Listening for the Harmony of ApolloPart II
In the first part of this article we explored the concept of qualitative number, in which each number has a unique character in relation to unity. We used this concept to develop a theory of beauty as the memory of unity and then to tell a story of coming into being, illustrated by the four sciences of qualitative number—arithmetic, geometry, music and astronomy. To approach the achievement of beauty in a material design, we might wish to use methods analogous to our ontogenic story. The following example is offered in the spirit of illustrating such methods.
The design presented here is intended to develop an understanding of timeless principles and explore the possibility of their expression in the present moment. The goal is not to imitate or reconstruct past forms specifically, and no claim is made that this particular method was used by the ancients. The subject of our design is the four-columned facçade or tetrastyle portico, an archetype of traditional architecture, which often forms a core or seed of more complex compositions. The design process may be divided into four stages which recapitulate the stages of our ontogeny.
1. Unity—We begin by restating the Pythagorean geometrical model—monad, dyad, triad, tetrad (figure 4a). From the inner tetrad, we construct an outer square, a material representation of unity, which is the limit of the height and width of our design (figure 4b).
2. Division—The essential scission of unity. The square is divided by means of its diagonals and the diagonals of its half (figure 4c). The intersection of these two sets of diagonals divides the sides of the square at the one-third points. These points will mark the centers of the two inner columns and also their height.
3. Structure—The disposition and demarcation of the major parts of the design (figure 4d). By the canon of Vignola,1 if the height of the columns, including base and cap, is divided into four units, the height of the entablature is one additional unit. The lower diameter of the columns will be 1/8th of their height. In our design this is 1/12 of the side of the enclosing square. The sizes of other elements are derived from the lower diameter. Here we may see an integration of incommensurable geometry and the linear measures of the canon. The decimal equivalent of 1/3 is .333…; that of half the side of the square is .500. The difference between them is 1/6 or .166…, half of which is 0.083…, or 1/12. Thus, geometrical manipulation alone could be used to size and position the inner columns.
4. Manifestation—Full articulation of the details and ornaments (figure 4e). The order selected for this design is Doric. Of the three Greek orders, Doric may be most associated with the Apollonian or rational function, the initial phase of the triadic cycle, the proceeding or coming forth. According to Vitruvius,2 it may be selected to represent senior male divinities such as Ares, heroes such as Herakles, or certain female divinities such as Athena. We see in figure 4d that the Doric capital may be constructed on the same geometrical pattern as that of the facade as a whole.
Figure 4. Portico of Apollo
Because this design is undertaken in the spirit of communication, bringing that which was hidden into light, and because the design uses the solar number twelve, it may be dedicated to the enlightened sun god Apollo. As it is based on the geometry of the musical octave (figure 4e), and remembering that Apollo was given the lyre by Hermes, teacher of that which is hard to find, we may call this design the portico of Apollo, solar musician.
The design, in its twelve-fold division, may be said to have used the powers of three and four; as 3×4 = 12, so 3 + 4 + 5 = 12. Among the many discoveries ascribed to Pythagoras is that the numbers 3, 4, and 5, taken as intervals of length, are the smallest whole numbers that will form a right triangle whose sides satisfy the famous equation a2 + b2 = c2. This 3-4-5 archetype forms the basis of an ontogeny of its own, that of the triadic cycle. Four may be taken here to represent the power of two which is associated with the proceeding aspect of the cycle, as it is the √ 2 that governs the doubling of the area of the square, symbolizing the essential division of unity, the first step toward manifestation. The power of three is associated with the maintaining aspect through the relation of the √ 3 to the altitude of the equilateral triangle, symbolic of both mediation and structure. The power of five represents the returning aspect because, through its relation to the so-called Golden Section, it may be said to return to unity through division.
Contemporary commentator Robert Lawlor characterizes these three aspects as generative, formative and regenerative, adding that the Golden Section was “used by the Platonic philosophers as a support for the ideal of divine or universal love. It is through the Golden Division that we can contemplate the fact that the Creator planted a regenerative seed which will lift the mortal realms of duality and confusion back toward the image of God.”3 Known in ancient Greece as the extreme and mean ratio, it was later called the Golden Section [ratio, proportion or division], and in more recent times it has been denominated by the Greek letter phi, Ø.
