BEAUTY MEMORY UNITY

The foremost ancient explorers of number as a guide and analog to the shaping of material form were the Pythagoreans. Pythagoras was a widely revered philosopher – mathematician who is said to have coined the term ‘philosopher’ meaning ‘lover of wisdom’. He was born about 600 BC on the Greek island of Samos. In his teens he traveled in Ionia, studying with several teachers including Thales of Miletus, who advised him to continue his studies in Egypt, with the priests of Memphis. Pythagoras is said to have studied in the Egyptian temple school system for twenty-two years, then perhaps for another twelve years in Mesopotamia, where he may also have picked up influences from Persia and India. Returning to Greece, now middle aged, Pythagoras set up a school on Samos. As his fame spread his disciples formed schools in other cities in Greece, Southern Italy and Sicily. But as they gained prominence and political influence opposition arose. Adherents of the school were persecuted, many were killed and the survivors dispersed. During the next century the teachings were passed down privately through families. In this period aspects of the philosophy were recorded in writing for the first time and it is said that the books of one of these authors, Philolaus, that Plato used as the basis for his mathematical creation myth in the dialog of Timaeus.

Number as quality, or state of being, is central to Pythagorean philosophy. David Fideler in the introduction to Guthrie’s ‘Pythagorean Sourcebook’ writes, (PS p 20) “… a metaphysical philosophy of Number lay at the heart of his thought and teaching, permeating … even the domains of psychology, ethics and political philosophy.”
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. Finally, qualitative number refers to an essential differentiating aspect of each number, without reference to mathematical manipulation or to actual objects. It is in this sense that number may become an archetype of Unity and in this sense number was used by Pythagoreans to model the coming into being of an orderly cosmos. Qualitative, 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; its unity a refraction image of Unity on the cosmic level. Its function is to vitalize and govern matter. The principle 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. It is thus not without foundation to link the two and speak of psyche as soul-mind.
According to Plato the means divinity used to create psyche is harmonic division. Such a pattern of division of Unity which allows for its own re-integration is known as a Logos. The sensible order of the cosmos arises in psyche by the result of the impression of Logos upon matter. To make such an impression is the function of psyche. This is how we come to see an orderly world from the chaotic mix of raw sense input.
Timaeus, a personage in the Platonic dialog of the same name who is taken to represent the Pythagorean world view, tells us that [TIM 30] “…this cosmos has come into existence as a living creature endowed with soul and reason … and [TIM 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 centre to centre; and the soul was woven right through from the centre to the outermost heaven …”
The form of the cosmos and the soul are spherical [TIM 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. Therefore the Divinity turned it into a rounded spherical shape…” The two dimensional representation of a sphere is a circle. The centre of the circle is a point equidistant from all points on the perimeter. Such a centre point can be said to have location in space, but by definition, no measurable dimension.
In addition, Timaeus [TIM 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’, nonphysical unchanging principal. Difference may be equated with the constantly changing, perceived, physical world. The median or balance between these two extremes is called intermediate being, which may be equated here with the circle centre described above. Contemporary architect and philosopher Keith Critchlow provides a likely geometric model in the diagrams in figure 1.

In describing figure 1a he writes [SSA, p 27], “The Soul is the essential intermediary in the Platonic tradition. …it is to be thought of as having a three fold 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 (dashed line) 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.” We may also see this joining of the heavenly and earthly within soul as an act of memory, as access by the individual to the Nous, allowing complete awareness of our ultimate nature.

In figure 1b [TSS, p 55] different terms are applied to the same geometric symbol. Figure 1b’s terms may be thought of at an individual scale and 1a’s at a cosmic scale. In contemporary terms we might call the three circles body, mind and spirit. In figure 1a the three circles are shown within the world soul called the One, the principle of unity itself, Nous can be taken to mean pattern and logoi as copy.

In the Pythagorean tradition, the Soul makes a journey from a divine realm to its merger with or wedding to materiality at birth, and another on the dissolution of its materiality at death. Timaeus [TIM 42] associates the individual souls with the stars, and in Plato’s Republic [REP 614-21], the myth of Er describes the soul’s journey from the stars through the seven planetary spheres to earth. Timaeus further relates that as the soul descends it becomes denser, more immersed in materiality until, upon reaching the earth, it has forgotten its original divine nature. In this view, the task of education actually is one of remembrance, or in Greek, anamnesis. Plato provides a demonstration of anamnesis in the Menno dialog, in which he uses geometrical pattern to elicit a mathematical proof from an uneducated boy.
Harmonic 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 of the presence of unity presumed to exist in the individual soul.
Memory is thus a key factor in the soul’s re-ascension. The content of this memory is unity. This is the transcendent unity that differentiates into multiplicity where it can be apprehended by mind, through the memory of geometrical archetypes. Sense information is then compared by mind to these archetypes of unity, called ‘Ideas’ or ‘forms’ by Plato, to create our image of the physical world.
For Pythagoreans, one’s actions may be judged against the need to maintain a dynamic balance or harmony of elements in the soul to facilitate remembrance. Study of the harmonic patterns in the physical world would lead by analogy to increased understanding of the harmony of elements in the individual’s soul; such remembrance being possible because the soul of the world and the individual soul were constructed on the same harmonic pattern, the Logos, allowing the individual to be considered a microcosm.

