Listening for the Harmony of ApolloPart I

Unity, Geometry and Beauty in Architecture

by Steven Bass

In the classical system of the five orders of architecture, even the most complex or intricate elements can be specified by numerical patterns of modules and parts. Such patterns, known as canons of proportion, have come down to us in various architectural treatises which designers may use as references. This is a rational approach to achieving beauty, an Apollonian path, fully visible and based in text. But we might remember that Apollo also follows another path to beauty, as leader of the Muses and master of the lyre, an instrument he received from Hermes and gave to Orpheus. In this article we will explore this less traveled though no less worthy path, which is more often associated with Orpheus, Dionysus or Hermes than with Apollo. This path is not irrational but rather hyper-rational, one that includes the rational and more, what might be called mythopoeic or even ecstatic. Such a journey is justified not on romantic or hedonistic grounds but is necessary to explore inner, hard-to-see aspects of number.

The foremost explorers of number as a guide and analogue to the shaping of material form were the ancient Pythagoreans. Pythagoras ( 6th century B.C.) was a widely revered philosopher-mathematician who is said to have coined the term philosopher, meaning “lover of wisdom.” Number as quality, or state of being, was central to Pythagorean philosophy. In the century and a half between Pythagoras and Plato, aspects of the philosophy were recorded in writing for the first time. It was the book of one of these authors, Philolaus, that Plato is said to have used as the basis for his mathematical creation myth in his dialogue Timaeus.

For Pythagoreans there existed three kinds of number. The first, quantitative number, is the number of counted things, the kind of number we use when taking inventory. The second, mathematical number, is what we manipulate without reference to actual objects, as in addition or subtraction. Lastly, qualitative number refers to an essential differentiating aspect of each number which establishes a unique relation to unity, without reference to mathematical manipulation or to actual objects. Pythagorians used this kind of number to model the coming into being of an orderly cosmos. Qualitative or archetypal number may be a guide to understanding the principles that inform sensible matter because it has the ability to link the apparent separateness of objects to unity.

The term soul, in the context of qualitative number, means that aspect of ourselves which is our connection to unity. Soul is the presence of unity on an individual level, a kind of personal cosmos, a refraction image of unity on the cosmic level. Its function is to vitalize and govern matter. The principal Greek philosophical term for soul is psyche, meaning non-physical animating principle. Psyche is also the root of our word psychological, meaning related to mind. Thus we can link the two in the term soul-mind. According to Plato, divinity used harmonic division to create soul-mind. Such a pattern of division of unity, which allows for its own re-integration, is known by the Greek term logos. The sensible order of the cosmos arises in the realm of soul-mind through the impression of logos upon matter. To make such an impression is part of the function of soul-mind. This is how we come to see an orderly world from the chaotic mix of raw sense input.


The Creator, Bible Moralisée, French c. 1250 Bodleian Library, Oxford, England
Christ using compasses to demonstrate the creation of the universe. Geometry represents the archetypal order underlying the physical world.

In ancient Greece, ratio and proportion were called logos, a word with a range of meanings concerned with comparative relationships and reasoning, with or without the aid of speech. The Neo-Platonist Theon of Smyrna writes: “The relation of proportion or ratio is also called [logos] and it is in this sense that it is said that there is a relationship of one thing to another thing… according to Plato, the word logos is used with four different meanings: for mental thought without words; for discourse proceeding from the mind and expressed by the voice, for the explanation of the elements of the universe, and the ratio of proportion….”1 One may thus say that logos is the proportional relationship which gives meaning or “the difference that makes a difference.”2