In seeking unity, whatever points the way from the many to the one is of value. Just as the golden sun and the metal gold have been symbolically related to oneness by Plotinus,4 the Golden Section embodies an approach to oneness in terms of geometric construction. Pursuing such a search for unity, we may ask: Is there a rectangular figure, symbolizing multiplicity, that can be proportionally divided to produce unity, symbolized by the square? Such a rectangle (figure 5a) has its sides in the geometric progression a2 : a : 1, such that a + a2 = 1. That is, root plus square equals unity. Phi is the only number that satisfies this condition.
Defined mathematically, Ø = [√ 5-1]/2; the square root of 5 equals 2.236…, subtracting 1, we have 1.236… ; and dividing by 2 gives .618… . Numerically substituting Ø for “a” in the equation a + a2 = 1, we have .618… + .381… = .999… ˜ 1.0. Phi, as a transcendental, does not actually reach 1 but makes what may be called a near approach. Keeping in mind that a ratio is a comparison of two numbers and a proportion is a relationship of equality between ratios, Lawlor reminds us, “The Golden Proportion is a constant ratio derived from a ‘geometric’ relationship …. ....it is first and foremost a proportion, not a number….”5
Geometrically, the relationship of the Pentad and Ø is shown in figure 5b. Just as the diagonal of the square is related to the √2 and the altitude of the equilateral triangle is related to the √ 3, if the side of the pentagon is equal to 1, its diagonal is 1 + Ø or 1.618….
The Latin word Venustas, usually translated as “beauty” or “delight”—one of the three Vitruvian criteria for evaluating architecture, “firmitas, utilitas, Venustas”6—may also be taken to mean pertaining to Venus. Best known as the irresistible goddess of physical love and beauty, her star is brilliant and unmistakable in the evening or the morning sky. In the spirit of remembrance we lift our eyes to observe her. The planet Venus alternates between being an evening and a morning star. One synodic period of Venus, for example, from one furthest morning, or western elongation to the next, takes an average of 584 days or 1.625 years, close to the ratio of Ø, 1 :1.618. Venus completes five synodic periods in eight years, and this is also a whole number approximation of Ø, 1: .625. Venus’s movements over an eight-year cycle divide the circle of the zodiac into five parts, thus constructing the pentagonal star, the pentagram.
Figure 5. Phi, Ø, the Golden Section
Venus was called Aphrodite in Greece. Robert Graves recounts her origin: “Aphrodite, Goddess of Desire, rose naked from the foam of the sea and, riding on a scallop shell, stepped ashore first on the island of Cythera … passed on to the Peloponese and eventually … [to Paphos in Cyprus where] the Seasons … hastened to clothe and adorn her . … Grass and flowers sprang from the soil wherever she trod, ... all agree she takes the air accompanied by doves and sparrows.”7 Graves further informs us: “The fates assigned to Aphrodite one divine duty only, namely to make love; but one day, Athena catching her surreptitiously at work on a loom, complained that her own prerogatives had been infringed and threatened to abandon them altogether. Aphrodite apologized profusely and has never done a handís turn of work since.”8
Aphrodite is not so much the goddess of love as she is the goddess of making love. As the movements of the planet
literally draw out the pentagram in the sky above our heads, and the geometry of the pentagram reveals, also literally, the Golden
Section, so Venus-Aphrodite is the ruling deity of worldly beauty and reproductive activity. Graves’s charming comment that Aphrodite “has never done a hand’s turn of work” may be a hint that this myth is less concerned with the generative or formative aspects of the cosmic cycle than with the regenerative. It is appropriate that “ …grass and flowers sprang from the soil wherever she trod..,” as plants embody the phi ratio in the phenomenon of philotaxis, which governs branching and distribution of leaves. In addition Lawlor reminds us, “ …The rose family is one of those based on five, as are all the flowers of the edible fruit bearing plants. … the flowers of love such as the orchid, the azalea and the passion flower, are all governed by pentagonal symmetry.”9
The use of the shell symbol in the myth is also apt. The growth intervals of shells such as the nautilus are related to the arithmetic of the Fibbonacci series, whose adjacent terms are whole number approximations of Ø. The spiral thus symbolizes the expression of archetypal number in time. By satisfying the expression a + a2 = 1, organisms can grow without changing their overall form through the process known as gnomonic expansion.
