Golden Section and Golden Rectangle

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adjustable dividers used to set out or measure golden sections and golden rectangles

Over the last year, several articles on the internet, and the comments that they have generated, have been scathing about the use of the Golden Section or Golden Rectangle as an underlying geometric system for designing a building or designing a piece of furniture or for setting out the main elements in a painting as if it was either just an odd intellectual game - and therefore obviously suspicious - or something that was so malleable that it has hardly any real meaning.  

However, it was only with developments in both mathematics and in engineering following the work of Isaac Newton in the 17th century, that architects and engineers have been able to calculate momentum and force with ever greater accuracy to determine if a building will stand up under the weight of the structure itself and will cope with thepressure from weight loaded on the floors or withstand the pressure of weather … so heavy snow on the roof or strong gale-force winds or worse blowing against the walls. 

Architects and masons in the medieval period and through the Renaissance understood that weight had to be carried down to the ground and tall thin walls, if not self-supporting, had to be propped up … so a buttresses is primarily a support for the structure and decorative only as a bonus … but they could not actually calculate loads or force. Admittedly, that hardly inhibited what they built when you look at the dome of St Peter’s or the spire of a great Gothic cathedral. 

But when architects and builders in the past wanted to push the limits - to build bigger or higher - or if they wanted to undermine the strength of a wall by inserting large doorways or windows or load the walls with a heavier form of roof or ceiling structure - there were two arguments that could be used to convince unsure clients: architects cited either precedence … something similar had been built somewhere else and it was still standing … or they used geometry by using a shape or underlying arrangement of parts that appeared to be inherently stable … so, for instance, by using an equilateral triangle as the basic element of a cross section.

From the Renaissance onwards, but actually derived from Greek and Roman theories of beauty, there was a third way to justify a design and that was to relate the proportions and the geometry of a design to the human form … preferably the perfect human form … so, Leonardo da Vinci demonstrated the underlying geometry and therefore the beauty of the human body by drawing a man, hands stretched out, fitting both within a circle and within a square. This is not the place to extend the discussion to the ideas of Divine Beauty or of God making man in his own image but it was certainly part of an attempt to analyse and categories and define what makes one shape or one building or one body beautiful and another not.

The clear and rather more practical advantage that came from using geometry or a grid to design plans and elevations was that builders, masons, brick layers or carpenters, could produce a plan or design using known geometric shapes and proportions and then lay out their work using ropes and pegs rather than using a unit of measurement.

Using a Golden Rectangle as the primary geometric control for a design has considerable appeal, in part because it is the expression of a proportional relationship but it can also, again in a very practical way, be set out with dividers or compasses on the drawing board or laid out on the ground with ropes and pegs without using measurements.

Expressed as a proportional relationship it is where a line that can be of any length is divided into two parts so that mathematically the larger part divided by the smaller part is equal to the overall length divided by the larger part.

 

When the Golden Section is used to form a rectangle by taking a Golden Section and making the shorter side of the rectangle the same dimension as the larger part of the Golden Section to - a Golden Rectangle - the subdivisions seem to take on almost magical properties because if you start with a Golden Rectangle and then divide it into two by drawing a line between the long sides to form a square then the rectangle in the other part is itself a Golden Rectangle and that can be divided again into a square and an even smaller Golden Rectangle and so on and so on down to smaller and smaller squares and rectangles. You can go in the opposite direction, expanding outwards, so by adding a square to the side of a Golden Rectangle you form a larger rectangle of exactly the same proportions … so another Golden Rectangle.

This trick of drawing a series of larger or smaller rectangle that are directly related to each other by their proportions is something that appeals to architects and designers but that same precise relationship … the larger part divided by the smaller part being equal to the whole divided by the larger part … is also found in a mathematical progression or sequence first published in 1202 by the Italian Leonardo Fibonacci in his book Liber Abaci. 

 
 
 

Arne Jacobsen was certainly not the only architect in Denmark in the 20th century to use the Golden Rectangle as a tool when he was drawing out a design. The garden designer G N Brandt divided his own garden in Ordrup - a garden Jacobsen living nearby must have known - into areas with the proportions of a Golden Rectangle and in 1925, in his design for the grounds of the Cathedral School in Viborg, Brandt set out the large area of grass below the main building with the proportions of a Golden Rectangle.

Drawings by the French architect Le Corbusier - fourteen or fifteen years older than Jacobsen - show that he used the Golden Rectangle to determine rational subdivisions of height and width for a modular system for architecture, for interiors and for furniture - and that was work that was certainly known about and discussed at the Royal Academy in Copenhagen by Kaare Klint and his pupils. And, perhaps even less obvious in the finished work, Charles and Ray Eames in the United States used a Golden Rectangle to determine the proportions of their LCW Chair from 1945/1946 so when seen from the front, the height of the seat is the same as its width, fitting tightly within a square, but, overall, the width and the overall height of the back define a Golden rectangle.

Geometry, proportion and scale were and are useful tools for the rational designer.