The Geometry of Art
and Life

Symphonic Composition - Periodic rediscovery of the Golden Section. Seurat's divisionism. Revival or Pythagorean doctrine in science and rat. Modern applications of dynamic symmetry in architecture, painting, and decorative art. Symphonic composition.

Introduction
And it was then that all these kinds of things thus established received their shapes from the Ordering One, through the action of Ideas and Numbers. [Plato, Timaeus ]

It is not generally suspected how much the above pronouncement of Plato--or in a more general way, his conception of Aesthetics--has influenced European [or, let us say, Western] Thought and Art, especially Architecture. In the same way that Plato conceived the "great Ordering One" [or "the One ordering with art," (Grk text ---)] as arranging the Cosmos harmoniously according to the preexisting, eternal, paradigma, archetypes or ideas, so the Platonic--or rather, neo-Platonic--view of Art conceived the Artist as planning his work of Art according to a pre-existing system of proportions, as a "symphonic" composition, ruled by a "dynamic symmetry" corresponding in space to musical eurhythmy in time. This technique of correlated proportions was in fact transposed from the Pythagorean conception of musical harmony: the intervals between notes being measured by the lengths of the strings of the lyra, not by the frequencies of the tones [but the result is the same, as length and numbers of vibrations are inversely proportional], so that the chords produce comparisons or combinations of ratios, that is, systems of proportions. In the same way Plato's Aesthetics, his conception of Beauty, evolved out of Harmony and Rhythm, the role of Numbers therein, and the final correlation between Beauty and Love, were also bodily taken from the Pythagorean doctrine, and then developed by Plato and his School. A great factor in Plato's Mathematical Philosophy--and, in a subsidiary manner, in his [p. ix] system of Aesthetics--was the importance given to the five regular bodies and the interplay of proportions which they reveal; we shall see this point of view transmitted all through the Middle Ages to the Renaissance and beyond, with the study, and the application to artistic composition, of the same proportions.

The name of the geometrical proportion (1/b=c/d) was in Greek, and in Vitruvius, analogia; harmoniously ordered or rhythmically repeated proportions or "analogies" introduced "symmetry" and analogical recurrences in all consciously composed plans. Let us point out at once that "symmetry" as defined by Greek and Roman architects as well as the Gothic Master Builders, and by the architects and painters of the Renaissance, from Leonardo to Palladio, is quite different from our modern term symmetry [identical disposition on either side of an axis or plane "of symmetry"]. We cannot do better than to give the definition of Vitruvius: "Symmetry resides in the correlation by measurement between the various elements of the plan, and between each of these elements and the whole. . . . As in the human body . . . it proceeds from proportion--the proportion which the Greeks called analogia--(it achieves) consonance between every part and the whole. . . . This symmetry is regulated by the modulus, the standard of common measure [belonging to the work considered], which the Greeks called the number. . . . When every important part of the building is thus conveniently set in proportion by the right correlation between height and width, between width and depth, and when all these parts have also their place in the total symmetry of the building, we obtain eurhythmy."

For this notion of symmetry seen as correlating through the interplay of proportions the elements of the parts and of the whole, the Renaissance coined the suggestive words "commodulatio" and "concinnitas" The mention of eurhythmy as a result of well-applied "symmetry" underlines the affinity between this correct [also etymologically sound] interpretation of symmetry [as opposed to the modern, static meaning usually applied to the [p. x] word] and rhythm; an old but correct definition of rhythm was: "Rhythm is in time what symmetry is in space."

This classical meaning of the word "symmetry" has been, with the technique itself, brought into light again within the last thirty years, and will henceforth be used in this work; we will meet later on the expression "dynamic symmetry" found in Plato's Theaetetus, and examine the special" eurhythmical" planning system covered by this term. It is also quite recently that, in the field of biology too, it was found that certain morphological intuitions of the Pythagorean and Platonist schools, and their interpretation by the Neo-Platonist thinkers and artists of the Renaissance, are confirmed by modern research. The Pythagorean creed that "everything is arranged according to Number" [taken up by Plato] is justified not only in Art [it was a Gothic Master Builder who in 1898 said, "Ars Sine Scientia Nihil" ] but also in the realm of Nature. The use of Geometry in the study and classification of crystals is obvious, but it is only lately that its role in the study of Life and Living Growth has begun to be recognized.

Curiously enough, the patterns, themes of symmetry, spirals, discovered in living forms and living growth, show those same themes of proportion which in Art seem to have been used by Greek and Gothic architects, and, paramount amongst them, the ratio of proportion called by Leonardo's friend Luca Pacioli "the Divine Proportion," by Kepler "one of the two Jewels of Geometry," and commonly known as "The Golden Section," appears to be the principal "invariant" [to use an expression popular in [p. xi] modern Mathematical Physics], as remarkable by its algebraical and geometrical properties as by this role in Biology and in Aesthetics. There are then such things as "The Mathematics of Life" and "The Mathematics of Art," and the two coincide. The present work tries to present in a condensed form what we may call a "Geometry of Art and Life" [p. xii]

Chapter 1. Proportion in Space and Time
The notion of proportion is, in logic as well as in Aesthetics, one of the most elementary, most important, and most difficult to sort out with precision; it is either confused with the notion of ratio, which comes logically before it, or [especially when talking of proportions in the plural] with the notion of a chain of characteristic ratios linked together by a modulus, a common sub-multiple; we have then the more complex concept which the Greeks and Vitruvius called "Symmetria," and the Renaissance architects, "Commodulatio." Let us start from the definition of ratio.

