From the Greeks to the Golden Rectangle
Co-authored by Todd Murdock
When it comes to rules of proportion, I never understood the whole picture. At least not until recently, not before spending the last three years studying the classical orders with Todd Murdock—one SketchUp rendition at a time. Now I know why I had such a hard time understanding the rules of proportion. There are none. They don’t exist. But there are guidelines.
I’ve noticed a growing interest in the classical orders—the root of most architectural design. But it wasn’t always so. When I started work in construction, in the early 1970s, modernism still held the attention of many architects, homebuilders, and homeowners. By that time, modernism—from Frank Lloyd Wright to Walter Gropius and the Bauhaus movement—had influenced traditional building styles, introducing smaller and more streamline moldings, large picture windows and sliding glass doors that brought the outdoors indoors, along with cantilevers, odd-shaped columns, and a host of other affects.
A decade later, while trying to understand how to design a colonial mantelpiece for a customer in Los Angeles, I learned how daunting it was to study the classical orders. First I discovered there wasn’t one single simple-to-understand system of design or proportion. In fact, what I ran into at my local university library was a confusing collection of systems and approaches to proportion. But as a carpenter, what I needed was just one! And everyone kept telling me that the classical orders were based on Greek temples, on the Parthenon. If that were true, how could there be so many different interpretations?
Like most carpenters, I had to get the job done, so I picked an author with the most titles, Asher Benjamin, and what appeared to be an easy-to-follow system. Benjamin’s approach seemed…well…approachable. By that time, I’d built several mantelpieces, but none of them ‘felt right.’ For once, I wanted to start with a truly pleasing design and needed to know how wide and how deep the pilasters should be, how tall the entablature should be, how far the cornice should extend. Yes, I wanted to know the rules. Benjamin had no shortage of rules. And for some silly reason, I still thought they all originated with the Parthenon!
I appreciated Benjamin’s simple system of breaking the classical orders down into a small number of parts—from 11 parts for the Tuscan to 14 parts for the Corinthian. I could handle that. But then I made the mistake of reading the next book by Asher Benjamin (The American Builder’s Companion, 1827), where he changed the rules completely and broke the Tuscan order into THIRTY-NINE parts. And to make it even more confusing, he described the size of each molding detail using a system based on minutes (The Architect, or Practical House Carpenter, 1830). Minutes! What does time have to do with proportion?
All of those parts (and minutes, too!) were too much for my arithmetic-challenged mind. Besides, other than some fancy decorative stuff, like acanthus leaves, flutes and volutes, triglyphs, metopes, and modillion blocks, all the orders looked pretty much the same to me! But still, I was relieved to have found rules, to discover the final word, to know how the Greeks proportioned the Parthenon, the Romans designed the Pantheon, and all of it straight from Asher Benjamin, the true source of American architecture.
About five years later I was back at the library and read a re-print copy of Abraham Swan’s The British Architect (1758). What a stunner! Swan broke the Tuscan order into only NINE parts, making the arithmetic much easier! And Swan shattered any ideas I had about universal rules for proportion!
|With only NINE parts, Abraham Swan’s system for proportioning the Tuscan order appears to be much easier than Benjamin’s—at least at first!|
|But Swan’s system isn’t so easy when you drill down to the size of each molding, which still requires multiple and time-consuming divisions. Instead of dividing the column’s base diameter into 60 minutes, he divides it into 48 equal parts.|
So how could there be two or even three different systems and measurements for proportioning the classical orders? Simple. None of these examples are based on specific Greek or Roman ruins. All of them are dependent on the opinion and taste of the individual author. Really, the only thing they have in common is that they owed their approach to Vitruvius (see History below).
Understanding systems of proportion
Few carpenters have an opportunity to study the classical orders, and those who have attempted even a cursory examination are immediately confused by drawings dating back several centuries that render each order in ascending height, from the ‘short’ Tuscan to the ‘tall’ Composite.
That illustration—with the orders drawn using the same base diameter—has caused more confusion than comprehension! The first impression is that the Composite and Corinthian orders are taller than the Tuscan order. Those old diameter-based illustrations lead us to believe that the Tuscan order isn’t heavier, stronger, and more substantial than any of the succeeding orders—which isn’t true at all!