What is meant by beauty has certainly been a contentious topic for a long period of time. Current popular opinion seems to be that beauty is subjective, ‘in the eye of the beholder’, with no possibility for firm or fixed objective criteria. In this view beauty is relative and judgments can only be made on the basis of personal ‘taste’. Beauty can have no ‘function’ today except pleasure. For a Pythagorean view of beauty we might look at a statement on the subject by the ‘Neo-Platonic’ philosopher Plotinus written in the third century AD [ENN I.6].
Plotinus first asks what we find beautiful in the material world. [ENN 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.”
Plotinus next points out that there is also 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’ 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’. A proportion is an equality of ratios and ratios are comparisons of numbers, thus proportions are composite entities.
Plotinus next says that the soul is drawn towards beauty and repelled by ugliness [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 … . The form, then, approaches and composes that which is to come into being from many parts into a single ordered whole, … for since [form] is one itself, that which is shaped by it must also be one, as far as a thing can be which is composed of many parts. So beauty rests upon the material thing when it has been brought into unity, and [beauty] gives itself to parts and wholes alike. …the beautiful body comes into being by sharing in a formative power which comes from the divine forms.”
The forms of which Plotinus speaks 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 1a, and they give shape to material objects , which otherwise would have none. In traditional terms unformed matter is called ‘chaos’, in Greek, ‘hule’ .
In a conversation with your present author, Critchlow noted, “Form is what raises chaos to cosmos.” Cosmos, the Greek term for the sensible universe, is related to the English word cosmetic, meaning a surface covering or layer. When your present author asked Critchlow, ‘What is the relationship between cosmos and the cosmetic’, he replied by holding up a table-tennis ball, saying (in paraphrase), ‘this sensible ball is the cosmetic aspect of the form called sphere’. By extension, cosmos is a surface covering on the form of the One; or we might say physical appearances ‘adorn’ the One, or prime principle. The One, which corresponds to the outer circle in figure 1a, might also be thought of as God, or the Divinity, called in Greek Theos. The relationships here are that 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 Hule.
Plotinus, in finishing this section says [3.34], “So much then for beauties of the realm of sense, images and shadows which, so to speak, sally out and come into matter, adorn it, and excite us when they appear.”

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. … [6.01] 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 … so for God [or in the realm of the Divine Intellect], the qualities of goodness and beauty are the same … and, we must posit beauty which is also the good; from this [good] immediately comes intellect, which is beauty, and soul is given beauty by intellect. Everything else is beautiful by the shaping of soul …”.
In describing this state of beauty fused with intellect or truth, and with goodness which can be viewed when the soul is pure, Plotinus writes, [7.01] “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 are converted and strip off what we put on in our descent; (just as for those who go up to the celebrations of the sacred rites there are purifications, and stripping off of the clothes they wore before, and going up naked) until, passing in the ascent all that is alien to the God, 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! …[the person] who has seen it glories in its beauty, is full of wonder and delight, enduring a shock which causes no hurt … [7.29] Here the greatest, the ultimate contest is set before our souls; all our toil and trouble is for this, not to be left without a share in the best of visions.”
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 on its holy seat’. “
So, in the end, study and work in the outer material world is, for Plotinus, just the prelude to the ‘real’ work, the ascent of the soul, which must take place within one’s self, using only that which is really yours, your memory. Here Eros, desire, stimulated by beauty is elevated to Love, a desire to join eternally with the highest Good [SYM 209-11, p 86]. This, however, is an inner occult road. We, as designers, will approach beauty by looking at its manifestation in the outer world through the four traditional sciences of number.

The study of number as guide to the soul came to be taught in seven aspects or subjects in the schools of the Pythagoreans and as imparted in Plato’s dialogs of the Republic, Timaeus, Laws and others. The three ‘verbal’ subjects are, Grammar, Rhetoric and Dialectic; the four ‘mathematical’ subjects are, Arithmetic, Geometry, Music and Astronomy. As the study of these subjects was meant to help harmonize the soul and assist its return journey to the divine, they became known as the Liberal Arts. Liberal, used in this original sense, implies that these studies help to liberate the soul from entrapment in the material realm. This system became the basis of the curriculum at the Platonic academy in Athens, founded in the fourth century BC. This is the system in which the early Christian fathers studied and it was refined in the Middle Ages, especially at the School of Chartres. It was the same system that was reinterpreted in the Italian Renaissance.
The classical Greek works of Plato and others of his time are the earliest written formulations of this study of number and harmony. But reason and history point to these archetypal subjects as existing outside of any particular time or place. They seem to have been developed over the entire span of known human existence, and awareness of them may well be seen as evidence of our emerging humanness.
The four mathematical subjects of Plato are known as the Quadrivium. In each of the subjects it is the harmonic, qualitative aspects that we are to study and not to overly focus on the mechanical details. Arithmetic is Number in concept. Geometry is Number in space. Music or Harmony is Number in time. Astronomy or Cosmology is Number in time and space. Each subject constitutes an ontogeny, a story of coming into being. This allows us to recount a combined numerical and geometrical story of how things come to be.

A Brief Theology of Arithmetic and Geome

All manifestation must have a starting point. This point contains a fundamental mystery – it has location without dimension. The point contains the All in potential as it is the necessary initial manifestation of anything. But the point 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. It is a kind of inflation of the point that allows the potential of the point to be actualized. It embodies the mystery of the point because its shape is entirely guided by the point. The circumference is a single line whose curvature is uniform, undifferentiated thus continuing the mystery of the point. Perhaps more important is that the relation between the circumference and the area it encloses, though fully visible, is not expressible in rational numerical units.
This first stage of our ontogeny, our story of coming into being, may be thought of as that of transcendent Unity. Unity as an applied unit of measure consistent with its character of Sameness, is called the Monad. In architecture such a key unit of measure is called the Module. An early image of the primal undifferentiated state of unity is the Uroboros, a snake with its tail in its mouth, symbolizing continuous self generation.
To further illustrate the nature of these ontological levels we will look at descriptions of them that appear in religious, philosophical and mythological literature.
In the first book of Genesis we read: [01-2] 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.”
Plato’s Timaeus tells us: [TIM 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 ultimate unity is not communicable through words, which are composite. Then he adds, [29-31] “Let us declare then 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.”

