The last few posts have all been to some extent about how drawing can be
locked into a debate about mathematics, rightness and underlying concepts of
beauty. These are concerns that have a long history. One aspect of beauty is
historically associated with the idea of a perceptual experience of pleasure or
intellectual satisfaction. This is often associated with the idea of a
harmony that should exist between the components of the perceived object
if it is to be seen to be beautiful. Alongside this concept of harmony is a
much bigger picture, one that sees beauty as representing the underlying
harmony of the universe as a whole. This is traditionally represented by
mathematical order, this order being a 'supreme value', one that is basic to
the whole universe.

In ancient Greece the Pythagoreans believed that harmony was associated
with mathematical order and balance, aesthetic experiences in art were
therefore closely tied to mathematical ratios and the key components
of harmony were symmetry and proportion.

One particular geometric ratio has been used by philosophers and artists
over and over again, as the supreme example of a harmonic ratio in both
art and nature, this is 'the golden ratio' or 'golden section. It was defined
by Eclid as; "A straight line is cut in extreme and mean
ratio when, as the whole line is to the greater segment, so is the greater
to the lesser."

The classic illustration of a golden section demonstrates how a spiral can be constructed by drawing a curve through the corners of squares built around its geometry, and it is often pointed out that this spiral is the same one many natural objects exhibit as they grow.

Theorists have tried to use the harmonic ratios associated with the Golden section and the related Fibonacci series of numbers to illustrate concepts of visual beauty, and the illustrations below are typical of the way in which this sort of geometry is laid on top of images to 'prove' why they are 'beautiful'.

The Golden section is so popular it even has its own website, See http://www.goldennumber.net If you were interested in the fact that the Bezier curve was developed as an aid to car design, see the section on Aston Martin design.
However the search for underlying rightness has been done in many ways and as art itself changes focus, so the methods of looking for this rightness change.

In the 1950s art educationalists such as Harry Thubron introduced a square into circle exercise. It was still being used when I began teaching on the Foundation Course at LCA in 1974/5. It was one of a series of related exercises designed to get students to find objects and shapes that lay in an indeterminate position between one thing and another. See. The circle to square exercise was designed to generate a series of forms that were neither circles nor squares, their very indeterminacy being a property that was looked for. These drawings were done by hand, (this was before the introduction of computers into the college), students would gradually adjust their squares and circles until they arrived at a mid point, in a similar way to the layout below, except of course the drawings would be much more organic and far less smooth in their transitions.

The point was to find a series of forms that were visually 'searching' for their 'rightness'. Being neither one thing nor another, they were usually 'active', their 'life' depending on our perceptual need for rightness, these mid-way forms were seen as vital to an understanding of a certain sort of 'organic abstraction' that was very popular during this time. The image below is of a Harry Thubron construction, (in the private collection of Glynn Thompson I believe). The one below that is in the collection of the Leeds City Art Gallery, and like the white shape above it, its red central form was designed by Thubron to lie between a square and a circle.

When Leeds College of Art became re-named the Jacob Kramer College, they needed a symbol or logo for the new college and this was based on Thubron's shape. I couldn't find a colour photograph, but you can just about see the 'red spot' in the centre of a white rectangle, in this image of one of the college annexes below. It's above the cars parked behind the iron railings.

You can see similar forms of 'organic abstraction' in the work of Victor Pasmore, (below), another artist from the same time period, who worked very closely with Thubron in the development of art education.

Like the Bezier curve, these forms give a certain visual satisfaction, however they point to another issue, one that is perhaps central to the difference between art and mathematics. These forms are interesting because they generate movement, the visual interest in the Golden Section it could be argued is because it does the same, movement being generated because the eyes are switching attention between the various divisions as they see the similarity of relationships between the part and the whole. Artists are looking for 'life' within the purity of mathematical form, once 'right' a form is perhaps too 'finished' or unable to be connected with as it is not alive.

However the need for some sort of underpinning structure continues, this might come from a particular closed logic, a rationale that runs through the individual elements of a construction or a tight relationship between the artwork and a particular piece of research. All of these things represent a desire for order of some sort. The problem is that for many of us life is as much an emotional dilemma as a thing that can be logically understood, therefore art needs to straddle the two if it is to reflect this duality of the human condition.

Plato saw our everyday world as an imperfect reflection of a supreme order that lay somewhere 'beyond', he would therefore have seen the sense in using a mathematical underlying order as a symbol for this, however Aristotle was more grounded in reality and believed we could find order by examining what was around us, not by looking for it in some realm of perfection. You can still see this division in how different cultures have used geometric form to express these ideas. Islamic decorative art is deeply connected with the possibilities of geometry to develop symbolic form. In geometric pattern an Islamic artist can signify God's will expressed through His Creation. (See) In the Christian art of the Renaissance, underlying geometry was used by artists to signify the relationship between the Trinity and worldly order. (See)

Perhaps the grid could be read as an archetypal symbol and very useable tool to both illustrate how mathematical order can lie beneath an image and put that order in place.

See also:

Mathematical shapes of interest
On growth and form
The weaving of grids
Maps