Take a look at Drayton Hall, one of the most iconic examples of Georgian architecture in America, built in approximately 1738 and largely untouched since.
|The portico is supported at the first floor with robust Tuscan columns—able to carry the weight of the floor and the roof above, while lighter Ionic columns support the second floor and roof.|
|Also notice that the Ionic columns rest on pedestals, and the railing terminates attractively just beneath the pedestal cap. Sadly, it is this type of detail or “order” that is now missing from much of architectural design.|
Classical proportions make a lot more sense if the orders are drawn to the same height, not the same base diameter. After all, carpenters (like architects and designers), have to work with fixed elevations—the height of an exterior wall, the height of an interior ceiling. That’s why an illustration like this one, based on a shared elevation, makes a lot more sense.
Now that Todd has provided an easy-to-follow animated guide to understanding proportion, let’s go back to the Parthenon, an ancient temple that many of us mistakenly point to as the source of proportional rules.
Life is funny. The stuff you avoid in your youth becomes, at an older age, far more appealing and attractive, like finding beauty in old people—you know, folks in their 50s and 60s—something you can’t imagine when you’re 20 years-old. Architecture is like that. To appreciate the classical orders, and value the variety of systems used to determine proper proportions, it helps to appreciate history, and I mean a lot farther back than 50 or 60 years.
Though it may not be on the list of the Seven Wonders of the World, the Parthenon is the most important architectural structure in the western world, built by the Greeks somewhere around 447 BCE. The history of the building—from design and construction, through wars, fires, and pillaging—is extraordinary. For video viewers, I recommend the PBS Nova program on the Secrets of the Parthenon. For readers, try Mary Beard’s, The Parthenon.
Vitruvius, a Roman military engineer and architect, published the earliest known work on architecture, De Architectura (The Ten Books on Architecture), sometime around 1st century BCE, and he studied the Parthenon closely. He particularly described minute deviations from perfectly straight lines, found in the stylobate or steps leading up to the temple, in the entasis of the columns, and what we recognize today as a slightly curved architrave, including a curvature in the metope panels themselves.
Vitruvius devised the first known system of proportions, using ‘parts’ to define each section of an order. Yet even Vitruvius describes how time has affected proportion: “It is true that posterity, having made progress in refinement and delicacy of feeling, and finding pleasure in more slender proportions, has established seven diameters of the thickness as the height of the Doric column, and nine as that of the Iconic” (Morgan 104).[i]
In 1452, Leon Batista Alberti, an author and architect, published the first major work on architecture of the Italian renaissance, De re aedificatoria (On the Art of Building). Alberti based many of his rules of proportion on Vitruvius’ work, written nearly fifteen hundred years earlier.
In 1562, slightly more than a hundred years later, Giacomo Barozzi da Vignola, an artist and architect who worked at one time with Michelangelo, measured Roman ruins and monuments, then published the Canon of the Five Orders of Architecture. As Todd previously explained, Vignola established the simplest and most influential system for proportioning classical columns and architectural ornamentation.
Less than ten years later, Palladio measured the Greek ruins and published his findings on proportion in The Four Books of Architecture (1570), using a system of proportion nearly the same as Vignola: for instance, Palladio’s Ionic column is “nine modules high…a module (being) the lower diameter of the column” (Palladio 19).[ii]
Because of Palladio’s supposedly careful measurements—but largely due to a prowess for publishing—he is often given primary credit for developing a true system of proportional rules based entirely on the classical orders. But it is Vignola’s system that has influenced most architectural authors.
And though Palladio is credited with being one of the first to measure Greek ruins, carpenters and contractors should pause to imagine just how careful those measurements might have been accomplished. As Calder Loth writes: “…a close look at some of the details reveals that Palladio was not always accurate in his depictions of many features. We can speculate that Palladio relied on assistants to do some of the actual measuring when he wrote down the notes and made sketches based on the information being conveyed to him” (Loth 7355).[iii]
|Henry Parke provides us with a pretty good clue about Palladio’s system of measuring the Parthenon.[iv]|
After Palladio, a raft of architectural authors penned and illustrated their own versions of the orders. In some respects, it is astounding how many men devoted so much time and energy toward establishing their vision of the classical orders and their system of proportion. These architect/author/illustrators included James Gibbs (who worked at the same period of time as Sir Christopher Wren and Inigo Jones), Abraham Swan, and William Chambers, all of whom influenced—often directly—the work of Asher Benjamin.