Timaeus tells us: [TIM 28] “We must … distinguish between 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 same and the different.
Robert Graves tells this Orphic creation story: [GM 2a-b, p 30] “… black winged Night, a goddess of whom even Zeus stands in awe, was courted by the wind and laid a silver egg in the womb of darkness; and Eros, whom some call Phanes [light], was hatched from this egg and set the world in motion.”
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 the light from the darkness. This is the ontological level of moral choice. It is the polar division within unity, the good vs the not good. For Pythagoreans the Dyad symbolized polar opposites such as: limit – unlimited; odd – even; one – many; right – left; rest – motion; same – different. Order is connected to the idea of limit for recognizable order must fall within some limit. Disorder and the not-good are unlimited. In Taoism the polar opposite qualities of the Dyad are known as yang and yin, the creative and receptive.

Any two separate things must have a relationship. Thus arises 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 structural stability a triangulation is required. The triangle resists deformation because none of its inner angles can change without a change in the length of the opposite side.
The Triad is the ontological level of the symbolic. The triangle is the first plane 2-dimensional figure. Geometrically we may represent the Triad by placing a third circle between those of the same and the different. This circle governs the creation of the equilateral triangle.

In Genesis we read: [06-8] “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. And the evening and the morning were the second day.”
Timaeus tells us [TIM 35]: “And he 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.”

Pythagoreans considered the Pentad the ‘balance point’ as it stands midway between the first four numbers of the decade and the last four.
In Genesis we read: [11-13]”And God said, let the earth bring forth grass, the herb yielding seed, and the fruit tree yielding fruit after his kind, whose seed is in itself … and God saw that it was good.” On the relation of this image Lawlor [SG p58] reminds us, ” … The rose family is one of those based on five, as are all the flowers of the edible fruit bearing plants. Thus five signals to man his proper foods. … The very exotic flowers, … the flowers of love such as the orchid, the azalea and the passion flower, are all governed by pentagonal symmetry.”
Timaeus [TIM 55c], after assigning each of the four elements to one of the Platonic solids, seems reticent to mention the Pentad saying only, according to Taylor, “… There is also a certain fifth composition, which divinity used in the fabrication of the universe …”. Or as Bury puts it, “And seeing that there still remained one other compound figure, the fifth, God used it up for the Universe in his decoration thereof.” Lee gives us, “There still remained a fifth construction, which the god used for embroidering the constellations on the whole of heaven.” All these references are to the three-dimensional figure called the dodecahedron, which has twelve pentagonal faces.

 Also ascribed to Pythagoras is the discovery that the numbers 3, 4, and 5, taken as intervals of length, will form a right triangle whose sides satisfy the famous equation A2 + B2 = C2. The 3-4-5 archetype can also be expressed as √2, √3 and √5. In this expression it forms the basis of a family of rectangles starting with the square. The diagonal of the square is equal to the √2. A rectangle whose short side is 1 and whose long side is √2 is called a ‘root 2 rectangle’. The diagonal of a root 2 rectangle is equal to the √3. A rectangle 1 by √3 is called a root 3 rectangle. Its diagonal is equal to the √4 = 2. A rectangle of 1 by 2 is a double square, its diagonal is equal to the √5. If this double square is considered a single half-square the ø ratio may be determined geometrically.
Expressed as roots the 3-4-5 archetype constitutes three formative powers that symbolize the cosmic cycle of proceeding, maintaining and returning. For Lawlor, √2 is characterized as Generative because it [SG, p 28], “… contains the power which instantaneously transforms 1 into 2.” and establishes the principle of Alternation; the √3 is said to be Formative [SG, p 34)], “The √3 contained within the Vesica Piscis is the formative power giving rise to the polygonal ‘world’.” ; and the √5 can be considered as Regenerative, because through its relation to phi, it may be said to return to unity through division, [SG, p 37], “The √5 is the proportion which opens the way for the family of relationships called the Golden Proportion. The Golden Proportion generates a set of symbols which were 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.”

Of the remaining numbers of the decade we can speak only briefly here.
Sixness – the Hexad is symbolic of perfection and Pythagoreans considered six to be a perfect number. Perfect numbers in this sense are equal to the sum of their factors. For example, 6 is equally divisible into six parts of one unit each, or two parts of three units each, or three parts of two units each; 6/6 = 1, 6/ 2= 3, 6/3 = 2, and 1+2+3 = 6. Note that 6/4 and 6/5 do not yield whole numbers. In the case of 28 we have, 28/28 = 1, 28/14=2, 28/7=4, 28/4=7, 28/2=14 and 1+2+4+7+14 = 28.
Sevenness – the Heptad is considered a ‘virgin’ because it does not ‘multiply’ within the decade, that is, seven as a prime is created only by addition, not multiplication and there is no whole number multiple of seven that is ten or less. Seven is associated with wisdom, especially with the virgin goddess Athena. Seven is also associated with the idea of ascension through the seven planetary spheres.
Eightness – the Octad represents the eighth sphere of the fixed stars. It also symbolizes harmonia as the octave is complete at eight notes.
Nineness – the Ennead symbolizes completion as nine is the last integer. It is the all inclusive, the boundary. It also is associated with the ocean as in ancient times the ocean was presumed to completely surround the land.
Tenness – The Decade represents the new Monad. It is the realization of the potential of the Monad. Its most important representation is called the Tetraktys.

