Lecture 5

The Universal Quest for Harmony and Unity

Part II: Geometry, Mathematics, and “the Old Way of Seeing”

Reading:

The Old Way of Seeing, p. 45 – 66  

...it is the mark of an educated man to look for precision in each class of things just so far as the nature of the subject admits...
      — Aristotle, Nicomachean Ethics, Book I, para. iii., as translated by W.D. Ross.

The principles that underlie harmonious design are found everywhere and in every time before our own; they are the historic norm.  They are the same in the eighteenth-century houses of Newburyport, Massachusetts, in the buildings of old Japan, in Italian villages, in the cathedrals of France, in the ruins of the Yucatán.  The same kinds of patterns organize Frank Lloyd Wright’s Robie House and Michelangelo’s Capitol.  The disharmony we see around us is the exception. 
      — Jonathan Hale, The Old Way of Seeing: How Architecture Lost its Magic (Boston: Houghton Mifflin, 1995), p. 2

Kepler's "Crystal Spheres" which he theorized as the geometry of the universe   

Kepler's Platonic solid model of the solar system  from Mysterium Cosmographicum (1596). 

The ‘Quest for Harmony and Unity’ that we looked at last time revealed a virtually universal sense that if something we do changes nature in some way, it should be done in accordance with some sort of natural order.  The mythical Fu Xi, you’ll recall, invented the primary ideograms of Chinese writing using nature as paradigm.  Tessie Naranio of the Santa Clara Pueblo told of healing ceremonies for their villages to “bring the human and cosmic forces together.”  And ancient Romans plowed a furrow around what would be their city wall in the founding ceremony of a castrum to establish a sacred domain within, only lifting the plow from the earth as they passed what would be the city’s gates.  These, and countless other acts, recognized the man-made as part of nature.

Geometric and mathematical relationships have been important to the notion of harmony long before the modern era.  Regulating lines—especially squares and “golden section rectangles” have long been used to ensure visual harmony. 

Proportional relationships in Andrea Palladio’s villa Maser

Images by Norman Crowe in Nature and the Idea of a Man-Made World, courtesy of MIT Press. Used with permission. All rights reserved.

While human visual perception is relatively constant from one person to another, the idea of good proportions and beauty necessarily varies, based on experience and knowledge.  There are some constants however, such as a sense of visual balance derived from body-related proportions (see explanation in text from last week’s reading), but even these vary according to cultural and other experiential circumstances.  Expression in architecture varies:  Compare, for instance, high Victorian with Gothic, or Gothic with Renaissance classicism, or Chinese Buddhist architecture with the West’s high Gothic.

So is this to say that all such things are relative and therefore rules to ensure beauty and proportions are essentially arbitrary and therefore futile?  Another way to approach the problem is through the internal logic of a given order.  Imagining a high Gothic window on a Classical façade, or a curving Chinese roof on a crisp Modernist glass box of a building.  This may provide a clue.  Families of elements and proportions provide the key.  All things within a family are derived from the same proportional relationships and general aesthetic character that includes materials, color, texture, rhythm or repetition of like elements, and so on.

The Golden Ratio

According to Euclid,  “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser.”

Golden ratio line. Image courtesy of EIsnel.

Or, the ratio of the whole line to its larger segment is the same as the ratio of the larger segment to the smaller.  This ratio is approximately 1.61803:1. A rectangle with sides in this proportion is a Golden Rectangle:

These relationships have been argued for centuries.  They emerge in treatises on geometry, in religious dogma, and in practical means to ensure “harmonious” visual proportions for things, and in various mystical philosophies.

• The Sufi tradition in Islam
• Pythagorean and Platonic geometries
• Renaissance and neo-Renaissance art and architecture

Examples:

Sufi Geometry	Image by Norman Crowe in Nature and the Idea of a Man-Made World, courtesy of MIT Press. Used with permission. All rights reserved.
Renaissance application of column diameters to coordinate proportional relationships	Image by Norman Crowe in Nature and the Idea of a Man-Made World, courtesy of MIT Press. Used with permission. All rights reserved.
A neo-Renaissance building with geometry based on squares and rectangles derived from the square	Images courtesy of Norman Crowe.
Victorian residence, carefully proportioned according to squares and rectangles derived from the square, understood by its carpenter-builder.	Photo courtesy of Norman Crowe.

• The same geometric and mathematical relationships may be expanded to produce the logarithmic, or “golden” spiral, a pentagram in which segments of a, b, c, d in order of decreasing lengths are in a ratio of 1.618 … or φ.
• Corresponding or related geometric ratios may be found in the many-chambered nautilus, in the growth patterns of flowers, reproduction rates for rabbits and bees, the spiral of pineapples, the DNA spiral …
• Is this proof that Euclid, Pythagoras, Plato, as well as Fibonacci, Kepler, Leonardo da Vinci, Michelangelo, Le Corbusier, William Blake, and others, understood the geometry of the cosmos?  What are we to make of this?  Selected things from nature seem to fit, but others do not.

The world, harmoniously confused,
Where order in variety we see,
And where, though all things differ, all agree.
      —Alexander Pope, Windsor Forest, l. 14-16

Nature, the mysterious forces that create the world we live in, is an experience of such beauty and awesome complexity that it is simply amazing to stumble across patterns that we can recognize and reduce to numbers and written rules.  And yet it happens.  As we peer deeply into the smallest parts of what makes up the whole, we find endless reflections of a proportion that describes so perfectly our place in it all—a perfect harmony of dynamics and balance, of known and unknown, of who we are and what possibilities we hold.
 — Priya Hemenway, Divine Proportion: φ (Phi) In Art, Nature, and Science (New York: Sterling Publishing, 2005), p. 141

Photos courtesy of Norman Crowe

• Does it make any difference whether these ratios and their geometric and mathematical properties are “the key to the universe” or not?  Might it be said that they are all part of a beautiful aesthetic, one that encompasses related mathematical and geometric puzzles, the history of human discovery (and it does so in a very cross-cultural way) and they can be found as manifest in both human artifacts and natural objects across time and space?

Discussion Session

For this session, each student should draw or reproduce in a series of photocopies, the facade and floor plan of a building, and overlay in ink lines, proportional relationships.  (For examples of this type of drawing, see Architecture and Geometry in the Age of the Baroque by George Hersey, and Architectural Principles in the Age of Humanism by Rudolf Wittkower.  Both are on the reserve shelf in the Architecture Library.