Ratio
The mental operation producing "ratio" is the quantitative comparision between two things or aggregates belonging to the same kind of species. If we are dealing with segments of straight lines, the ratio between two segments AC and CB will be symbolized by AC/CB, or a/b if a and b are the lengths of these segments measured with the same unit. This ratio a/b, which has not only the appearance but all the properties of a fraction, is also the measure of the segment AC = a if CB = b is taken as unit of length. [p. 1.]

Geometrical Proportion
The notion of proportion follows immediately that of ratio. To quote Euclid:
"Proportion is the equality of two ratios." If we have established two ratios A/B, C/D, between the two "magnitudes" [comparable objects or quantities] A and B on one side, and the two magnitudes C and D on the other, the equality A/B = C/D [A is to B as C is to D] means that the four magnitudes A, B, C, D are connected by a proportion. If A, B, C, D are segments of straight lines measurd by the lengths a, b, c, d, we have between these measurements, these numbers, the equality a/b = c/d; this is the geometrical proportion, called discontinous in the general case when a, b, c, d, are different, and continuous geometrical proportion if two of these numbers are identical. The typical continuous proportion is therefore a/b = b/c, or b2 = ac.

b = ̀ac [note: see document source for correct symbol in this equation] is called the proportional or geometrical mean between a and c. It is the geometrical proportion, discontinous or continous, which is generally used or mentioned in Aesthetics, specially in architecture.

The equation of proportion can have any number of terms. a/b = c/d = e/f = h/g, et cetera, or a/b = b/c = c/d = d/e, et cetera; we have always the permancy of a characteristic ratio [this explains why the notions of ratio and proportion are often confused, but the concept of proportion introduces besides the simple comparison or measurement the idea of a new permanent quality, which is transmited from one ratio to the other; it is this analogical invariant which besides the measurement brings an ordering principle, a relation between the different magnitudes and their measures]. The second series of equalities a/b = b/c = c/d = d/e, et cetera, represents the characteristic continuous proportion, geometrical progression or series, like 1, 2, 4, 8, 16, 32, et cetera.

The Golden Section
The simplest asymmetrical section and the corresponding continuous proportion: The Golden Section.

The Golden Section. -The Greeks had already noticed that three terms at least are necessary in order to express a proportion; such is the case of continuous proportion a/b = b/c. But we can try to obtain a greater simplification by reducing to two the number of the terms, and making c = a + b. So that [if for example a and b are the two segments of a straight line of length c) the continous proportion becomes: a/b = b/a + b or (b/a)2 = (b/a) + 1.

If one makes b/a = x, one sees that x, positive root of th equation x2 = x+1, is equal to 1=̀5/2 [note: see document source for correct symbol in this equation] [the other, negative, root being 1-̀5/2] [note: see document source for correct symbol in this equation] . This is the ratio known as "Golden Section"; [p. 3] when it exists between the two parts of a whole [here the segments a and b, the sum of which equals the segment c] it determines between the whole and its two parts a proportion such that "the ratio between the greater and the smaller part is equal to the ratio between the whole and the greater part."

This proportion, called in the text-books "division into mean and extreme ratio," has got, as we will see in the next chapter, the most remarkable algebraical properties. The Greek geometers of the Platonic school called it [GK text --], "the section" par excellence, as reported by Proclus [On Euclid]; Luca Paciolli, Leonardo's friend, called it "The Divine Proportion."

Generalization of The Concept of Proportion
The gemetrical proportion [resultng from the equality between two or several ratios] is only a particular case of a more general concept, which is "a combination or relation between two or several ratios ."

The more us ual proportions, besides the geometrical, are:
(1) The arithmetic proportion in wch the middle term [if we take the minimum of three terms for a proportion] overlaps the first term by a quantity equal to that by which it is itself overlapped by the last term, or [if a, b, c, are the three terms] c - b/b - a = 1 [example: 1, 2, 3]

(2) The harmonic proportion, in which the middle term overlaps the first one by a fraction of the later equal to the fraction of the last term by which the last term overlaps it, or b - a = (c - b) a/c equivalent to c - b/b - a = c/a [example: 2, 3, 6, or 6, 8, 12].

There are in all ten terms of proportions, established by the neo-Pythagorean School.