The decade is the sum of the first four numbers, 1+2+3+4 = 10. If each integer is represented as a dot and the ten are arranged in a triangular pattern, as in figure 3, the image of the Decade so formed is known as the Tetraktys. This kind of morphic or geometric arithmetic was common in ancient times. We should remember that the number sigals we take for granted were not used in Europe until the early Renaissance. Prior to that time arithmetic was taught with small round pebbles similar to marbles called calci. The term is related to our word calculation.
The four levels of the Tetraktys represent four stages of coming into being: unity, division, structure and full manifestation; as well as the four subjects of the Quadrivium, Number, Geometry, Music and Astronomy. The Tetraktys is a combination of number and geometry. Musically, the first four integers are the only numbers needed to obtain the ratios of the Pythagorean tetrachord. The fundamental is 1:1, the octave is 2:1, the fifth is 3:2, and the fourth is 4:3.
In his Forward to the Theology Of Arithmetic [an ancient anonymous manuscript of philosophical notes associated with the Platonic academy] Critchlow comments on the Tetracktys [TOA p 9], “That ten is ‘complete at four’ is a well known Pythagorean paradox based on the simple cumulative progression of 1+2+3+4 = 10; or in the direct manner of those who had no separate number symbols” [as in figure 3a]. He adds that, “… we have a most interesting Pythagorean idea to contend with – that ideal number is not necessarily subject to a sequential or causal progression from one through to ten, but is rather a unity with ten essential or potential qualities, simultaneously present in the Decade or Tetraktys.”
Critchlow continues [TOA p 10], “In this way we can see a progression that embraces Plato’s doctrine that ‘images’ in the human mind reflect the universal source in the mind of the Divine Artificer. So we get one reflecting as a single dot or shadow on a surface for us to ‘imagine’ or remember oneness from. If we should take the Pythagorean tradition literally, the first and comprehensive evidence of ‘ideal’ or archetypal number is an array of dots in triangular form [see figure 3b]. This pattern might be called, in current usage, a refraction pattern of a single triangle. … This Decade or pattern can be taken both as a graphic (differentiated) form of ten dots in four rows and/or as one triangular form outlined in nine dots with a center point. From the above pattern we can see how the Tetraktys of the Pythagoreans may represent both a minimal oneness and a maximal tenness simultaneously.”
Some correspondences to the tetraktys as an ontological model are indicated in table 1 below:

A ratio may be described as a comparative relationship between two numbers such as 2:1. A proportion, Theon says [MUP, p 49], “…is the relationship of ratios to each other, as 2 is to 1 so 8 is to 4.” That is to say, proportion is a relationship of equality between ratios. Theon’s example is of a four term, discontinuous proportion, a:b :: c:d. It is also possible to have a three term, continuous proportion such as a:b :: b:c, which may be written a:b:c. In such a proportion the first and last term are called the ‘extremes’ and the middle term is called the ‘mean’. There are three types of continuous proportions used in ancient musical theory, which generate the significant ratios found to exist within the octave and which are relevant to visual design.
1. In the Geometric proportion, the product of the extremes equals the square of the mean, b2 = ac, b = √ac. For example, 2:4:8, where 4 = √2×8. Or, it may be described verbally, that ‘the mean differs from each extreme by the same ratio’; in this example, the double ratio, 4 = 2×2 and 8 = 4×2.
2. In the Arithmetic proportion, the sum of the extremes is equal to twice the mean, 2b = a+c or b = [a+c]/2, for example, 2:3:4 where 3 = [2+4]/2. Or it can also be said that ‘the mean differs from each extreme by the same number’. In the above example 2, 3 and 4 differ from each other by 1. In this way the arithmetic mean may be said to ‘give rise’ to the ‘natural’ series, 1, 2, 3, 4… . For this reason Taylor [TA, p 97)] considers the arithmetic mean as primary. In the context of music however, the geometric mean may be considered primary because it generates continuous octaval relationships; 2:4:8:16… .
3. The Harmonic proportion can be said to combine the qualities of the other two means. The harmonic mean is equal to the product of the extremes divided by the arithmetic mean, b = 2ac/a+c, for example, 2:3:6 where 3 = 2x2x6/[2+6] = 24/8. It can also be said to be ‘the mean which exceeds one extreme, and is exceeded by the other extreme, by the same fraction of each extreme’. In our example, 3 is greater than 2 by 1/2 of 2 and 6 is greater than 3 by 1/2 of 6.
In discussing the three means, Lawlor [SG, p 81)] calls our attention to their inter-relationships. First, he notes they contain “the general formula for all our basic mathematical operations. The arithmetic proportion contains the law for addition and its inverse, subtraction, and describes the relationship which gives the natural series of cardinal numbers 1, 2, 3, 4, 5, 6, etc. The geometric proportion contains the law for multiplication and its inversion, division, and describes the relationship which gives any series of geometric progressions”.
He adds “…the arithmetic proportion shows an equality of difference, but an inequality of ratio. Thus in the arithmetic proportion 3: 5:7, 7-5 = 5-3, but 7/5 does not equal 5/3. A geometric proportion on the other hand, is characterized by an equality of ratio but an inequality of difference. Therefore, in the geometric proportion, 2: 4: 8, 4/2 = 8/4 but 4-2 = 2 does not equal 8-4 = 4. … The most important and mysterious character of the harmonic proportion is the fact that the inverse of every harmonic progression is an arithmetic progression. Thus 2, 3, 4, 5, is an ascending arithmetic progression while the inverse series, 1/2, 1/3, 1/4, 1/5, is a descending harmonic progression. In music it is the insertion of the harmonic and arithmetic means between the two extremes in double ratios – representing the octave double – which gives us the progression known as the ‘musical’ proportion, that is, 1, 4/3, 3/2, 2.”
Ratio and proportion were called ‘logos’ in ancient Greece. This word had a range of meanings all having to do with comparative relationships. The basic meaning was reasoning, with or without the aid of speech. Theon [MUP p 48] writes, “The relation of proportion or ratio is also called this [logos] and it is in this sense that it is said that there is a relationship of one thing to another thing. It also has the meaning of the explanation of the elements of the Universe: the recognition of things which honor and are honored, and it is in this sense that we say: taking account of something, or not taking account of it (word of honor). The calculation of bankers is also called logos, as is the discourse of Demosthenes and Lysias in their written works (speech); the definition of things which explains their essence (reason), since it is to this that it applies; the syllogism and introduction; the tales of Lybius and fables. The name logos is also given to the eulogy, to the tale and to the proverb. … But, 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. … The ratio of proportion of two terms is a certain relationship that these terms have to each other, such as the double or triple.”
One may thus define logos as ‘proportional relationship which gives meaning’ or, to use a phrase attributed to Gregory Bateson, ‘the difference that makes a difference’. The passage from Theon also indicates a common ground, in ancient times, of proportional relationship, music, geometry, number, and “the explanation of the elements of the universe”, to be found through rational discourse.

