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# Applications of the Golden Mean to Architecture

The Golden Mean, an irrational number related to the Fibonacci sequence, arises in the study of biological growth and hierarchical systems. It does apply to architectural composition in the context of scaling hierarchy but, this design methodology is unrelated to a rectangle’s aspect ratio1.618:1. And, simply considering rectangular aspect ratios does not guarantee good design.

What is called the Golden Mean (or Golden Ratio) is an irrational number approximately equal to 1.618 and usually

denoted f (the Greek letter Phi). This number arises as the solution to the problem of subdividing a rectangle into a square x2 and a remaining, smaller rectangle that is similar to (i.e. has the same aspect ratio as) the original large rectangle.

Design is linked mathematically with natural growth through hierarchical subdivisions at distinct scales, which are found in a majority of natural structures. There is further more a regular geometrical relation among different scales of substructure and in many cases, the scales are related by a single scaling factor.

A crucial lesson that comes from understanding natural structure is to realize that scales in a natural hierarchy are skewed towards the smallest sizes. Natural growth begins at the infinitesimal scale and develops through an ordered hierarchy up to the largest size. The spacing of different scales is therefore not uniform. There are proportionately more small levels of scale than large scales. In actual design, the brief and human dimensions fix the larger scales, then the smaller scales are computed from those.

Design can be guided by knowing a sequence of sizes that should be defined very approximately by the tectonics of the structure itself. Where structural members don’t provide a required scale, the architect creates ornament. This is a key point. What probably happened throughout history is that builders simply created subdivisions that “felt right” because those mimicked natural forms.

For example, the diagram illustrates two important features.

First, it generates a hierarchy of scales — not ratios of sides — that can then be used to approximately regulate a structure’s subdivisions. The second point is to recognize fractal scaling, where similar components (four rectangles in this simplified case, but in practice any shape at all) repeat at different magnification. Scaling similarity is the main characteristic of all fractals, and can be found in many of the world’s most beloved historical buildings.

To summarize: the Golden Mean f is useful in human creations in the same way it is found to occur in nature, where it related to the hierarchical scaling that is a consequence of organic growth. Any design that is meant to appeal to human users in the same mathematical sense as complex structures ought to exhibit a natural hierarchy of scales and those can be generated by either an exponential or a Fibonacci sequence.

Nowadays, the Golden Mean continues to be misused for its purely mystical value, to sell monstrous or otherwise bizarre structures that fail to connect emotionally with the user.

Aesthetic response to a Golden Mean rectangle

The widely-circulated claim that a rectangle with aspect ratio f gives maximum visual and aesthetic pleasure to an observer is false. Eminent architectural historians, Rudolf Wittkower and distinguished architect and thinker, Christopher Alexander simultaneously and independently disproved nineteenth-century claims that the Golden Mean is responsible for a unique aesthetic experience. Visual experiments establish that people can’t distinguish between rectangles with ratio the Golden Mean f and one whose aspect ratio differs by up to 6%.

So, why do architects continue to uphold historic misconceptions about the Golden Mean? The reason is that people will believe myths while remaining oblivious to both mathematical proof and scientific experiments. The major contemporary projects try to use the Golden Mean as one of their selling points.