Rules or guidelines?
As Benjamin said in the preface to the third edition of The American Builder’s Companion 1827:
I have next treated on the origin of building, of mouldings, and of the orders. I have endeavored to explain them so clearly and fully, that they cannot be misunderstood. At first they were selected from several authors, drawn at large, and wrought. After careful examination, such parts as I did not approve, were altered, by drawing and working them over again, and repeating this process several times, till after the most minute and careful examination of every part of the four first orders, I was confirmed in the opinion, that no further alterations for the better could be made; for the result of which experiments see these orders are they are severally laid down.[v]
Yes, he does sound a bit over confident. A little later in the book, Benjamin explains part of his reasoning for his rendering of the classical orders: “The design, here annexed, and also the Doric, Ionic, Corinthian and Composite orders, I have selected from several authors, and have made all the alterations, that in my opinion, were necessary to rend them conformable to the practice of the present time” (33).
We know without a doubt that one of the authors Benjamin alludes to is Sir William Chambers, because it is from Chambers’ earlier work that Benjamin most egregiously plagiarizes, sometimes multiple pages at a time. In fact Benjamin was a serial plagiarizer, he knew no limits to stealing work from other authors and suggesting it was his own. Anyone interested in the full story of Asher Benjamin and his sources should read Abbott Lowell Cummings’ in depth study.[vi]
Though he was no plagiarizer, Swan also admitted to creating his own designs and incorporating unique proportions for the classical orders:
For I observe the designs which have been published by others, have, for the most part, been grand and pompous; which, though they may be excellent in their kind, will but seldom come into use, as being only proper for very large buildings.
But as there are more gentlemen of moderate fortunes than of great estates who may be inclined to build houses, I suppose some less expensive designs may be acceptable to the public, as being of more general use, such as will be found in several of the following plates” (Swan iii).[vii]
The point is that what we may have thought were direct measurements taken from ancient ruins are nothing of the kind. All of these influential authors made their own rules, and chose their own systems of proportion based upon what they thought looked best for each occasion.
Careful what you read: myths & misapprehensions
When I was in school, a professor once told me: “Don’t believe any writer who refers to a recognized or iconic work of literature. Most authors never read the actual source material; most authors just read other authors who have quoted from the source material. Don’t be a sloppy researcher,” he told me, “always read the original source material yourself. You might be the only one who does.”
Let’s start with that myth about the human body being the source for all architectural proportions. Countless writers and speakers and internet voices repeat that story. Instead of repeating it again, let’s look at the original source—Vitruvius, who wrote: “Without symmetry and proportion there can be no principles in the design of any temple; that is, if there is no precise relation between its members, as in the case of a well-shaped man” (Vitruvius 72).
So what did our Roman teacher mean? Vitruvius explains it well, without ambiguity. He describes how the human body is “designed by nature” and each part is perfectly proportioned to create a symmetrical whole, “from the chin to the top of the forehead and the lowest roots of the hair.” In the second paragraph of On Symmetry, Vitruvius even notes the proportional size of human feet: “The length of the foot is one sixth of the height of the body; of the forearm, one fourth; and the breadth of the breast is also one fourth.”
But nowhere does the author establish or even hint at a link or relationship between human proportions and the proportions of the classical orders.
Vitruvius simply uses the symmetry and proportion of the human body as an example of natural beauty, an analogy: “Similarly, in the members of a temple there ought to be the greatest harmony in the symmetrical relations of the different parts to the general magnitude of the whole” (73).
Marcus Frings describes Vitruvius’ appreciation of natural proportion in the human body as “a model by virtue of its perfection of symmetrical shaping first and foremost and not in its inherent proportions, which is often misunderstood; cf. [Frings 1998]. The human body, as an example of modular creation from nature, is chosen by Vitruvius as a paradigm for the required rules of proportion” (Frings).[viii]
One possible reason folks get so confused by Vitruvius may be because the Roman draws our attention to the language of measurements, how our terminology is derived from descriptions of human anatomy: “Further, it was from the members of the body that they derived the fundamental ideas of the measures which are obviously necessary in all works, as the finger, palm, foot, and cubit. These they apportioned so as to form the ‘perfect number, called in Greek τέλειον, and as the perfect number the ancients fixed upon ten. For it is from the number of the fingers of the hand that the palm is found, and the foot from the palm” (pg. 73).