Timaeus, in Plato’s dialogue, who represents the Pythagorean worldview, tells us [30]3 “...this cosmos has come into existence as a living creature endowed with soul and reason …”and [36] “... when the structure of the [world] soul had been finished … the Divinity proceeded to fashion the whole corporeal world within it; fitting the two together center to center—and the soul was woven right through from the center to the outermost heaven….” The form of the cosmos and the soul are spherical, as [33] “...A suitable shape for a living being that was to contain within itself all living beings would be a figure that contains all possible figures within itself….” The two-dimensional representation of a sphere is a circle. The center of the circle is a point equidistant from all points on the circumference. Such a center point can be said to have location in space, but by definition, no measurable dimension. In addition, Timaeus [35] tells us that the soul has a three-part structure composed of “sameness, difference and intermediate being.” Sameness may be equated with the realm of cosmic mind expressed by the Greek word nous, unchanging principle. Difference may be equated with the constantly changing, sensible world. The median or balance between these two extremes is called “intermediate being,” which may be equated here with the individual psyche. Contemporary British architect and philosopher Keith Critchlow provides a likely geometric model of the circular three-fold Platonic soul (figure 1). He writes:

The Soul is the essential intermediary in the Platonic tradition … it is to be thought of as having a threefold nature. This can be conceived diagrammatically as comprising three circles. Firstly, a higher circle represents the Heavenly domain of transcendental principles, home of the “eide” or formal ideas. Secondly, a lower circle (touching the upper one at its lowest point) which represents the earthly domain of the created order, the immanent enactment of the principles in matter—our world of sensory experience. Thirdly, a joining circle representing the domain of the soul. This latter circle is centered on the meeting point of the heavenly and earthly circles with its topmost point reaching to the center of the heavenly and its lowest point reaching to the center of the earthly circle. It thus symbolizes the threeness of Soul, as well as demonstrating its balancing role. The upper part of this Soul sphere can be taken as the most subtle in nature and as like its heavenly prototype as is conceivable. The lower part animates matter and totally permeates it. The center being a balance between the two.4

In contemporary terms we might call the three circles spirit, mind and body. In figure 1 the three circles are shown within a larger circle called theos, the divine, the world soul, or the one. In this model the world soul and the individual soul share a common center, thus establishing an analogous relationship, that is, like the logos.

In the Pythagorean tradition, the soul makes one journey in its merger or wedding to materiality at birth, and another on the dissolution of its materiality at death. Timaeus [42] associates the individual souls with the stars, and in Plato’s Republic <sup51< sup=""> [614–21], the myth of Er describes the soul’s journey from the stars through the seven planetary spheres to earth. Timaeus [43] further relates that, as the soul descends, it becomes denser, more immersed in materiality until, upon reaching the earth, it has forgotten its original nature. The joining of the heavenly and earthly by soul may be seen as an act of memory, as access to the nous or divine intelligence that allows complete awareness of our ultimate nature. The task of education in this context is one of remembrance, or in Greek, anamnesis. Plato provides a demonstration of anamnesis in the Meno dialogue, using geometrical pattern to elicit a mathematical proof from an uneducated boy.

Figure 1. Geometrical representation of the Platonic soul

Mathematical pattern thus appears, for Plato, to be a primary prompt for the soul to recall its original state of unity. This remembrance of the true self, anamnesis, is an essential part of the process of the soul’s re-ascension. The anamnesic model assumes that you already have the knowledge of the logos within you and the power to recall it, because the soul of the world and of the individual are constructed on the same harmonic pattern, the logos. The individual may thus be considered a microcosm. Memory is a key factor in the soul’s re-ascension. The content of this memory is unity, the transcendent unity that differentiates into multiplicity, but can be apprehended by mind through reference to geometrical archetypes. The mind compares sense information to these archetypes of unity, or Ideas, to form images of the physical world and allow the experience of beauty.