Graves adds: “Some hold that Aphrodite sprang from the foam which gathered about the genitals of Uranus, when Cronus threw them into the sea.”10 Cronus, whose victory over Uranus initiates Aphrodite’s birth, is better known as Father Time, the planet Saturn. Moments in time may be brought out of eternity in the same way that segments may be marked off or cut from the continuous circumference of the circle. Saturn, as the outermost visible planet, performed this action for souls descending from the timeless realm of Uranus, the fixed stars. Thus Father Time holds the sickle, perhaps a source of the castration image in the myth. The image suggests that emergence into time is the father of beauty; if we had never separated from unity, we would not need to return.
That physical beauty is linked through love to the vital energy of life is illustrated in the myth of Pygmalion, in which Aphrodite brings to life or animates the perfectly wrought statue with which the artist has fallen in love. The myth implies a connection between love, beauty and soul, because soul, in the Platonic system, is that which animates or moves us. As the geometrical symbol of Aphrodite is the pentagon, which her star Venus marks off in its dance across the zodiac, and as the pentagon divides internally to generate the phi ratio, we may speculate that the phi ratio is an intermediary through which inert matter may be animated.
Figure 6. Portico of Venustas
The method of division of the circle into five parts and the internal divisions of the pentagram provide appropriate geometrical keys for another tetrastyle design. As in the previous design, the process can be seen to unfold in four stages:
1. Unity—As in the first example, we repeat the steps of our ontogeny, arriving at the square (figure 6a).
2. Division—As a fundamental act of division, we apply the Ø ratio to the height of the square and use this distance as the radius of a circle which will mark the apex of the pediment (figure 6b).
3. Structure—To obtain the size and placement of the two inner columns, we divide the circumference of the circle into five parts using a procedure related to Ø. Connecting these points, we construct the pentagram. Intersecting points on the diagonals mark the column diameter and its position (figure 6c). The horizontal diagonal will mark the lower edge of the entablature. It may be noted here that a square with a side of 1.0 and a circle with diameter of 1.236… have perimeters that are equal within 3%, as 1×4 = 4; and pi D = 3.14… x 1.236… = 3.88… which is 97% of 4. Such a relationship accomplishes a squaring of the circle, symbolizing a harmonic union of spirit and matter.
4. Materialization—Having placed and sized the major elements, we now complete the elevation (figure 6d), and move to detailing of the elements. We select the Corinthian order for this design because of its inner relationship to the pentagon and phi, which is hinted at in the myth of Callimachus as told by Vitruvius.11 The artist meditates at the grave of a maiden of Corinth. His design for a capital is inspired by leaves, flowers and volutes which have grown up around a basket of the maiden’s grave objects, which was weighted down with a stone tile. The maiden may be taken as a reference to Venus-Aphrodite. Though outwardly the spirit of the maiden has taken the form of vegetation, the inner structure of the capital’s composition is based on the double pentagon (figure 6e). As this design explores the relation of number to beauty in the sensible world, it may be dedicated to Venus-Aphrodite, goddess of physical beauty. But we should not fool ourselves into thinking that we can learn her innermost secrets in design school. We are only at the doorway of her temple, on the Portico of Venustas. We cannot now explore the mysteries within. Looking back through history, however, we can see several methods of applying cosmological geometries to design.
1. Geometrical Division—Beginning with the circle as monad, we may divide it within itself, initially by the ray or axis of creation, which establishes a direction for all further differentiation. Each of the regular polygons, beginning with the triangle, may then be seen as a monad in its own right. The manner in which each polygon divides the circumference, and aspects of its internal structure, may be used to locate parts or fix measures of a design.
2. Harmonic, or Proportional Division is similar to Vitruvian symmetria. In such systems all measures of the design have some relation of equality to a key measure or module, that is to say, a proportional relation. When the parts have measures which are symmetrical in this sense, with each other and with the whole, they are said by Vitruvius to have Order, or “due measure”; and the whole is said to achieve “Eurythmy” or ….“beauty and fitness in the adjustment of the members .…”12 Proportional division or Vitruvian symmetria can be achieved by ratios with musical relations. These are expressed in the writings of Alberti, Palladio and others as groups of rectangles whose sides are in the ratios of 1:1, 2:1, 3:2, and 4:3, the ratios of the Pythagorean tetrachord; 5: 3, a Ø approximate; and 5:4, approximately 2Ø, marking the tempered third. Palladio, for example, recommends determining the height of a room by selecting one of the three means, Arithmetic, Geometric or Harmonic, between its length and width. The musical ratios themselves arise from division of unity by these three means. In many plants and animals, and in the human body, symmetry is conferred by the phi ratio. Phi may be said to be the ultimate giver of Vitruvian symmetria, the ultimate proportional division, because it expresses geometrically the pattern of division that returns to unity [a + a2 = 1]. It thus establishes the possibility of beauty as the shock of delight in the anamnesic experience, the remembrance of unity in multiplicity.
