of the other elements is fastened upon by fire
and is cut by the sharpness of its angles and sides, it coalesces with the fire, and then ceases to be
cut by them any longer" (57b).
The two right triangles I characterized as the fundamental or elementary particles of
Plato's universe. The four elements can similarly be conceived as analogous to what we would
call molecules. Plato thinks of them in a similar way: "We must imagine all these to be so small
that not single particle of any of the four kinds is seen by us on account of their smallness, but
when many of them are collected together, their aggregates are seen" (56c). The workings of
Plato's chemistry is easy to understand when thought of in light of these two analogies. To begin
with, it is possible to construct a periodic table of sorts for Plato's elements, detailing the shape,
type of elementary triangle used in the shape's construction, and the number of triangles needed
to put the shape together. Such a chart would look like this:
Element
earth
fire
air
water
Plato's Periodic Table
Type of Triangle
# of Triangles
45-45-90
24
30-60-90
24
30-60-90
48
30-60-90
120
Shape
cube
tetrahedron
octahedron
icosahedron
This information is all one needs both to visualize and compute any interaction between
the elements. If you remember back to those horrid exercises from high school chemistry that
required us to balance chemical equation after chemical equation, it might be comforting to note
that the students at Plato's academy may have done something awfully similar 2300 years ago.
They made us balance the number and type of atoms in a chemical reaction, say, dropping
vinegar onto baking soda to make carbon dioxide gas and water, by making sure that each side of
the equation had the same number of C's, H's, Na's, and O's.
e.g. CH3COOH + NaHCO3 ==> CO2 (g) + H2O (l) + Na+ + CH3COO-
Plato's students would have had to make sure that when describing what happens when fire
elements crash into water elements (i.e. boiling water), each side of the equation had the same
number and type of elementary triangles.
e.g. 1 Water + 1 Fire ==> 2 Fire + 2 Air
In this interaction, the only triangles involved are of the 30-60-90 type. Looking at Plato's
periodic table one can read off the relevant information: a water element contains 120 such
triangles, a fire element 24, and an air element 48. The reason that, as Plato says, "water, when
divided by fire or by air, on re-forming, may become one part fire and two parts air" is that this is
the combination which preserves the number and type of fundamental triangles. One immediate
consequence of this view is that earth elements can never be transformed into any of the others
because the earth elements are made from different elementary triangles. This is a consequence
Plato is aware of, for he says that earth, when smashed by one of the other particles, "is borne
hither and thither until its parts, meeting together and mutually harmonizing, again become earth,
for they can never take any other form" (Timaeus, 54d).
One potential objection to this theory, as Gregory Vlastos notes, is that the volumes of
the enclosed shapes do not stay the same in these interactions. Vlastos' solution is that Plato's
theory is not meant to conserve the volume of matter because "matter only exists in the form of
space encapsulated by polyhedra" (Plato's Universe, 90). Vlastos says that "[Plato's] theory
commits him to the conservation of matter by surface, not by volume; what it keeps constant in
every transformation is the aggregate surface area of the corpuscles" (Plato's Universe, 89-90).
While true, to single out the surface area of the polyhedra as the property conserved in
these interactions seems artificial. The surface area of Plato's elements is not a basic property; it
is one that follows from the number of elementary triangles used in an element's construction. It
leads Vlastos to say that "had Plato made [the equilateral triangle] the atoms of his theory, along
with the squares . . . then all allowable transformations . . . could be represented as the
dismantling of n-faced and reassembling them as m-faced ones" (71). Vlastos' scheme to
describe the particle interactions works as follows. He begins by labeling the equilateral triangle
faces, t's, and the square faces of the cube, s's. Using our earlier example of water boiling,
Vlastos represents this as,
1W <==> (1F + 2A)
20t
= [4t + (2 x 8t)] (71).
In this manner, the number of equilateral triangles (t's) and squares (s'), or in other words the
number of faces of the solids, is conserved. Since the faces are conserved so is the total surface
area. One can place Vlastos' scheme into a chart similar to the one drawn above:
Molecule
fire
air
water
earth
Type Of Atom
equilat. triangle (t)
equilat. triangle (t)
equilat. triangle (t)
square (s)
# of Atoms
4
8
20
6
Shape
tetrahedron
octahedron
icosahedron
cube
Plato's theory, which takes the two right triangles "to be the original elements of fire and the
other bodies" (Timaeus, 53d), already produces the exact same scheme Vlastos derives from
imagining the equilateral triangles and squares as the atoms of the theory. The only advantage of
Vlastos' scheme is computational, and it is only an advantage for someone who has difficulty
multiplying by six. Had Plato ignored the elementary triangles and started with the equilateral
triangle and the square as the atoms of his theory, there would be no good reason for stopping the
number of atoms at two instead of including a regular pentagon, a regular hexagon, and so on,
among the atoms. Another reason why Plato uses the two triangles instead of the two faces as his
atoms becomes evident from a consideration of another problem in the Timaeus, that of the
different kinds of earth, fire, air, and water.