Practical constraints prevent any detailed discussion of astronomy, however, the key concept here is the cycle of light and dark, particularly of the sun but also of other planets and fixed stars. The undifferentiated light of the sun is followed by a period of darkness. A simple yet fundamentally correct way to measure the changing nature of the light source, the sun, is by observing a shadow. One method of controlling the measurement of a shadow is by setting up a shadow stick vertically in flat ground. The stick may be thought of as a mediating third element in the sun – shadow, or light-dark relationship. As the sun changes its position in the sky, the shadow changes its position inversely on the ground. As the sun represents the source of light, the shadow represents the absence of light and its movement is inverse to the source of light. As the sun moves to the right the shadow moves to the left, as in figure 08a.
Observation of the movement of shadows, whether natural or artificial, yields information in pattern form. An elemental piece of information obtained from the shadow stick is the east-west axis. The procedure begins with the shadow stick at the center of the 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, shown in figure 08b. 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 e-w axis, figure 08c. This is done by using the two points of shadow intersection as centers and the length of the e-w line as a radius. This generates the Vesica Pisces whose two ‘points’ position the n-s axis. By repeating the Vesica construction along the e-w axis, the intersection of the four arcs positions a square, figure 08d.
Burckhardt [SAE, p 23] and Critchlow [TSS, p 29], both of whom refer to a Hindu sacred book on temple building called the ‘Manasara Shilpa Shastra’, describe a traditional Indian procedure to find this axis, figure 8a. Interestingly, a very similar method is described by the ancient Roman architect Vitruvius [VIT I.VI.6], figure 9. Burckhardt [SAE, p 24] says, “The reader will have noticed that the three phases of this rite correspond to three fundamental geometrical figures: the circle, image of the solar cycle, the cross formed by the cardinal axes and the square derived from it. These are the symbols of the Far Eastern Great Triad, Heaven – Man – Earth. Man appears in this hierarchy as the intermediary between heaven and earth, the active principle and the passive principle, just as the cross of the cardinal axes is the intermediary between the limitless cycle of the heavens and the terrestrial ‘square’.” Another view might be that the second geometric figure to appear after the circle, is the triangle; formed by the initial center and the two shadow points on the circumference, as in figure 08b. Burckhardt’s analogy, though, brings into correspondence the ‘ritual sequences’ of the first four numbers, the first four geometric forms, and the first four musical consonances, with four basic astronomical observations:

number geometry music astronomy Platonic term
1 • sphere-point fundamental 1:1 the sun same
2 • • line octave 2:1 the shadow different
3 • • • shape – triangle fifth 3:2 e – w axis being
4 • • • • volume – cube fourth 4:3 n – s axis manifestation

The theme of the division of the circle is important in the relation of time to astronomy. As suggested by Burckhardt in the quote above, segmenting the continuous circumference of the circle is analogous to order being brought out of chaos, or to establishing a moment in eternity. In modern terms we say we have become ‘oriented’ when we know where we are. Orient means east, where the sun rises, so to be oriented implies being aligned with the sunrise and therefore with the patterns of movement of the heavenly bodies. The point of sunrise is the point of scission of the limitless circumference and it establishes the axis.
The vertical shadow marker itself represents the third directional axis. This third element, or mean, can be taken to establish the structural relation between heaven and earth. It is characterized by Critchlow [SSA, p18] as, “The potency of creation or vertical ray, axis of the lightning flash; the division of light from darkness, the ontological axis of being. It is related to the Greek letter Ø, phi as in phyla, to bring forth, and physis, nature, and phylon, race, sex. One version of this symbol – the Hittite – is used as the determinative for ‘the gods’.”
A good natural shadow marker would be a free standing tree, recalling the symbol of the tree of knowledge. In many traditions the teacher sits beneath a tree; the Buddha, for example, receives enlightenment beneath the Bodhy tree. A good artificial marker would be a stick, which is a part of a tree. If the stick was marked or notched to record observations, it recalls the Biblical symbol of Jacob’s ladder.
In ancient Greek times the shadow marker was called a ‘gnomon’; a word related to ‘gnosis’ meaning experienced knowledge. The gnomon is associated with proportional transformation through time, and here we see it associated with transformation of light and dark through time. Its relation to the concept of gnosis indicates that the experience of pattern, and the study of transformation, are a pathways to knowledge

