Article by Eden Muir for le Tour Sutton, October 2010.


The Search for Harmony in Architecture

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Pythagoras Let's travel back in time 2,500 years to the Greek island of Samos. A mathematician is investigating musical intervals and harmonics by tuning a single-string instrument, the monochord. He discovers that if the string length is reduced by half, its pitch is exactly one octave higher. Certain ratios such as 2:3 and 3:4 produce pleasing sounds, others create irritating disharmony. Pythagoras (570 - 495 BCE) had discovered the mathematics of consonance and dissonance, the foundation of musical harmony.

Golden Section Ever since that brilliant insight, Pythagoras and others speculated on the existence of a similar mathematical basis for harmony, proportion and beauty in art and architecture. We all remember the Pythagorean theorem explained by the familiar triangle with sides of 3 and 4 and hypotenuse of 5 units. Could there be a geometry as simple as that for aesthetic coherence and harmony? Pythagoras is said to have advanced the Golden Section, a 1.618 to 1 ratio which corresponds to certain fundamental proportions of the human body. The regular pentagram, which also contains that ratio, became the Pythagoreans' symbol.

Parthenon For centuries researchers have superimposed Golden Sections over building plans to prove that architectural beauty is derived from invisible regulating lines. The Roman architect Vitruvius (80-15 BCE) encouraged a more holistic approach: harmonious design must consider multiple factors: the building, site, orientation, wind, light, shade, elevation and materials. Vitruvius was also fascinated by architectural units and numbers. He documented the proportions of the classical orders: the height of a Tuscan column was 7 times its lower diameter; Doric 8; Ionic 9. Vitruvius saw the human figure as the ultimate inspiration for proportional design systems just as it had generated dimensional units since the time of the Egyptians: the palm equals 4 fingers; the foot is 4 palms; the cubit six palms; the man's height 4 cubits.

 Vitruvian Man Fast-forward 12 centuries: mathematician Fibonacci (1170-1250) introduces his famous number series (1, 1, 2, 3, 5, 8, 13, 21, etc.) which approaches the Golden Ratio. It has the added advantage of being expressed as an attractive spiral within the Golden Rectangle. Later, Leonardo da Vinci (1452-1519) investigates human anatomy directly, through dissections of cadavers, and sketches the Vitruvian Man to illustrate his discussion of symmetry and harmony in temple architecture. Alberti looks to the human frame to inspire his system of proportions, and Pacioli describes the "divine" properties of the Golden Ratio, which had already gathered mystical connotations.

Orders For many centuries, harmony and beauty in a building were judged by how carefully the architect followed the classical Vitruvian rules which had evolved into a sophisticated proportional system. But with the 20th Century came the revolutionary new ideas of Frank Lloyd Wright (1875-1950) who rejected the past and developed his own "organic" architecture with proportions, ornament and forms inspired by landscape and plant life.

Modulor Le Corbusier (1887-1965) went further, stripping away all ornament on his white stucco villas perched on stilts. But this did not mark the end of "magic" ratios. In 1948 Le Corbusier published his own proportional system based on the body, summing it up as "a range of harmonious measurements to suit the human scale, universally applicable to architecture." This radical Swiss Modernist felt compelled to pursue a mathematical basis for architecture and he obsessively traced Golden Rectangles to guide his design decisions.

Landsberg Building In the late 1870s, with the railroad coming to Frelighsburg, the visionary Mayor Joseph Landsberg decided to build what would be for a time the finest imported goods store in eastern Canada. The masons laid up the thick brick walls using a foolproof geometry worthy of the Greeks--the building was 80 by 60 feet in plan, forming a perfect 3/4/5 triangle with a convenient 100 foot hypotenuse. Perhaps more surprisingly, the main facade also measures 60 feet to the top of the cupola. This sturdy tribute to Pythagoras still sits proudly on rue Principale!

CN Tower Today, most architects would say that harmony is not based on numbers but on design intuition, or simply having a "good eye" for form, materials, colours and context. But architects and master builders have always found it hard to resist a juicy mathematical ratio. If you look carefully, you'll find 1.618 to 1 proportions in the oddest places, from Toronto's CN Tower to small town commercial facades. Check out the older buildings on your own Main Street and see how many ancient Golden Sections are hiding in plain sight!

1. Pythagorean theorem demonstrated with a 3/4/5 triangle

2. Golden Section constructed geometrically

3. Golden Section with Fibonacci spiral superimposed on the Parthenon

4. The classical orders aimed at harmony down to the smallest detail

5. Vitruvian Man by Leonardo da Vinci

6. Le Modulor, Le Corbusier's architectural design guide based on the Golden Ratio

7. The Landsberg Building, Frelighsburg, 1879, built on a Pythagorean plan

8. CN Tower. Golden Ratio between heights of tower (555.3 m) and glass floor (342 m). Using the official published heights, we get 555.3 / 342 = 1.62 / 1, or the Golden Ratio.

© Tous droits réservés Eden Greig Muir, architecte

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