Unfortunately, it has become common for writers to mistakenly assume that Vitruvius is describing a direct correlation between the size of a man’s foot and the size of a column’s diameter. Nonsense. It’s just an analogy.
The golden rectangle: one more myth
Since we are back to Vitruvius, let’s take a closer look at the origin of Leonardo da Vinci’s Vitruvian Man and that illustration’s supposed link to the golden rectangle or golden mean. And then we’ll tackle the King Kong of all myths—the golden rectangle’s relationship to the Parthenon.
The same section from Vitruvius’ The Ten Books of Architecture inspired Leonardo da Vinci’s drawing of the Vitruvian Man. In the midst of discussing the significance of symmetry and proportion, Vitruvius wrote:
For if a man be placed flat on his back, with his hands and feet extended, and a pair of compasses centered at his navel, the fingers and toes of his two hands and feet will touch the circumference of a circle described therefrom. And just as the human body yields a circular outline, so too a square figure may be found from it. For if we measure the distance from the soles of the feet to the top of the head, and then apply that measure to the outstretched arms, the breadth will be found to be the same as the height, as in the case of plane surfaces which are perfectly square. (73)
Leonardo da Vinci’s drawing is a close replication of Vitruvius’ description of symmetry and proportion found in the human body. Though nowhere does Vitruvius refer to the golden mean, nor does he rely on irrational numbers or ratios for any of system of measurements. Further, while discussing the ancient’s rule that “in perfect buildings the different members must be in exact symmetrical relations to the whole general scheme,” Vitruvius discusses why the Greeks settled on ten as the “perfect number,” partly because that is the number of fingers and toes on the human body and because it is easily divisible, but “as soon as eleven or twelve is reached, the numbers, being excessive, cannot be perfect until they come to ten for the second time” (pg. 73).
And in da Vinci’s own notes, which are written in mirrored script on the illustration, there is no mention of the golden ratio. The artist simply quotes from Vitruvius’ original work, leaving his purpose in no doubt—a rendering of Vitruvius’ original work.
Of all the works of man attributed to the golden rectangle—from the Great Pyramids of Giza to Notre Dame, from The Last Supper to The Mona Lisa—the Parthenon is probably the most prominent example, and likewise, the most distinguished misapprehension. To track down the relationship between the golden rectangle and the Parthenon, I dug through every book on architecture I could lay my hands on.
Mary Beard’s 2010 study, The Parthenon, never mentions the Golden Rectangle.
Fiske Kimball, an American architect, historian, director of the Philadelphia Museum of Art for 30 years, and highly respected author of A History of Architecture (1918)—one of my favorite and most accessible books on historic architectural styles—never mentions the Golden Rectangle.
Spiro Kostof, author of A History of Architecture—a reference common to college courses on architectural history, and one that I turned to when I first started studying the subject—never mentions the Golden Rectangle.
And though he seems to have an illustration for every architectural feature, concept, and detail in his must-have Illustrated Dictionary of Historic Architecture, Cyril Harris doesn’t include any information on the Golden Rectangle.
Marianne Cusato, in her remarkable work Get Your House Right (a must-have guide book for builders), does discuss the Golden Section (at different times described as the Golden Mean, Divine Proportion, etc). According to Cusato, “The Golden Section was first discovered and used in antiquity during the Golden Age of Greece. The numerical approximation of the ration was embodied in the architecture and theory of the Italian Renaissance and rediscovered in the nineteenth century” (pg. 28).
However, Robert Adam, in his influential work, Classical Architecture, discusses the Golden Rectangle in more depth, and agrees—at least in spirit—with Cusato when he explains that the golden section “has some unique and remarkable characteristics. Many of these were known by mathematicians in antiquity and the Renaissance, but there is no documentary evidence of any architectural interest before the nineteenth century. …While no proof of the eternal beauty of this figure or its conscious use in any period but the recent past can be found, its characteristics are of some interest” (pg. 116).