It is hoped that this discussion of philosophical number begins to answer the first of the two questions, ‘How could number have the power to affect us?’ In brief we may say it is because number is the means by which ultimate unity is divided to become the sensible world; and because that aspect of ourselves that connects us to that unity is based in the same principles, it is constructed by analogy. As to our second question, the relevance of this power for us may be that construction by analogy of the individual and the cosmos leaves open the possibility of self dignification through participation with unity; and the communication of that elevated state to others by one’s actions or work.
In Plato’s Republic, Socrates puts it this way [REP 500-01], “And the philosopher holding converse with the divine order, becomes orderly and divine, as far as the nature of man allows; … And if a necessity be laid
upon him of fashioning, not only himself, but human nature generally, whether in states or individuals, into that which he beholds elsewhere, will he, think you, be an unskillful artificer of justice, temperance, and every civil virtue? … Will [the world] disbelieve us, when we tell them that 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 … they will first look at absolute justice, beauty and temperance, and again at the human copy; and will mingle and temper the various elements into the image of a man; and thus they will conceive according to that other image, which, when existing among men, Homer calls the form and likeness of the Divine”

 We have descended from the Unity of Ideal number through the four subjects, each like a step on the Tetraktys and we are now in the realm of the sensible, day to day world, what might be called the world of action. From here we must begin the return journey of the Platonic soul. If we are to act in the spirit of Platonic remembrance we might base our approach to design on the Pythagorean ontological process just described. As designers we act through the shaping of matter. That shaping may be guided by a remembrance through re-enactment of the geometric archetypes we experienced in our descent.
The designs presented here were undertaken in the spirit of attempting to understand and explore possible relationships between architecture and traditional patterns of number and geometry. The goal of the designs is not to specifically imitate or reconstruct past forms and no claim is made that this exact method was used by the ancients. Rather the goal is to develop an understanding of timeless principles and explore the possibility of their concrete expression in the present moment.

As the subject of our design we select the four columned facade or tetrastyle portico, figure 10, a genera piece of traditional architectural composition which often forms a core or seed of more complex compositions. We may see in it an integration of incommensurable geometry and specific linear measure.
The design process used here may be divided into four stages:
1. Unity – We begin by restating the Pythagorean geometrical model – Monad, Dyad, Triad, Tetrad. From the inner tetrad we construct an outer square, a material representative of Unity, which will be the limit of the height and width of our design, figure 10.1.
2. Division – The essential scission of Unity. The square, our representative of Unity is divided by means of its diagonals and semi-diagonals, figure 10.2. The intersection of these two sets of diagonals divides the sides of the square at the third points. These points will mark the centers of the two inner columns and also their height. This geometrical division to obtain an architectural tetrastyle is similar to that which obtains the musical tetrachord, figure 7.
3. Structure – The disposition and demarcation of the major parts of the design. Using the third points we further subdivide the height of the square into 12 units, recalling the 12 semi-tones of the octave. We set the top of the columns at the 2/3rds point making them 8 units high. By the rules of Vignola for the Doric order [AV p 32], if the height of the columns, including base and cap, is divided into 8 units the height of the entablature is 2 additional units. The lower diameter of the columns will be 1/8th of their height, making them 1 unit or 1/12th of the overall width of the facade. The sizes of the other elements are derived from lower D, figure 10.3.

4. Manifestation – Full articulation of the details and ornaments, figure 10.4. The order selected for this design is Doric. Of the three Greek orders, Doric may be associated with the Apollonian or rational function, and expressed through the √2 system; it is the initial phase of the triadic cycle, the proceeding or coming forth. It may be selected to represent senior male divinities such as Zeus, heroes like Heracles, or male identified female divinities like Athena. The Ionic order may be associated with the Dionysiac or non-rational aspect, expressed through the √3 system, the maintaining aspect of the cycle. It might be appropriate for a goddess like Vesta, guardian of the hearth, Juno the first lady of the Olympians or Diana huntress of the moon. The Corinthian order may represent the synthesis or harmony of these two Dyadic aspects, which is necessary for the full flowering of conscious remembrance. It is expressed through the √5 system, including phi; it is the, returning, regenerative aspect of the cycle. It may be appropriate for Venus, goddess of beauty, or Hermes, guide of souls and teacher of subtle wisdom [VIT I.II.5].
Figure 11 illustrates that the Doric capital may be constructed on the same geometrical background as that of the facade as a whole. In this figure we also see a further connection of the geometrical method and the linear canons. The decimal equivalent of 2/3 is .666… , that of half the diagonal of a square is .707… . The difference between them is .041… . In our design lower D is equal to 1/12th of the overall length or .0803, very close to twice .041. Thus the manipulation of the diagonal of the square could have been used to size and position the inner columns.
As this design is undertaken in the spirit of communication and of bringing that which was hidden into light; and also because of our design’s use of the solar number 12, it may be dedicated to the enlightened sun god Apollo. As it is based on the geometry of the musical octave, and remembering that Apollo is governor of the Muses, it may be called the portico of ‘Apollo – Sun Musician’.

