The Golden Section and Composition

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A lot of nonsense has been written about composition, probably because it depends for its success upon the sensitivity of the artist and viewer, not upon fixed rules.  But, people do want to find principles to guide them, and when the principles have a mysterious aura of wholeness and harmony they often become irresistible.  That is the case of the Golden Section.  First a little history.

 

As early as the 5th century BCE, educated Greeks were expected to have breadth of knowledge.  Mathematics was important to them, in part because of its apparent perfection as a system of reasoning, and Plato is said to have had a sign over the door of his school that read, “Let no-one ignorant of geometry enter here.”  The Greeks brought geometry to such a high level of perfection that they solved very complicated problems, some involving polyhedra.  Plato wrote about the five regular polyhedra, the Platonic Solids, in the Timaeus, so they are named after him.  These solids are the tetrahedron, hexahedron (cube), octahedron, dodecahedron, and icosahedron.  They are beautiful, surprising forms ; the dodecahedron, for example, is made up of twelve pentagons, assembled into a spherical shape.

 

This interest in mathematics continued in the Renaissance, and with the development of perspective—which might be viewed as the use of geometry in drawing—the Platonic Solids garnered renewed interest.  This is where the Golden Section (aka. Golden Ratio, Golden Number, Golden Mean, Phi, Φ) comes in, although it wasn’t called by any of these names then, but by the term Euclid used:  “extreme and mean ratio.”  It is important in the construction of pentagons and other geometrical figures, including the dodecahedron.

 

Phi is an amazing number, and the easiest way to introduce it might be to look at the Fibonacci sequence, a sequence that finds favor with those who are grasping for some kind of ultimate harmony.  It is the sequence in which each successive number other than the first two is the sum of the two immediately preceding numbers:  1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 . . . .  That gives us a series of ratios:   1/1, 2/1, 3/2, 5/3, 8/5, 13/8, 21/13, 34/21, 55/34, 89/55, 144/89 . . . .  And this gives us a decimal sequence (each number here is rounded off to four places):  1.0000, 2.0000, 1.5000, 1.6667, 1.6000, 1.6250, 1.6154, 1.6190, 1.6176, 1.6182, 1.6180 . . . .  You will guess, rightly, that the successive terms of this sequence approach—but never reach—an irrational number (one that cannot be expressed precisely by a fraction).  Each successive number is closer to this number than the previous one, and they are alternately larger or smaller than the one the sequence is approaching.  That number is Φ, and for practical purposes it is regarded as 1.6180.  (Its reciprocal [1/Φ] is 0.6180 and its square [Φ2] is 2.6180, and that is quite remarkable!)

 

The Greeks apparently discovered phi in the 5th century BCE while studying the construction of geometrical figures.  This is easiest to see in the pentagram (five-pointed star).  If you draw a star with a continuous series of five straight line-segments, as we all learned to do as children, the length of any of these segments divided by the distance between two adjacent points of the star is phi.  If you connect all of the points by straight lines you have a pentagon.  The segments you drew to make the star are the “diagonals” of the pentagon, and the straight lines you drew to connect the points were the “sides” of the pentagon, so phi can be described as the length of a diagonal of a pentagon divided by the length of a side of the pentagon.  Thus, phi is important in the construction of a pentagon, and Euclid defines his “extreme and mean ratio” as a line divided into two segments so that the ratio of the whole line to the larger segment is the same as that of the larger segment to the smaller.

 

The first volume of Divina Proportione (Divine Proportion), a three-volume book published by Luca Pacioli in 1509, marvels at the properties of phi which he calls the “Divine Proportion.” He then goes on to discuss the Platonic Solids and other polyhedra.  And although he says that he wants to reveal the secret of harmonic forms through the discussion of phi, this volume is solely concerned with geometry; thus, the harmonic forms he referred to are surely the Platonic Solids.  The second volume discusses proportion as it applies to architecture, and the proportion that he writes about here is Vitruvian proportion, the proportion represented in Leonardo’s famous drawing of a man with outstretched arms standing in a circle overlapped by a square (which is actually, an illustration for the book).  The last volume is a translation (unacknowledged) of a book on the Platonic Solids by the famous painter Piero della Francesca’s.  Nowhere does Pacioli talk about the use of phi in constructing either buildings or paintings.  The term “Golden Section” doesn’t appear in print until 1835, when it is used in a book about elementary mathematics written by Martin Ohm, and that book doesn’t deal with composition, architecture, or painting, either.