The nature of beauty has long been a contentious topic. For a Pythagorean view, we might look at a statement by the Neo-Platonic philosopher Plotinus written in the fourth century A.D. Plotinus first asks what we find beautiful in the material world: [1.21] “Nearly everyone says it is good proportion of the parts to each other and to the whole … which produces visible beauty, ... being beautiful is being well proportioned and measured. On this theory, nothing single and simple, but only a composite thing will have beauty. It will be the whole which is beautiful and the parts will not have the property of beauty by themselves … But if the whole is beautiful the parts must be beautiful too…And when the same face sometimes appears beautiful and sometimes does not, though the same good proportion is there all the time, surely we must say that being beautiful is something else over and above good proportion, and good proportion is beautiful because of something else.”6 Plotinus next points to the beauty of virtue, but, he asks: [1.43] “... what can be meant by good proportion in beautiful ways of life, laws, studies or branches of knowledge? How can speculations be well proportioned in relation to each other? If it is because they agree, there can be concord and agreement between bad ideas. ... virtue is a beauty of the soul, a truer beauty than those mentioned before; but how is virtue well proportioned?” Plotinus’s questions here are rhetorical, as he does not consider virtue to be composite, and therefore it would not be beautiful in the theory of beauty as proportion.

Plotinus then notes that the soul is drawn towards beauty and repelled from ugliness. He continues: [2.08] “Our explanation of this is that the soul … is related to the higher kind of reality in the realm of being, and when it sees… a trace of its kindred reality … remembers itself. But how are both the things in that [higher] world and the things in this beautiful? We maintain that the things in this world are beautiful by participating in form; for every shapeless thing … is ugly as long as it has no share in formative power….” Plotinus’s forms are the same as those described by Plato in the Timaeus and other dialogues. These forms exist as Ideas in the intelligible realm, called nous in figure 1, and through the work of mind they give form to matter, which otherwise would have none. In traditional terms unformed matter is called “chaos,” in Greek, hyle.

Form is what raises chaos to cosmos. Kosmos, the Greek term for the sensible universe, is related to the English word cosmetic, meaning a surface covering or layer. By extension, kosmos is a surface covering on the form of the one, or we might say physical appearances adorn the one, or prime principle, which corresponds to the outer circle in figure 1. Nous, divine intelligence or ideal form, is a part of the one, which, when projected through psyche, gives rise to perceptible order in the otherwise chaotic, or formless material ground, the hyle. For Plotinus, the value of beauty is that it turns the attention of the soul upward from the changing material realm to the unchanging realm of the divine intellect. But perception of beauty is only possible when the soul is pure, as purity is related to unity: [5.49] “This is the soul’s ugliness, not being pure and unmixed, like gold, but full of earthiness; if anyone takes the earthy stuff away the gold is left and is beautiful, when it is singled out from other things and is alone by itself … [6.0] and every virtue is a purification, so is even wisdom itself … [6.12] the soul when it is purified becomes form and formative power, altogether bodiless and intellectual, entirely belonging to the divine, whence beauty springs.…” In the realm of nous, [6.23] “... beautifulness is reality and … the qualities of goodness and beauty are the same….”

In describing this state of beauty fused with truth (intellect) and the good, which can be viewed when the soul is pure, Plotinus writes: [7.0] “Anyone who has seen it knows what I mean when I say it is beautiful. It is desired as good and the desire for it is directed to good, and the attainment of it is for those who go up to the higher world … and strip off what we put on in our descent; ... until, passing in the ascent all that is alien to [the divine], one sees with one’s self alone, that alone, simple, single and pure, from which all depends, to which all look and are and live and think,—for it is the cause of life and mind and being. If anyone sees it, what passion they will feel,... what a shock of delight!” Finally, Plotinus offers this guidance for our journey: [8.26] “Shut your eyes, and … wake to another way of seeing, which everyone has but few use. ... [9.07] Go back into yourself and look; if you do not yet see yourself beautiful, then, just as someone making a statue which has to be beautiful cuts away here, polishes there, makes one part smooth and clears another till the statue has a beautiful face, so you too must cut away excess, straighten the crooked, clear the dark and make it bright, and never stop ‘working on your statue’ till the divine glory of virtue shines out on you, till you see ‘self mastery enthroned upon its holy seat.’”

Study and work in the outer world of action is, for Plotinus, just the prelude to the real work, the ascent of the soul, which must take place within one’s self utilizing memory of unity. Elsewhere Plotinus argues that beauty is the outward appearance of the attractive power of love which binds us to unity. This, however, is an inner, occult road. As designers, we may approach beauty by looking at its manifestation in the outer world through the four traditional sciences of number.

