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The Plastic Number and the Divine Proportion
by Juan C. Dürsteler [message n 145]

Scale and proportion are key concepts of visual representations. The divine proportion during centuries and, more recently, Van der Laaan’s plastic number have been proposed as aesthetic choices for proportion. Nevertheless it’s not clear whether they really simplify understanding or aesthetics nor whether they are connected to our nature or not.
Modulor.jpg (73518 bytes)
Modulor by Le Corbusier. The Swiss architect created this schema about proportions based in the golden section, that you can find in the human body. For example the ratio between the distance of the head and navel to the ground is approximately Phi (1.618...). 

Since ancient times there has existed the idea that certain serial arrangements of numbers reflect certain properties of nature either better or worse. In fact this is the underlying concept of scale. A scale is a sequence of ordered numbers that usually serves as a comparison in order to define proportions between the real universe and the one we are willing to represent [possibly in graphic form].

The complexity that the representation of the real world imposes has given rise to the appearance of many different choices of scales. Among them you can find the many musical scales (diatonic, chromatic, tempered, sorog hirajoshi…) where sounds (sound frequencies) that are perceived as equivalent are fractions or multiples (proportions in the end) of other sounds.

In architecture, proportions are important and for many centuries architects have wondered which relations between sizes of the different architectonical elements are most appropriate, i.e. most aesthetically or functionally pleasant . Not in vain Goethe defined architecture as “frozen music”. 

 Which are the ideal proportions for graphic representations? Does there exist a perfect proportion between height and width of a visualisation?. 

Over the last few centuries many people have considered that the Phi number, better known as the divine proportion or the golden section is a standard for balance and beauty in regards to proportions. Phi is 1.618033988..., or the limit that the ratio between any two elements of the Fibonacci sequence tend to. 

The Fibonacci sequence is very easily constructed. Each term is just the sum of the two preceding ones, beginning with 0 and 1.

0  1  1  2  3  5  8  13  21  34  55  89  144  233 ...

The nice thing about the golden section is that it is a proportion that appears with certain frequency in nature, especially in geometry, but also in the approximate proportions of the human body.

In the website of Ron Knotts of the University of Surrey you can find many examples. Many other (yet much more disputable) ones are available at Goldennumber.net

But there are also a lot of misinterpretations around Phi like, for example, it’s a common mistake that in the Nautilus shell (a sea cephalopod) Phi plays an important role. This is not true, its shell is constructed around a logarithmic spiral, not around a golden spiral, as can be seen at "Spirals and the Golden Section" by John Sharp. Many attributions to golden proportions are only wishful approximations. 

But let’s come back to our interest. Phi is a member of the so called “morphic numbers” that have the interesting properties that you can find two values k and l which satisfy that

Morphic_en.gif (9319 bytes)
Morphic number condition. k=2 and l=1 give the golden section, k=3 and l=4, produce the plastic number. Tha chart shows the interesting properties of these two numbers. When p is the golden section, 1+p=p2 and p-1=1/p. When p is the plastic number you get p-1= p-4 y p3=p+1.
Click on the image to enlarge it
Source: article about "Morphic Numbers" (see the text) 

A question immediately arises: is there any other morphic number besides the golden section?. Arts, Fokkink and Kruijtzer from the University of Delft demonstrate in their article “Morphic numbers” that there are only two morphic numbers, the divine proportion and the “plastic number” discovered in 1928 by the architect and Benedictine monk Hans van der Laan, who used it as a base for the proportion of his architectural constructions. The plastic numbers gives birth to the Van der Laan scale that was used in the construction of the chapel of St. Benedictusberg a Benedictine abbey.

VderLaanBnd01.jpg (32294 bytes) VderLaanBnd02.jpg (31766 bytes)
Interior of the chapel of the Benedictine abbey  of Sint Benedictusberg, designed by Hans van der Laan (1904-1991) using the plastic number as the basis of its scale.
Click on the image to enlarge it
See the photo gallery of the same.

Answering our question, could it be that the golden section or the plastic number is the ideal proportion to make graphic representations? There’s no indisputable evidence about this. Sr Wiliiam Playfair, reputed as one of the first to make bar charts in the 18th century, used predominantly close to the golden section proportions in his graphics, although he made use of other proportions too.

Edward Tufte points out that human preferences for proportions in rectangular shapes have been the subject of study since 1860 by psychologists that have found a mild preference for proportions around the golden section, but with a variation that goes from 1.2 up to 2.2.

The existence of a “natural” proportion connecting to the perceptual roots of the human nature is not nonsense. Should it exist, it would provide a basis on which to construct harmonious scales and probably less cumbersome graphics. A related idea states that given the fractal nature of the world, information visualisation in fractal form could be closer to our natural way of perceiving the world, thus being a more advantageous one.

Although the idea is very appealing, unfortunately there isn’t indisputable evidence about it. The way we humans process perceptual information is still largely a mystery. Structuralists consider, for example, that each and every representation is of an arbitrary-conventional nature, rejecting the possibility of sensorial, representation that can be understood without the need to learn a particular convention.

When faced with this situation pragmatism is the choice. Following Tufte, if the nature of the representation suggests its shape, follow it. If not, preferably use a wider rather than a taller shape with a proportion that appears useful or pleasant to you.

In my personal opinion, consistently using a coherent scale, be it the golden section, the plastic number or whatever else is always a good choice to build harmonious representations. But here the key is consistency, not the proportion itself.

I Owe the inspiration for this article to an interesting discussion with the architect Manuel Couceiro da Costa and Jim Wise, cognitive psychologist and expert in information visualisation, during a cold and rainy Saturday morning at the Circulo de Bellas Artes in Madrid.

Links of this issue:

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html   Ron Knotts' Website about the golden section
http://goldennumber.net/   Golden number net
http://www.nexusjournal.com/Sharp_v4n1-pt04.html   Spirals and the Golden Section" by John Sharp
http://www.math.leidenuniv.nl/~naw/serie5/deel02/mrt2001/pdf/archi.pdf   Artícle about Morphic Numbers
http://www.vijlen.net/kerk/content/foto's%20abdij%20st%20benedictusberg.html   Photo gallery about Sint Benedictusberg
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