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  • Scalebound or Scaling Shapes:A Useful Distinction in the Visual Arts and in the Natural Sciences
  • Benoit B. Mandelbrot, Mathematician

It is often said that 20th-century 'modern' buildings are sterile, not built to human scale and, in fact, unnatural. The more I ponder such statements and their variants, the more elusive I find their meaning and the more I feel that a discussion of the logic and of the aesthetics of the notion of scale needs to be resumed. Clearly, a building's absolute or even relative height and number of stories are incidental and other aspects of scale are more important: I propose that it might be interesting to introduce into aesthetics the distinction between scalebound objects and scaling objects, a broad distinction that is proving increasingly useful in several scientific contexts. One of my conclusions is that it is fruitful to call Mies van der Rohe's buildings scalebound-a term a physicist would use to describe a flawless crystal and the solar system-and to call the Paris Opera House a scaling building-the term scaling also being applicable to typical views of the Alps and to the visual characteristics of many other objects in nature, some of them visible (large or small) and others invisible to the naked eye.

Before I elaborate on this dichotomy, I must emphasize it has limits. I realize that numerous examples both in the sciences and in the visual arts combine scalebound and scaling features. In addition, one must keep in mind the pitfalls of analogy; mine are unlikely to be entirely new and are certain to be found eventually unsatisfactory through counterexample and contradiction.

In very rough outline (important features will be introduced gradually later), I propose the term scalebound to denote any object, whether in nature or one made by an engineer or an artist, for which characteristic elements of scale, such as length and width, are few in number and each with a clearly distinct size.

A spherical radome for sheltering a radar assembly is particularly scalebound, since its perceived scale is determined by the smallest possible number of measurements: a single intrinsic measure, its radius, and a single extrinsic one, the distance from which it is viewed; if a scalebound object is on the human scale, it will not, in general, be on the scale of a fly. Consider another good example: a Bauhaus style, glass cubetype building. If its windows are identical rectangles delimited by metal grids, its only characteristic lengths are the height and width of the whole and of the windows, where the smaller is a harmonic (whole divisor) of the bigger fundamental one. Its basic structure is so spare that as the result of slight changes, such as interrupting the façade by a few floors of different design or by changing the divisions between the windows from thin metal to thick concrete, the overall structure is enormously enriched. Some Post-Bauhaus architects are experts at such enrichment. In any event, any scalebound object does have a scale of its own, and the nearest it can come to being on a human scale is when it has the same size as a human standing next to it.

A scaling object, by contrast, includes as its defining characteristic the presence of very many different elements whose scales are of any imaginable size. There are so many different scales, and their harmonics are so interlaced and interact so confusingly that they are not really distinct from each other, but merge into a continuum. For practical purposes, a scaling object does not have a scale that characterizes it. Its scales vary also depending upon the viewing points of beholders. The same scaling object may be considered as being of a human's dimension or of a fly's dimension.

In several natural sciences, developing independently of each other, the distinction between scalebound and scaling objects has recently acquired much importance. For example, this distinction is basic to my analysis of shapes that I call fractal sets. This leads to fractal geometry, a new geometry arising next to Euclid and devoted to shapes with many scales of length [1]. In addition, I believe that fractals can...


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