Number as guide to the harmony of the soul was taught in seven aspects or subjects in the schools of the Pythagoreans. In each of the subjects it is the harmonic, qualitative aspects that we are to study; we are not to focus overly on the mechanical details. The Trivium of three verbal subjects are Grammar, Rhetoric and Dialectic. The four mathematical subjects, known as the Quadrivium, are Arithmetic, number in concept; Geometry, number in space; Music, number in time; and Astronomy, number in time and space. As the study of these subjects was intended to help harmonize the orbits of the soul and assist its return journey to unity, they became known as the Liberal Arts, implying that they would help liberate the soul from entrapment in matter. The Liberal Arts became the basis for the curriculum at the Platonic academy in Athens, founded in the fourth century B.C. The early Christian fathers studied the system, which was further refined in medieval schools such as Chartres. This same system was reinterpreted in the Italian Renaissance.

The works of Plato and his contemporaries are the earliest written formulations of this study of number and harmony in the West. But reason and history point to these archetypal subjects as existing outside of any particular time or place. Indeed, they seem to have been developed over the entire span of known human existence. Such numerical archetypes may be used to construct an ontogeny, that is, to tell a story of the coming into being of the sensible world. What follows is a brief ontogeny using arithmetic and geometry (figure 2). It might be thought of as a Pythagorean-based creation myth.

Figure 2. Ontological stages and correspondences

Oneness—the Monad. All manifestation must have a starting point, which contains a fundamental mystery—it has location without dimension. Though the point contains everything in potential, it cannot be perceived with the senses as it is unextended in dimension. It is represented in geometry by the dot. The first shape is the circle, an inflation of the point that allows its potential to be actualized. The circle embodies the mystery of the point, as its circumference is a single line, entirely guided by the point, whose curvature is uniform and undifferentiated. Perhaps more importantly, the relation between the circumference and the area it encloses is not fully expressible in rational units. This first stage of our story may be thought of as that of transcendent unity. When taken as an applied unit of measure consistent with its character of sameness, unity is called the monad. In architecture such a key unit of measure is called the module.

This ontological level of trancendent unity is described in Genesis: “In the beginning God created the heaven and the earth. And the earth was without form and void—and darkness was upon the face of the deep. And the spirit of God moved upon the face of the waters.” (1: 1–2) We can read this as alluding to the center point, the circumference and their trans-rational relationship. Plato’s Timaeus tells us: [29] “Now to discover the maker and father of the universe is a task indeed, and when discovered, to reveal him to all through the ministry of discourse is a thing impossible.” The experience of transcendent unity is not fully communicable through words, just as the mystery of the circle is not fully communicable through rational numbers. Then he adds: [29-31] “Let us declare on what account the composing artificer constituted generation and the universe. The artificer was good… Hence … he was willing to produce all things as much as possible similar to himself … Do we rightly conclude that there is but one universe; or is it more right to assert that there are many and infinite? But indeed there can be but one if it is fabricated according to its exemplar. For that which comprehends all intelligible living creatures can never be second to any other.”

Twoness—the Dyad. If unity contains the all, everything manifest and unmanifest, then it must also contain its own opposite, separation from unity. For such a separation to occur there must be a direction prior to any movement. This initial form of extension is known variously as the ray of creation, lightning flash, world tree or axis mundi. Geometrically, this is the line. This movement is not an external separation but is division of unity from within. It is represented here by two equal circles centered on the axial ray.

In Genesis we read: “And God said, Let there be light: and there was light. And God saw the light, that it was good: And God divided the light from the darkness.” (1: 3–4) For something to be perceivable with the senses, a difference must exist. The beginning of difference in the dyad is symbolized by the separation of light from darkness. Timaeus tells us: [28] “We must… distinguish that which always is and never becomes from that which is always becoming but never is. The one is apprehensible by intelligence with the aid of reasoning, being eternally the same, the other is the object of opinion and irrational sensation, coming to be and ceasing to be, but never fully real.” These are the real and the actual. The dyad is the ontological level of moral choice, the separation from unity, the good versus the not good. For Pythagoreans, the dyad symbolized polar opposites such as limited-unlimited, odd-even, one-many, rest-motion, same-different. Order is connected to the idea of limit, for recognizable order must fall within some limit. Disorder and chaos are unlimited.