Marcus Frings, who I have already mentioned, jumps into the argument with both feet in an extensively footnoted article—The Golden Section in Architectural History. Frings, a German art historian and museum curator, finds no actual use of the ratio until possibly the nineteenth century. He traces the mythical theory back to Luca Pacioli, the Renaissance mathematician and intimate of Leonardo da Vinci:
For a long time the Golden Section does not occur in architectural theory. It first appears in the nineteenth century, through Zeising and Fechner, and then rises to a certain fashion in the third and fourth decade of the twentieth century, where Neufert and Le Corbusier get to know it. Neufert held out great hopes for a renewal of architecture through the Golden Mean, but he soon became sober. Nevertheless he presents it in extenso. After early experiments Le Corbusier uses the Golden Section to develop his catalogue of measures, which has – due to roundings and combinations – not much in common either with the Golden Mean or with the Fibonacci series. In fact, Neufert and Le Corbusier seem to use the Golden Section as a way to embellish their own subjective artistic creation by theory and ratio. In any case, the Golden Section certainly does play a role in the writings of these architectural theorists. Prior to the 19th century, however, the Golden Section is simply absent in written architectural theory.
Frings reminds readers of all the architectural authors who mention the Golden Rectangle as the foundation of their work, yet never once provide an example of how they have used the principal in practice.
If you don’t like to read or you don’t trust academics, watch the PBS Nova series. This program is easy to follow, especially when the narrator says with sincere simplicity: “For centuries many scholars believed the golden ratio gave the Parthenon its tremendous power and perfect proportions—most notably the ratio of height to width on its façade. Today the golden ratio’s use in the Parthenon has been largely discredited.”
The program explains with demonstrative detail that rather than the ratio of the golden ratio (1:1.618…), “the ratio most scholars find used in the construction of the Parthenon is 4:9, as found in the height of the face to its width, in the width of the columns, and the distance between their centers.”
The narrator of the Nova feature goes on to say:
Korres and his team have investigated every angle on the Parthenon. And although the building looks straight, they’ve discovered there’s barely a straight line on it.
These curves are no accident. They start with the foundation, or stylobate. Each of the 46 columns has a gently curving profile and leans inward. Even the architraves, marble beams that span the columns, as well as the architectural elements above them, are curved.
Therefore, each of the metope panels (the spaces between the triglyphs along the frieze) are also slightly curved and uniquely shaped and sized to fit between each column. This fact should also dispel the myth that there is a consistent golden ratio seen in the frieze details of the Parthenon.
But that’s not all! For those who want an in-depth video presentation on the golden ratio, don’t miss Keith Devlin’s academic and entertaining video debunking a panoply of myths associated with the golden rectangle, golden ratio, golden section, and golden mean.
Finally, a conclusion (of sorts)
So what does all this mean? Put simply, when it comes to classical proportioning, there are no definitive sources or mystical sets of rules to go by—only guidelines. Carpenters should be familiar with the “order” and hierarchy of the basic elements, but shouldn’t get hung up on the small details. Take a step back and look at the whole picture first. For instance, when you’re building a patio cover, don’t use 4×4 posts—they look too spindly, too skinny, they don’t provide the human eye with a sense of comfort or security at supporting a heavy load. At a minimum, use 6×6 posts. And don’t use a 6×6 beam, use a 6×8 or better yet, a 6×10, or even bigger, depending on the span!
But the classical orders are more than a set of guidelines. As Brent Hull is fond of saying, they provide us with a “grammar:” the classical orders are our original source material for details that are pleasing and symmetrical, like matching window sill and chair rail elevations, sizing baseboard and crown molding, and choosing proportions for projections like eave returns, mantelpiece shelves, and classically inspired door entablatures.
But enough already about the classical orders! The next article on architecture for carpenters will cover the Victorian period—the apotheosis of woodwork.
[i] Vitruvious. Vitruvious, The Ten Books On Architecture, translated by Morris Hicky Morgan. Dover, 1960.
[ii] Palladio, Andrea. The Four Books of Architecture. Dover, 1965.
[iv] Photo credit: Henry Parke, Royal Academy lecture drawing showing a student on a ladder, with a rod, measuring the Corinthian order of the Temple of Jupiter Strator, Rome, SM 23/9/3, Reproduced by Calder Loth with Permission of the Trustrees of the Soane Museum, London.
[v] Benjamin, Asher. The American Builders Companion 1827. Dover, 1969, Preface to third edition.
[vi] Cummings, Abbott Lowell. An Investigation of the Sources, Stylistic Evolution, and Influence of Asher Benjamin’s Builders’ Guides. Oberlin College, 1950.
[vii] Swan, Abraham. Georgian Architectural Designs and Details (1757). Dover, 2005.