 We may continue with another example to illustrate that the previous design is neither a fluke nor the exclusive method of guiding the creation of the tetrastyle archetype. As the previous design has used the qualitative properties of the numerical powers 3 and 4, we may also use the power of 5 to guide or design.
As a theme for this design we suggest the principle of Venustatis, one of the three Vitruvian criteria for evaluating architecture, ‘firmitatis, utilatis, Venustatis’ [VIT I.III.2]. The Latin word Venustatis, usually translated as beauty, grace, or delight, may be taken to mean pertaining to Venus, who is best known as the irresistible goddess of physical love and beauty, and as the brilliant unmistakable light body in the evening or the morning sky. In the spirit of remembrance we look up to observe her.

The movements of the planet Venus remain close to the sun with a maximum separation of a constellation and a half, about 48 degrees. 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 583 days or 1.6 years, close to the Ø ratio, 1 : 1.618. Expressed arithmetically, in whole numbers of years and synodic periods, 365d x 8y = 2920 days, 584d/period x 5 periods = 2920 days. The ratio of years to synodic periods is 8/5, a whole number approximation of Ø. The actual cycle is about 2&1/3 days short over 8 years, 584 days is an average.
Astronomer J. Schultz notes [MRS p 117], “Every sixth Venus period is therefore remarkably similar; the various stages are repeated almost in the same form. With the eight year period a kind of Venus calendar is given in which the total succession of phenomena can easily be surveyed for longer periods of time.”
Schultz’s diagrams for the geometry of Venus are shown in figures 12a-c. These figures hold the zodiac still and trace the path of the planet as it moves with reference to the zodiac. The earth is at the center. A synodic period is a cycle around the sun. Figure 12a shows two synodic periods in which Venus makes two retrograde loops, that is, backwards movements with reference to the zodiac. Figure 12b shows the five loops of the full eight year cycle, at which point Venus goes retrograde at the same place in the zodiac where the cycle started. Figure 12c shows the five fold geometry highlighted by combining the planet’s observed position and its linear distance from Earth, its geocentric orbit. By plotting Venus’ conjunctions with the sun over an eight year cycle we see it constructing the pentagonal star, the pentagram.

Apollo, nudging Hermes asked: ‘You would not mind being in Ares position, would you, net and all?’ Hermes swore by his whole head that he would not … both gods laughed uproariously. Zeus was so disgusted that he refused to … interfere in a vulgar dispute between a husband and wife.” The lovers are eventually released from the net and “… Aphrodite went to Paphos where she renewed her virginity in the sea.” After this escapade, according to Graves [GM p 69], come a seemingly endless series of affairs. The list of her lovers goes on the include Hermes, Dionysus, Adonis, Anchises and numerous gods and heroes.
Graves concludes the myth of Aphrodite by informing us [GM p 70], “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 prerogatives had been infringed and threatened to abandon them altogether. Aphrodite apologized profusely and has never done a hand’s turn of work since.”
Aphrodite is not so much the goddess of Love, as she is the goddess of making love. The main object of her affection is Ares, the planet Mars. The astronomical aspect of this becomes clearer if we remember that almost the entire cast of characters in the myth are planets; Cronus is Saturn, Zeus is Jupiter. Hephaestus is the element fire, which the ancients presumed the stars were made of. There is a witty double-entendre quality about the image of Apollo [the Sun] nudging Hermes [Mercury], asking if he would like to be in the position of Ares [Mars] conjuncting with Aphrodite [Venus]. Mars, in its cycle, makes eight loops before completing its cycle of synodic periods in about 17 years. This establishes a 5 to 8 relationship between the
number of loops each planet makes in a complete cycle. Since the Mars cycle is about twice as long as the Venus cycle, the two will appear to ‘meet’ regularly in predictable parts of the zodiac.

According to Graves [GM p 49], “Aphrodite [foam born] is the same wide ruling goddess who rose from chaos and danced on the sea, and who was worshipped … as Ishtar, or Ashtaroth.”, also as Innana and Isis. This deity has long been identified with the planet Venus. 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, or Ø ratio, figure 13a; so Venus-Aphrodite is the ruling deity of worldly beauty and reproductive activity. We may recall that in discussing the geometric root systems Lawlor [SG p 31] had told us, “Transformation can be seen to occur by means of three general processes: the Generative, symbolized by the √2; the Formative, symbolized by the √3; and the Regenerative, symbolized by the √5 and its related function of phi, the Golden Mean.” Graves 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 principles, than with the Regenerative.
The use of the shell symbol in the myth is quite apt. Shells as spirals, embody the principle of growth by gnomonic expansion, which can be seen as a physical expression of Ø, figure 13b. The growth spirals of shells are related to the arithmetic of the Fibonacci series, whose adjacent terms are whole number approximations of Ø. That “… grass and flowers sprang from the soil wherever she trod …” is also quite apt as plants too embody the Fibonacci series in the phenomenon of ‘Philotaxis’ which governs branching and distribution of leaves, figure 13c, and many flowers associated with love, such as the rose, are five petalled. All of these attributes are illustrated in Botticelli’s famous painting, figure 13d.
That physical beauty is linked through love to the vital quality 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 that there is 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 phi could be a numerical intermediary through which inert matter may be ensouled. Thus the pentagon and its internal divisions could provide an appropriate geometrical key for our design.
Cronus, who’s defeat by Uranus initiates Aphrodite’s birth, is better known as ‘Father Time’, the planet Saturn. A connection between time and the spiral is made by Lawlor [SG p 70],”The deeper purpose for this development of the spiraling of numbers comes from the face that we have a model for the way an indefinite cause [root] can express itself in a play of definable numbers and forms. The spiral is still our most profound image for the movement of time …” It may be assed here that 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, the outermost visible planet performed this action for souls descending from the realm of the fixed stars. Thus Father Time holds the sickle and this may also be a source of the castration image in the myth. This suggests the method of division of the circle into five parts as a guide to our design.

