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The Deep Structure in Apollo's Perfect Head

If you travel to Göttingen in Germany, you might visit a curious museum collection. Most of the objects are sculptures of mathematical shapes, which were created as teaching aids to help students understand complex geometry. One especially odd item is a copy of a famous bust of antiquity known as the "Belvedere Apollo". This version has pencil lines drawn across its surface, as though the head of Apollo had been deliberately defaced. Still more curiously, the lines were applied by students at the request of the famous mathematician Felix Klein. The lines mark particular geometric features of the curved surface-features that can be given precise definitions by mathematicians, in such a way that they could all agree on exactly where the lines should travel across the surface.

The mathematical shapes at Göttingen are now old and the mathematics that they describe is firmly in the syllabus of university courses around the world. Nonetheless, in recent years, pictures of this defaced statue have been on display at research conferences where psychology and the study of the brain are discussed. What do scientists hope to learn from this? The link with the statue of Apollo arises because psychologists would like to understand how our eyes and brains help us to see objects in the natural world. One especially tough problem is how we see visual shape.

Even a simple sketch with only a few black and white lines can be sufficient to give us a vivid impression of the shape of something. Artists practise their technique at length to get this just right. Newspapers have cartoons every day. How does so little convey so much? Felix Klein thought he might have the answer in a rather direct way from his mathematical geometry. He actually dared to think that he might discover the basic principals of aesthetic beauty simply by applying his mathematics. The "Belvedere Apollo" is a prime example of the ideal shape of classical beauty but judgements about aesthetics are highly individual and subjective. For Klein, the dilemma was to understand why so many people agree, when their individual judgements are subjective. The lines on the surface of Apollo, which follow the rules of geometry, might offer a solution. Perhaps, simply by getting the mathematics correct, the secret principals underlying classical beauty might be revealed. This now appears to have been desperately optimistic. Nonetheless, as so often in science, something rather useful emerges from the remains of a daring, if misconceived, idea.

Psychologists and brain scientists have increasingly come to realise that, in order to understand why our brains work in particular ways, we must derive our understanding from the natural world. This is a central idea in biology, which we owe to Charles Darwin. Organisms (plants, animals, bacteria, and viruses) are constantly challenged by their environment. Those that are best suited to their environment survive and have offspring. The offspring face similar challenges, so over a sequence of generations the population changes. Eventually, it comes to consist of individuals that are better equipped biologically to live in their surrounding environment than those many generations ago. Thus, the environment shapes the individuals who must inhabit it. Darwin's insight was natural selection: the environment will select individuals through the mechanism of inheritance from one generation to the next. Nowadays, we also recognise that the environment has a profound influence throughout the lifetime of single individuals, through the mechanism of learning.

Whether we are thinking about either genetic inheritance or learning, we need to understand our visual environment in order to understand shape perception. Here lies the connection with mathematics and Felix Klein. Since before the time of Euclid (some 2200 years ago), mathematicians have used their skill to describe the natural world. Indeed the word geometry derives from Ancient Greek and refers to "measurement of the land". If mathematicians have insight into the structure of the natural world through their understanding of geometry and shape, then psychologists and brain scientists need to understand their way of thinking. Obviously, not all of the mathematics will be relevant for understanding how we see. Furthermore, the steps from a mathematical description of shape to an understanding of the psychology of shape will be much more tortuous than Felix Klein supposed. The various intermediate steps need to take in brain science, optics, computer vision and robotics, neural networks and psychology. Hence, this involves gathering information from a wide and varied field. One of my aims with a 21st Century Fellowship is to write a book, which will evaluate how far we have got and attempt to achieve a consensus about what should happen next.

Quite reasonably at this point, you may be feeling a little sceptical, perhaps even hostile. Possibly, we could put together an account of geometric shape in the natural world. Possibly also, we could use that account to understand how and why nerve cells inside the brain respond to visual information when our eyes are pointing in a particular direction as we look at an object. Conceivably, the geometry that we have discussed so far may be satisfactory for dealing with solid objects, such as statues, pebbles, tree-trunks, chairs, tables and so on. It is much less clear that the same geometry would be adequate for dealing with other objects that are of profound interest to us. Think about the changes in the human face during a conversation. The facial gestures are fluid and continuously changing. Think about flowers or leaves on a single plant, where the flowers and leaves grow to the same basic pattern but differ from one another, if we inspect in detail. Think about a flowing garment or stream of water. With all of these examples, we face a common paradox. We are acquainted precisely with the objects. We may conjure them up to our imaginations. Nevertheless, if we are asked for a detailed description in words, the description is always less than adequate until the reader has experienced the objects visually for themselves.

Simply in order to discuss this alternative kind of shape, we need to give it a name and I propose here to refer to it as organic shape. To some degree, this label is no more than a statement of our ignorance. In another way, it captures an important aspect of these shapes. In all these cases, the shape is subject to a process that changes over time. It is not simply that the object moves around, whilst retaining its shape. The shape of the object is actually formed by a process that is extended in time. For example, flowers grow according to particular patterns that depend upon the species of plant. They often possess a degree of symmetry due to this process, although the symmetry is rarely exact because of variations in the rate of growth.

It would be a major achievement to understand how we psychologically perceive objects such as this, which have developed according to a pattern. One approach is to carry out visual experiments in which people match up shapes that possess the same underlying pattern. To do this, they will have to identify visually the common elements that arise from the underlying pattern, rather like a botanist who is studying new species of plants. For convenience, computer graphics will be used to prepare and display the individual shapes. Like flowers, these artificially created shapes will differ in their individual appearance because the computer will superimpose random variation on the underlying pattern. The human observers will have to find the pattern and ignore the random variation. With modern brain-scanning techniques, we can also attempt to discover which areas of the brain are most active when humans are responding to these shapes.

Does this mean that we should change the focus of Felix Klein's enquiry? Perhaps, the true nature of aesthetic judgement lies in the excitation of a particular state of the brain, rather than a mathematical description of ideal form. This proposal simply repeats the mistake in a new form. During their daily activity, people make judgements about the visual attractiveness of objects. Some of these objects are natural; others are human artefacts. Most people would feel that their evaluations are personal and unique, in a way that cannot be explained or probed by psychologists or brain scientists. Perhaps the most important lesson from Felix Klein's earlier attempt is to refine the question that science can legitimately ask. Whilst continuing to respect the personal element of aesthetic judgements, we can still ask what features of our brain and general psychology equip us with the ability to make judgements about shape and form. We can also enquire exactly how our experience of the structure of the natural world moulds our visual systems. This shared influence affects us all. Chaucer wrote, "ful wys is he that kan hymselven knowe!". As we become more "wys" about how our brains and bodies work, this increases rather than diminishes our appreciation of what it is to be human.