Threeness—the Triad. Any two separate things must have a relationship. This requires the triad. For sensible perception or any act of communication to take place, there must be at least a sender, a receiver and a medium of transmission. For any kind of physical structural stability, a spatial triangulation is required. The triad is the ontological level of the symbolic. We may represent the triad in our diagram by placing a third circle between those of the same and the different. This circle governs the creation of the equilateral triangle, the first plane figure.

In Genesis we read: “And God said, Let there be a firmament in the midst of the waters, and let it divide the waters from the waters. And God made the firmament, and divided the waters which were under the firmament from the waters which were above the firmament, and it was so. And God called the firmament Heaven.” (1: 6–8) Timaeus tells us [35] “And [the Demiurge] composed [the world soul] in the following manner and out of the following constituents. From the indivisible, eternally unchanging Existence and the divisible, changing Existence of the physical world he mixed a third kind of Existence intermediate between them, ... and taking these three components he mixed them into a single unity.”

Fourness—the Tetrad. A fully manifest object must have at least three dimensions, length, width and height. These three intervals are marked off by a minimum of four points: length width height. The tetrad, the ontological level of manifestation, may be visualized as a crossing of dyadic opposites, such as warp and weft which cross to create a fabric, or male and female polarities which must be crossed to allow physical birth. Geometrically, the tetrad may be represented by the square. It is in this sense that we still use the expression “the four corners of the world.”

In Genesis we read: “And God said, Let the waters under the Heaven be gathered together unto one place, and let the dry land appear—and it was so. And God called the dry land Earth: and the gathering together of the waters called he Seas.” (1: 9–10) Timaeus tells us: [31–32]

Anything that has come to be must be corporal, visible and tangible: but nothing can be visible without fire, nor tangible without solidity, and nothing can be solid without earth. So when the creator began to put together the body of the universe, he made it of fire and earth.

But it is not possible to combine two things properly without a third to act as a bond to hold them together. And the best bond is one that effects the closest unity between itself and the terms it is combining; and this is best done by a continued geometrical proportion [a:b :: b:c] .... If then the body of the universe were required to be a plane surface with no depth, one middle term would have been enough to connect it to the other terms, but in fact it needs to be solid and solids always need two connecting middle terms.

So the creator placed water and air between earth and fire, and made them so far as possible proportional to one another, ... and in this way he bound the world into a visible and tangible whole. So by these means and these four constituents the body of the universe was created to be at unity owing to proportion; in consequence it acquired concord, so that having once come together in unity with itself it is indissoluble by any but its compounder.

Our ontogeny has taken us from the mysterious point of emergence to full manifestation in four stages. Our building blocks have been the qualitative aspects of number and geometry. The stages of the story and some correspondences are summed up in figure 2. The two other subjects of the Quadrivium, Music and Astronomy, from the viewpoint of qualitative number, may also be treated ontogenically.

Figure 3. Number, geometry, music

As we have seen, the principle of unity, oneness can be taken as the beginning, as well as the all-encompassing, and may be represented by the circle. In terms of a ratio, unity may be stated 1:1; musically, unity might be called the fundamental (figure 3). But, as we have seen, unity is unmanifest. In order for something to be perceivable, there must be a difference within unity. The first step in this differentiation is for one to transform into two. In music this relationship of 2:1 is called the octave. The 2:1 ratio can be musically expressed through the relative length of strings or pipes, a distance known as an interval. A string of any given length, taken as an interval of one, is vibrated to produce the fundamental; a string twice as long, with an interval of two, will vibrate at a rate that produces a consonant sound one octave lower than the fundamental. The triad, represented in music by the 3:2 ratio, is known as a fifth. If the octave interval distance is from one to two, a string when plucked or struck at the 3/2 point will produce a consonant sound five notes lower than the fundamental. The tetrad is represented musically by the ratio 4:3, known as a fourth. With an octaval distance of one to two, a string when plucked or struck at the 4/3 point will produce a consonant sound four notes lower than the fundamental.