As the purpose of undertaking this design was to explore 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 could learn all of her innermost secrets here in design school. No, we are only at the outer gateway to her sacred temple and this presentation must end here on the Portico of Venustatis. For to explore the inner mysteries of beauty your present author is not qualified to guide and our present academic venue not well suited. In parting, this hint can be offered for your own journey: Center yourself and listen for the music.

There are three traditional methods of applying proportional ideas to design: pure geometry, linear arithmetic and harmonic ratios. Geometry is probably the oldest as it springs directly from movements of the human body and no calculation is necessary. It also maintains the closest relationship to unity but it may not be practical in actual construction situations. Arithmetic is easy to communicate but risks the loss of unified vision. An intermediate approach is to use ratios of small whole numbers.
The archetypal creation story using all these methods is recounted in Plato’s Timaeus. In describing the construction of the World Soul [TIM 35-36], posits number as the mediator between the manifest and the non-manifest. As this mediation is the function of Soul, number may be considered both the soul’s medium of creation and expression. The principle Greek word for soul is ‘psyche’, which in our language includes the concepts of ‘mind’ and the idea of ‘anima’, the cause of motion. As the manifest, sensible world is always in motion, qualitative number, that is number as monad or module, may be a key to its understanding.
Figure 17 is a suggested illustration of the process. Timaeus first tells us that the Creator [Theos],

motivated by the Good, constructed only one World Soul, and it is comprised of an indivisible aspect, called the Same and a divisible aspect, called the Different. The Creator mixes a third essence comprised of both; and after ‘having made one from the three’ divides this whole into modulated portions, or as Thomas Taylor calls them, ‘becoming [or beautiful] parts’.
The division is done according to the Geometric progressions starting from 2 and from 3. These are traditionally illustrated in the geometrical ‘Lambda’ diagram. The creator now ‘marks off’, ‘fills up’ or ‘receives’ the intervals between each term in both sequences, with the Arithmetic and Harmonic means. The ratio of the last two means [9/8] is then used to further subdivide the sequences; and when then the subdivided powers of 2 and 3 are recombined a sequence of ratios are formed that comprise the Pythagorean, or natural musical scale.
Finally, the Creator separates the material or ‘fabric’ of the soul into two strips, which are placed in the form of an X like the Greek letter chi. The strips are bent round into two circles, one of which becomes the celestial equator, carrying the fixed stars, the second becomes the ecliptic, carrying the sun and planets. This creation myth has utilized the four mathematical subjects of the Quadrivium; Number, Geometry, Music and Astronomy. The stages have here been plotted on the pattern of the Tetracktys

In many plants, animals and in the human body symmetry is conferred by the phi ratio, figure 20. When the parts have measures which are ‘symmetrical’ in this sense, with each other and with the measures of the whole, the work is said by Vitruvius to have both ‘Order’ and ‘Eurythmy’ which he defines as “… beauty and fitness in the adjustment of the members … when they all correspond symmetrically.” Note that the limit of measure is associated with the generation of order.
Proportional division in the Vitruvian ‘symmetria’ sense can also be achieved by systems of small whole number ratios, some of which are ‘musical’ relations and others are rounded off transcendental ratios. These are expressed in the form of groups of rectangles in the writings of Vitruvius, Alberti, Palladio, Serlio and others. Such families of rectangles are compared with each other and with the musical octave in figure 21.

specified or fractionated. In the Timaeus Plato has proposed a possible method, a likely story, by which such a fractionation could take place. This uses division by the three means, a pattern of number which gives rise to the geometry of the Lambda diagram and the intervals of the natural musical octave, then to the distribution of the stars and planets.
A key aspect of this ancient system is the concept of Anthropos, also called Cosmic Man, known as Purusha in India, Adam Kadmon in the Kabbalistic tradition or DaVinci’s Vitruvian man. This cosmic man stands [or lies, as the case may be] between earth and heaven, between the manifest and non-manifest, just as the individual embodied person stands between the ground and sky. Thus there may be an analogy between the individual human being and the cosmos as a whole. 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. It also may lie behind the Delphic-Socratic admonition ‘Know thyself’. If the individual is an analog of the cosmos, created on the same pattern of number, then creating in turn by the same principles serves to remind us of that one cosmos and our relation to it, that is to say, of unity in multiplicity. This, as Plotinus tells us, is the source of beauty. And as we become more beautiful the closer we are to that source, the better enabled we are to convey that experience to others through our work, whatever its subject may be. Since beauty, for Plotinus, is also the Good, and people are naturally attracted to the Good, we acquire the power of creating captivating work. This possibility of self-elevation through action based in knowledge, and the power it brings are, in brief, the reason why we should care about number and its nature. Some images of Anthropos in architecture are shown in figure 22.