Example ----------------------------------------------------------- Example
c - b/b - a = c/c (1,2,3) . . arithmetic proportion --- c - b/b - a = c/a (2,3,6) . . harmonic proportion
c - b/b - a = (1,2,4) . . geometrical proportion ------ b - a/c - b = c/a (3,5,6
b - a/c - b = b/a (2,4,5) -------------------------------- c - a/c - b = c/a (6,7,9)
b - a/c - b = c/b (1,4,6) -------------------------------- c - a/b - a = b/a (4,6,7)
c - a/b - a = c/a (6,8,9) --------------------------------- c - a/c - b = b/a (3,5,8) . . Fibonacci Series

The tenth corresponds to the additive series 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, . . . et cetera, in which each term is equal to the sum of the two preceding ones; it is intimately connected with the Golden Section and plays an eminent role in Botany. We will meet it again in the next chaper.

We have seen in the Introduction that the technique by which, in a complex plan or design, the proportions were linked so as to get the right correlation [or "commodulation"] between the whole and its parts was called by the Greek architects and Vitruvius "Symmetry"; and the result obtained where this technique was corectly applied was the "eurhythmy" of the design and of the building. We generally associate the terms of "rhythm" and "eurhythmy" with the Arts working in the time dimension [Poetry and Music] and the notion of Proportion with the "Arts of Space" [Architecture, Painting, Decorative Art]. The Greeks did not care for these distinctions; for them, for Plato in particular, Rhythm was a most general concept dominating not only Aesthetics but also Psychology and Metaphysics. And Rhythm and Number were one. [Rhythmos and Arithmos had the same root; rheî n = to flow.]

For them, indeed, Architecture was not only "rozen Music" [Schelling] but [p. 5] living Music. The notions of periodicity and proportion, and their interplay, can be used for succession in time as well as for spatial associations. If periodicity [static like a regular beat, or dynamic] is the characteristic of rhythm in time, and proportion the charactaeristic of what we may call rhythm or eurhythmy in space, it is obvious that in space, combinations of proportions can bring periodical reappearances of proportions and shapes, just as in a musical chord or in the successive notes or chords of a melody we may really perceive an interplay of proportions.

If Architecture is petrified or frozen Music, so is Music "Drawing in Time." But we will, in what follows, leave aside the "numbers" of Music and Poetry in order to elucidate how the Greek and Gothic master Builders applied their knowledge of proportion and "Symmetry," and how and where their Geometry of Art meets the Geometry of Life.

Greek and Gothic Canons of Proportion
All through the nineteenth century architects and archeologists have tried to find out explanations and keys for the beautiful proportions of Greek and Gothic monuments; to find, that is, whether their builders had explicit rules and canons of proportion and design, or whether the perfection of these monuments was just due to a mixture of luck and good taste.

We won't examine here the out-of-date and ponderous theories of Viollet-le-Duc and Dehio, but will only note that about 1850 Zeysing already had observed the obvious presence of the Golden Section in the frontal view of the Parthenon [AC/AD = DC/AC = * o n Plate LVII father down].

Modern research [in the last thirty years] has elaborated three principal theories concerning Greek and Gothic canons of proportion, resepctively expounded by the American J. Hambidge, the Norwegian F. M. Lund and the German Professor Moessel. The three theories are converging, agree about one main point, and when combined into one synthesis may be said to give the most probable solution and key to the problem under consideration; after what we have seen about the transmission of the Pythagorean Pentagram and the importance of the Decad, we shall not be surprised to discover that in all probability [p. 124] Pacioli's and Zeysing's intuitions were right, and that the secret of Greek "symmetry" and later of Gothic "harmonic" composition resides not only in the use of analogia [geometric proportion], but especially in the predominance, amongst the different possible analogies, of the analogia par excellence, the Golden Section.

The clue to the manipulation of proportion in agreement with the Greek concept of symmetry [Vitruvius mentiond the concept, but cautiously refrained from explaining the technique] was found by Mr. Hambidge in Plato's Theaetetus, and this in the sentence concerning numbers which are [GK text - -]; meaning irrational [non-commensurable] numbers like ̀2, ̀3, ̀5, [Note: see document source for correct symbol] such that the squares [or other surfaces] constructed on them produce sequences linked by comensurable proportions ["commensurable in the square" being the exact translation of Plato's expression]. The importance of the use of these irrational proportions [irrational in the first dimension, but rational, commensurable, consonant in the square], and the accuracy of Hambidge's interpretation, are proved by the sentence of Vitruvius advising the use of "geometrical" proportions [that is, irrational, as compared to the arithmetical, rational or aliquot ratios like 1/2, 2/3, 3/4, etc.] in delicate problems of symmetry, and confirmed by Pacioli's very explicit "che la proporzione si molto più ampia in la quantità continua che in la discreta" [Divina Proportione, lib. II, cap. XX], the discrete or "static" or "simple" [p. 125] symmetries being 1/2, 2/3, 3/4 and their likes, and the "continuous" or "dynamic" ones being ̀2, ̀3, ̀5, [Note: see document source for correct symbol] et cetera.