Plato admits the existence of different kinds of the same elements. Speaking of fire, Plato
gives the following threefold division: "first, flame, secondly, those emanations of flame which
do not burn but only give light to the eyes, and thirdly, the remains of fire, which are seen in redhot embers after the flame has been extinguished" (58c-d). "There are similar differences in air,"
Plato continues, "of which the brightest part is called the aether, and the most turbid sort mist and
darkness, and there are various other nameless kinds which arise from the inequality of the
triangles" (58d). Water and earth also admit various kinds (58d, 60b-c). The problem is to figure
out how these differences can be accounted for if the elements are constructed in the same
manner and possess identical shapes.
Two paragraphs earlier Plato tells us: "as to the subordinate species which are included in
the greater kinds, they are to be attributed to the varieties in the structure of the two original
triangles" (57c). So the difference evidently lies somewhere in the two elementary triangles. And
since the only way a 30-60-90 triangle can change its structure and still remain a 30-60-90
triangle, is to increase its size. Plato goes on to say "for each structure did not originally produce
the triangle of one size only, but some larger and some smaller, and there are as many sizes as
there are species of the four elements" (57c-d). If one reads "structure" as 'elementary 30-60-90
triangle' and "triangle" as 'equilateral triangle used as a face of a solid' in the prior sentence, one
can reformulate the question in the following way. Is there more than one way to make an
equilateral triangle out of smaller 30-60-90 triangles?
In fact there is an endless number of ways to make an equilateral triangle out of 30-60-90
triangles. The following diagram shows how this can be done with two, six, and eight 30-60-90
triangles.
Three ways to make an equilateral triangle out of 30-60-90 degree triangles
The same holds with true with the analogous question regarding the various ways to construct
square faces out of isosceles triangles. The following diagram shows how this can be done with
two, four, and eight of the 45-45-90 triangles.
Three ways to make a square out of 45-45-90 degree triangles
The elements constructed out of these various faces, what Vlastos calls "isotopes" of earth, air,
fire, and water, explain how each of the elements can admit various kinds.
Literary Device?
We have spent a lot of time this quarter noting the various literary techniques Plato
employs in the construction of his dialogues. So it should come as no surprise that the same sorts
of devices reoccur here. And while they may not be as elaborate as in the Symposium, what
Plato is up to in the Timaeus may be just as rewarding to look at. Consider the opening line of
the dialogue, where Socrates begins by saying: "One, two, three, but where, my dear Timaeus, is
the fourth of those who were yesterday my guests and are to be my entertainers today?" (17a).
Significantly, and maybe after reading Plato all quarter, not all that much of a surprise, this is the
only Platonic dialogue to start with a number. Unfortunately, I have yet to find a consistent
meaning of "one, two, three, but where, my dear Timaues, is the fourth" that meshes with all the
mathematics in the dialogue. It is something I am still searching for and confident exists.
Conclusion
Moving ahead about 1800 years, I want to conclude by briefly looking at the Timaeus in
connection with the German scientist Johannes Kepler, famous for discovering the three laws of
planetary motion which still bear his name. In a book entitled Mysterium Cosmographicum,
originally published in 1596, Kepler proposed a model for understanding our solar system. The
idea is to use the five regular polyhedrons, by this time known as the Platonic solids, to explain
the distances between the planets and the sun. As Kepler says:
The Earth's orbit is the measure of all things; circumscribe around it a dodecahedron, and the
circle containing this will be Mars. Circumscribe around Mars a tetrahedron, and the circle
containing this will be Jupiter. Circumscribe around Jupiter a cube, and the circle containing this
will be Saturn. Now inscribe within the Earth an icosahedron, and the circle contained within it
will be Venus. Inscribe within Venus an octahedron, and the circle contained within it will be
Mercury. You now have the reason for the number of planets (Mysterium Cosmographicum, 69).
Kepler's model is amazing, if not for its ingenuity, then for its accuracy (his values for the
distances between the orbits of the planets fit the data remarkably well). Combining the two
creation stories in the Timaeus into a coherent whole, it is the sort of thing Plato would have
loved. But just as one can imagine how the discovery of Uranus in 1781 by Sir William Herschel
entirely undermined Kepler's model, incredible though it is, one can also see how this example
shows what is ultimately wrong with Plato's universe. The world, time and time again, proves to
be neither confined nor closed. When objects such as Uranus or a fifth regular polyhedron are
discovered, such schemes have a tendency to quickly fall apart.
Bibliography
Euclid. Elements. Tr: Heath. Encyclopaedia Britannica, Inc., 1952.
Kepler, Johannes. Mysterium Cosmographicum: The Secret of the Universe. Tr: Duncan. Abrais Books, 1981.
Mohr, Richard. The Platonic Cosmology. E.J. Brill, 1985.
Plato. Collected Dialogues. Ed: Hamilton. Princeton University Press, 1999.
Vlastos, Gregory. Plato's Universe. University of Washington Press, 1975.
Weiss, Piero. Music in the Western World. Schirmer Books, 1984.