The first four numbers form four ratios—1:1, 2:1, 3:2 and 4:3—which define the octave and its two essential musical divisions. These first four numbers can also represent the stages of geometrical becoming—point, line, plane, solid. Extending the parallel, we see four stages of musical consonance, a fundamental, its octave, its fifth consonance, and its fourth consonance. This grouping of notes, known as the Pythagorean tetrachord, formed the framework for the harmonies of Hermes, Apollo and Orpheus.

Considering astronomy from a similar ontological-harmonic viewpoint, light may represent the monad and its absence, darkness, the dyad. The triadic relationship of these elements may be established by setting up a shadow stick. Sunrise, the point of scission of the limitless circumference, establishes the axes of the four cardinal directions. Observation of the movement of shadows yields information in pattern form. As the sun changes its position in the sky, the shadow changes its position inversely on the ground. Such an observation may begin with the shadow stick at the center of a circle. The shadow is marked when it crosses the circumference of the circle just after sunrise and again just before sunset. Connecting the two points on the circumference gives the east-west axis. The north-south axis, which is marked by the high point of the sun’s movement through the sky, can be found by constructing a perpendicular to the east-west axis.

In ancient Greece the shadow marker was called a gnomon, a word related to gnosis, meaning experienced knowledge. The repository of such knowledge was the temple. The word temple is related to tempo—time pattern. The temple may be thought of as the concretion of the spacial-temporal pattern of the astronomical bodies, especially the Sun, sacred to Apollo. As space may be said to emerge with the triad and time with the tetrad, their product—twelveness, the duo-decade—may symbolize the manifest cosmic structure. Thus the year, equal to one solar cycle, is traditionally divided into twelve zodiacal signs.

We have described a descent from unity using the images of the four subjects, each like a step on an ontological pyramid. We are now in the realm of the sensible, day-to-day world, what might be called the world of action. It is from here that we must begin the soul’s return journey. If we wish to act in the spirit of Platonic remembrance, we might base our actions on the Pythagorean ontological process. Socrates offers this guidance to would-be governers of Plato’s Republic: “the philosopher holding converse with the divine order, becomes orderly and divine, as far as the nature of man allows …. ” [500-501] He adds, “no state can be happy which is not designed by artists who imitate the heavenly pattern. … and when they are filling in the work they will often turn their eyes upward and downward, … first looking at absolute justice, beauty and temperance, and then at the human copy … thus they will conceive according to … the form and likeness of the divine.”As designers, we act through the shaping of matter, and that shaping may be guided by remembrance through re-enactment of the geometric archetypes we experienced in our descent.

In the next part of this article we will continue on the pathway to beauty by developing two designs: one inspired by the twelvefold pattern of the sun and the musical octave, dedicated to Apollo, and another utilizing the golden section, dedicated to Aphrodite.


Steven Bass teaches a course titled "Theory and Practice of Proportion” which this article is based on.

American Arts Quarterly, Volume 18, number 2.

1. Theon of Smyrna, Math Useful for Understanding Plato, trans. by Robert Lawlor (San Diego: Wizard’s Bookshelf), p. 481.
2. Gregory Bateson, Mind and Nature (New York: Bantam, 1980), ch. III, p. 76. ch. IV, p. 110.
3. Plato, Timaeus, trans. by Thomas Taylor [1793], (Minneapols: Wizards Bookshelf, 1976). For a reading akin to this discussion, see R.G. Bury’s translation (Loeb Classical Library, 1981).
4. Crithlow Keith, Soul as Sphere and Androgyne (London: Golgonooza Press, 1980), p. 27.
5. Plato, The Republic, trans. by Benjamin Jowett (New York:Vintage, 1991).
6. Plotinus, Ennead 1.6, On Beauty, trans. by Armstrong ( Loeb Classical Library, 1966).