Though the study of biological structure has depended for centuries upon the visual appreciation of anatomies by the human eye and mind, it has rested for at least many decades upon the additional methods of measurement and analysis. In earlier times measurements were made with calipers, and simple univariate and bivariate statistics were the primary analytical methods available. In more recent times, measurements have come to be automated using electronic measuring devices, and analysis often proceeds much further employing such approaches as the multivariate statistical method, invented, indeed, years ago, but not a practical tool for biologists until the current...

However, as we think about the possible structures that data may take up, it is clear that the matter may be far more complex. For instance, the moment that our data are in more than two or three dimensions, the moment, in other words, that more than three variables describe each point in our data, we have difficulty in recognising what the data show. And the moment that the data take up forms in which groups may be of shapes other than spherical or elliptical (hyper-spherical or hyper-eliptical, as the case may be), the moment that groups have interfaces with...

Again, when we come to study the interfaces between groups, there are many different possibilities. Though figure 7 shows, left frame, an interface between two groups that is of simple, linear, form, it also demonstrates, middle frame, a more complex interdigitated arrangement that is easily seen in this two-dimensional example but which might be less easily recognised in the multi-dimensional situation. And it depicts, right frame, a further extension of this complexity resembling the type of interface that one might see between two immiscible liquids such as oil and water.

On the other hand, figure 8 shows that two groups...

It is clear, then, that if such data complexities truly exist within biological structures, we must have additional tools for coping with them. But if we are to save ourselves a great deal of work, we might first ask the question — do such complexities ever really exist in biology? The answer comes back loud and clear that they do; and I would like to present you with two examples from the rather early work on the primate shoulder that I carried out for my Ph.D. degree.

We were interested at that time in what the shape of the shoulder girdle...

For this reason, then, a large part of our efforts over the last few years have been aimed at trying to move from simple ways of looking at measurements defining structures to more complex ways of analysing measurements representing structures. In other words we have been attempting to reclaim information inherent in measures of structures that were lost because the methods for analysing the structures were overly simplistic.

However, it is equally evident to us that the very process of taking measurements from specially defined points already loses a great deal of the information inherent in shapes simply because the...

The answer may lie to some degree within a series of methods that have been evolved in recent years to cope with the problems of characterising the complexity of a picture. These methods allow holistic appreciation of pictures such as photographs of sections and radiographs. These, though not themselves three-dimensional, can provide the three-dimensional information.

The technique involves optical Fourier transformation. This is a process whereby a single complex wave is broken down into its component simple waves. It is what we achieve by passing a complex wave (white light) through a lens thus breaking it down into its component...

Such methods must be of interest to anatomists, of all scientists those most closely involved in the study of structure. And indeed, in our own work, they have proven most useful. We have been interested for many years in the relationship between internal architecture and stress bearing in bones. The chief view of this, an idea more than a century and a half old (see Culmann and Meyer as reported in Murray, 1936) is that the trabeculae in cancellous bone are arranged in orthogonal architectural networks that parallel the orthogonal mechanical networks of the linear principal stresses that can be...

We have examined a gamut of questions about shape and structure. We have moved from visual examination of shape, through characterization and comparison using individual measurements one by one, through measurements put together by multivariate analysis, to quasi-continuous characterizations of shape in which field definitions are obtained by extrapolation between measured points.

We have passed through methods that look at total definitions of shape as external form, whether of a two-dimensional outline or a three-dimensional envelope. We have moved towards definitions of shape that include internal structure and pattern as well as external form.

We have looked at examples of...