 

This completes our brief survey of the history of phi, itself, but it leaves us with a question we need to answer:  where did the bogus claims come from?  In fact, they are very, very recent, going back to Matila Ghyka who wrote two books, Esthétique des proportions dans la nature et dans les arts (1927) and Le Nombre d’Or: Rites et rythmes pythagoriciens dans le développement de la civilisation occidentale (1931).  (Aesthetics of Proportions in Nature and the Arts, and The Golden Number:  Pythagorean Rites and Rhythms in the Development of Western Civilization.) Note the dates!  Mario Livio says of them, “these books are composed of semimystical interpretations of mathematics” and “the books contain a collection of inaccurate anecdotal materials on the occurrence of the Golden Ratio in the arts.”  It may seem astonishing that such claptrap stemming from two books could become so widespread in such a relatively short time, but the Bermuda Triangle Myth has a similar history.  Nobody claimed that anything odd was happening in that area (which isn’t defined in a consistent way in the various myth-making accounts) until the 1950s.  And then Vincent Gaddis wrote two books in 1964 and 1965 and the myth-production industry was in full swing.

 

Of course, a few artists have been influenced by Ghyka’s claims, and have made deliberate use of the Golden Section since, but what about architects and artists who came before them:  did they actually use it?  Various modern claimants say that it can be found in the designs of Babylonian artifacts, in the Pyramids, and in the Parthenon  The foundation for these claims is laid by cherry-picking the examples, cherry-picking the measurements to be used, misrepresenting historical statements about these relics, and fabricating data.  (This is not unlike the support offered for the Bermuda Triangle and a host of other dubious notions.)  I will take only the Parthenon.  According to Livio, the width of the Parthenon divided by its height is not 1.6180 but about 2.25.  Even jiggery-pokery with the measurements doesn’t help much, here.  But we might ask, why was the Parthenon chosen?  There are many existing Greek and Roman temples with widely differing proportions, and that, of course, is the answer:  the Parthenon is the most famous, and if looked at casually, it comes closer to supporting the claims than most of the others.  And of course, you then pick the measurements which provide ratios that are closest to phi.  Are you going take the width including the base on which it sits, or just the width from the sides of the columns; and are you going to take the height from the bottom of the base to the top of the columns or from  . . . .

 

As for its use in the composition of drawings and paintings there are two things to look at:  1) is the ideal shape of a drawn or painted image, disregarding the frame, the ratio 1 to 1.6180 (a “Golden Rectangle”), and 2) can we locate points within the image which are derived from phi and which mark the center of interest, or other important items?  As for the first question, if you look at great paintings in the Uffizi, you have Rembrandt’s “Self-portrait as an Old Man” (1.11/0.97 = 1.144), Titian’s ‘Venus of Urbino” (1.65/1.19 = 1.3860), Durer’s “Adoration of the Magi” (1.14/1.00 = 1.14), and Predella’s “Last Supper” (2.10/0.32 = 6.5625).  An exhaustive search might turn up a few with a ratio close to 1.6 by pure chance.  As you can see, the actual dimensions of these paintings are highly variable:  I made a small and unscientific sample and found an average ratio of about 1.4, which has nothing to do with the Golden Section.

 

As for the second question: If you take a Golden Rectangle and divide it with a single line so that you have a square at one end, you will find that the remaining piece is also a Golden Rectangle.  And if you divide it in the same way you will produce a line that will intersect the first.  This point of intersection is said to be an aesthetically pleasing point in the first large rectangle, a point which should be used to locate the center of interest or some other very important item.   (Remember that we have already seen that the image size of a painting is quite unlikely to be a golden rectangle, so this idea is already irrelevant.)  And of course, we can continues such divisions for ever, dividing all of the various golden rectangles as they appear.  Moreover, we can divide the first Golden Rectangle starting at the opposite end, and we can do this with the smaller Golden Rectangles, too.  And we can divide the smaller rectangles at their opposite ends, etc., etc.  This gives us an infinite number of points to use in making our claims.  But we don’t even have to do that.  We can just locate the first point and exclaim in astonishment that there is something in the general vicinity that is important (and it doesn’t even have to be the center of interest) and that is what is usually done.

 

Let’s look at what we have if we just divide the first golden rectangle at both ends, and the smaller rectangles at both ends.  This gives us four points that are located 0.375% in from a short side and 0.375 % in from a long side.  Some current artists tell us to simply construct lines 38% from each side of the image–regardless of its shape—surely this renders the whole notion of the Golden Section absurd.  Such an approach essentially divide the sides into thirds, and such division into thirds is sometimes (wrongly) said to be an “abbreviated form” of the Golden Section.  In fact, the “Rule of Thirds” was enunciated by John Thomas Smith in 1797 in his book, “Remarks on Rural Scenery,” and he gave it its name.  He was elaborating on some comments made by Joshua Reynolds, and this is long before Ghyka’s claims about the Golden Section.  The idea behind the Rule of Thirds is very simple:  if you divide a canvas in half, the composition is likely to be static, and if you divide it near a margin, the composition is likely to be unbalanced, so you divide it about into thirds.  There is nothing absolute or highfalutin about this, and it makes perfectly good sense.