3. Arithmetic Division—in which an overall measure of the design is divided into some number of parts. In the case of the Vitruvian tetra style temple13(figure 7a), the width of the facade is divided into 11.5 parts; 9.5 of which comprise the height of the columns to the underside of the architrave. It is suggested here that arithmetic systems such as that given by Vitruvius may be public methods of recapitulating geometric constructions, which in ancient times were privately held by various family, craft and religious bodies. Such methods of approaching beauty are possible and understandable if we postulate architecture as a mesocosm, a middle ground in which the temple, like the human being, stands with the sky above and the earth below, ontologically joining the realms of the same and the different. These opposites are united and vitalized by the creative act, in which a flow is established between them. Similar to what is variously called prahna, chi or pneuma, such a vital flow may be likened to the Sheckenah of the Solomonic temple or the Egyptian Ka. The middle realm is also that of mind, psyche or soul, and it is within psyche that the creative act takes place. Thus, there may be analogies among the individual human being, the temple and the cosmos as a whole. Such an ideal analogue of the individual, called Anthropos or Cosmic Man, appears as Purusha in India, Adam Kadmon in the Kabbalistic tradition, and Leonardo’s Vitruvian man (figure 7b). This cosmic man symbolically sacrifices himself to give the pattern of measure for the temple. The concept of Anthropos informs the maxim of the Renaissance humanists that man is the measure of all things, as well as the Vitruvian myth of the Dorians measuring themselves to obtain the pattern for the Doric order.14 It also may lie behind the Delphic-Socratic imperative “know thyself.” If the individual is an analogue of the cosmos, created on the same pattern of number, then creating in turn by the same principles reminds us of that one cosmos and our relation to it, that is to say, of unity in multiplicity. This, as we have seen, is the source of beauty.
Figure 7. Anthropos, the Temple and the Squaring of the Circle
Since beauty, in the Platonic tradition, is also the Good and people are naturally attracted to the Good, by studying such methods we may acquire the power of creating captivating work, whatever its function, scale or material. But the designer should keep in mind that beauty is not guaranteed by geometry, because beauty is not in things or the measure of things. It is the joyous state of the individual psyche as it recalls unity. Its achievement in design is not all or nothing but is in keeping with our own degree of inner clarification.
The path to beauty is an inner one. It is not direct, well illuminated or clearly marked. Some even deny it exists at all. But don’t be discouraged. Plotinus advises: “keep working on your statue,” and Socrates admonishes us, “In heaven there is laid up a pattern of the ideal city which those who desire may behold, and beholding, may set their own house in order.”15 If the path seems difficult or lengthy, remember, look up and listen for the music.
2. Vitruvius, Ten Books on Architecture, trans. by Morgan (New York: Dover, 1960), I, 2, 5.
3. Robert Lawlor, Sacred Geometry (London: Thames & Hudson, 1982), p. 37.
4. Plotimus, Ennead, 1.6.5, trans. by Armstrong (Cambridge: Harvard, Loeb Classical Library, 1966).
5. Robert Lawlor, Sacred Geometry, p. 46.
6. Vitruvius, Ten Books on Architecture, I, 4, 2.
7. Robert Graves, The Greek Myths (New York: Penguin, 1982), p. 49.
8. Ibid., p. 70.
9. Robert Lawlor, Sacred Geometry, p. 58.
10. Robert Lawlor, Sacred Geometry, p. 49.
11. Vitruvius, Ten Books on Architecture, IV, 1, 8–11.
12. Vitruvius, Ten Books on Architecture, I, 2, 1–4.
13. Vitruvius, Ten Books on Architecture, III, 3–4.
14. Vitruvius, Ten Books on Architecture, IV, 1.6.
15. Plato, The Republic, trans. by Benjamin Jowett (New York: Vintage, 1991), p. 592.
American Arts Quarterly, Summer 2001, Volume 18, Number 3.