 

The “Eyes of the Rectangle” is another bizarre notion that is sometimes mentioned in the same breath as the Golden Section.  The “Eyes” are found by drawing diagonals from corner to corner in a rectangle, and then dividing in half each of the line-segments that run from the center to a corner .  These are said to be important points in a composition, but they have competition from a rival “Eyes of the Rectangle” notion.  These rival “power spots” are found by drawing diagonals in the rectangle and then drawing lines from each corner so that each meets a diagonal at a right-angle.  This gives us three sets of supposedly important but competing spots, and the best that can be said about them is that they are not too near the center or an edge—if the image is of about average proportions.  But that isn’t all.  We also have the “Rabatement of the Rectangle” the vertical found by marking off the length of the short side on the long side and erecting a vertical from that point.  (As the French word “rabatement” suggests, this amounts to rotating the short side down to touch the long side.)  This vertical is said to divide the painting in an important way.  Those devoted to this notion make the usual claims about how it has been used by great painters to create powerful compositions, and examples found by cherry-picking are offered as proof.  Doesn’t this collection of inconsistent approaches—all with the same sorts of claims—tell us something . . . ?

 

So, what should we do to compose our drawings or paintings?  It is easier to say what we shouldn’t do than what we should do.  Someone has summed this up by reference to the British flag:  Don’t divide the canvas in half horizontally or vertically or diagonally, and don’t put your center of interest in the very middle.  “Divide” doesn’t only mean “divide by lines”, but also by color, or value, or texture.  Then, Edgar Whitney offers a rule for what we should do:  the center of interest should be located at a point that is at a different distance from each of the four sides and not too close to an edge.  (I like this idea.)  The Rule of Thirds, applied loosely, as it was intended to be, is compatible with all of these notions.

 

These are only rules of thumb, but you shouldn’t break them unless doing so helps you to create an image that is more expressive.  Suppose you have a head-and-shoulders portrait (in which the head is gazing to your left) positioned near the right-hand edge of a horizontal rectangle, and that several long horizontal lines run across the canvas.  This might work , the lines (and the direction of the gaze) connecting the head to the area in which it is placed.  Now, to show how important the psychological element of the gaze is, imagine that the head is looking to your right, out of the frame!

 

Degas’ painting, The Absinthe Drinkers, is the best example I know of brilliant composition that defies simple formulas.  In it, the man on the right is looking out of the frame, but the woman who is the center of interest is immediately to the left and keeps us from following his gaze; moreover, the indifference to each other shown by the posture of the two subjects is important to the painting.  Consider also, the many portraits of the Madonna and Child made over the centuries.  These are appropriately placed in the middle of a painting in which they are surrounded by various saints and angels.  The wanted variety can come from the positions and expressions of the people represented.  As I said earlier, compositional rules are rules of thumb.

 

Notes

Mario Livio, The Golden Ratio, New York:  Broadway Books, 2002.  This marvelous book was my principle reference for the historical background on phi and Ghyka.  The author is Head of the Science Division of the Hubble Space Telescope Science Institute.  Livio writes very well, and the book is fascinating and fun.  If you are interested in mathematics, this is a wonderful inquiry into what may be the most remarkable of all numbers.  If you are interested in art, it reveals the story of the Golden Section and how it came to be regarded (erroneously) as a basic principle of composition.   

Harald Mante, Photo Design, New York:  Van Nostrand Reinhold Company, 1971.  This wonderful book is unfortunately out of print.  It consists primarily of black and white photographs which show the importance of format, spots, lines, shapes, and contrast in effective design.  The composition of these photos is dramatic and fascinating.  Thumbnail diagrams call attention to the points he is making in the very brief text.  (I recommend that you not take his references to the Bauhaus gurus seriously—the book stands on its own without them, and they are pretentious nonsense.)  I mention this book here because it shows by example that artificial, geometric methods of composition are irrelevant.  Artists have more important things to think about.  (But you should also note that photography is very different from drawing and painting, although the basic principles discussed here are important to both.)  Mante's more recent book, The Photograph:  Composition and Color Design, is interesting but not very helpful.  In it, Mante accepts the Golden Ratio as an important tool, and follows Goethe (a very dubious source) in talking about complementary colors.