In my lecture "On the Path to a General Theory of Proportion: Research Perspectives for the 21st Century," which I gave at the SFIA conference, "Ecological Architecture, The Unstoppable Wave", at Berkeley on July 4-7, 2002, I proposed strategic guidelines required for the establishment of a general theory of proportion. I am an economist and ecologist and hold a Ph.D. in geography. The lecture identified environmental aesthetics as a basic resource that is required for the ecological survival of our global civilization. Research in proportion should be established as a decisive means for gaining objective and operational tools for preserving and re-establishing this "basis resource", i.e. the aesthetics in man's environment.
The paper focused on the fact that the aesthetic qualities or beauty found in objects depends primarily on the balance between order and diversity and on a number of figurative levels and the patterns mediating between them. This "mediation by patterns" creates a unified or balanced perceptual effect for each figurative level of aesthetic shapes. Referring to proportion, the overall patterns are that of geometry, (primarily Euclidian and non-Euclidian geometry). Today there is also no doubt that fractal geometry is a proportion generating pattern. Since proportion refers to extensional ratios, it is bound to systems of geometrical or mathematical ratios. Patterns of symmetry and topology are, in principle, also patterns of visual math, but they differ from proportion because their patterns are related either to invariants in transformation and regularity (i.e. the 247 symmetry groups) or invariants of positions and networks (topology, i.e. knots).
Architectural aesthetics depends also on additional criteria like unity and diversity in other visual math criteria: texture, colour, rhythm, light & shadow, groupings, etc. The fulfilling of these additional criteria will often be worthless if the underlining proportions (geometric modules) have no balance - as is the case in the majority of modern buildings. In a book to be published in 2004 I will discuss proportion principles - based on elementary practical geometry - in context with many other principles discussed already by the classical Gestalt Psychology. All these traits can be objectified and can easily be handled. The buildings of the past, in addition to their balanced figurative or geometric qualities, also showed unified schemes of micro-fractality in material texture, colour, and in the play of light and shade. Rhythm and substructures were harmoniously balanced by proportional order, and the sequences of Gestalt laws were masterfully respected. Proportional geometry was based on practical geometry. Master carpenters may have played within these easy rules of geometry.
Perfect proportional beauty can be based on the simple geometric figures or regular polygons or solids. They comply simultaneously with the mathematical visual patterns of geometry, symmetry, and topology: examples include the regular triangle, square, and pentagon or their polytopes and duplications (hexagons, octagons, decagons etc.) and the regular polyhedra (Platonic and Archimedean solids). The beauty of shapes with simultaneously perfect geometrical, symmetrical, and topological patterns (i.e. knot patterns) is present in the classic Islamic architectural ornamentation (Arabesques); many of these decorations embody all 17 plane symmetry groups (H. GÖTZE 1991, MONTESINOS AMIBILIA 1987). The overlapping of these three forms of visual patterns, geometrical, symmetrical, and topological, may explain why simple polygonal geometry has such a strong aesthetic appeal. The three core polygons, (the equilateral triangle, square, and pentagon), were powerful design tools in pre-industrial architecture. The equilateral triangle, square, and pentagon can be used to develop approximately 300 proportion codes which are present in world architecture. Pre-industrial designers were able to derive these polygons using the basic geometric constructions of practical geometry, i.e. fig. 2 (as shown in RORICZER's Geometria deutsch 1486 and DÜRER's Underweysung 1525, whereby the pentagon is approximately constructed).
In order to create the diversity that is inherent in beauty, designers must create a mental environment that fosters the free flow of intuition and creativity. A proportional theory would facilitate this freedom by creating a base of simple rules and limits. This blending of intuition and the rules derived from proportion was characteristic in the buildings of the past. Pre-industrial designers understood the "the old way of seeing" (J. HALE 1994), the principles underlying harmonious design, and that "intuition is not [merely] raw feeling."
"The old buildings smiled, while our new buildings are faceless. The old buildings sang, while the buildings of our age have no music in them". . ."the principles that underlie harmonious design are found everywhere and in every time before our own; they are the historic norm." (ibid.).
The primary focus of this paper is on the way that this overall balance between order and diversity can be achieved. The predominance of order or diversity can be counterbalanced by four proportion-related functions and sub-functions, particularly in architecture. The gestalt pragnanz (beauty) of the shapes of pre-industrial art, architecture, artefacts, and landscapes depended on the balance of order and diversity. As mentioned, proportion represents the extensional part of the unifying principles (in 2D, 3D, 4D, XD), and may inhere some danger of overstressing the part of order within the figurative balance of order and diversity. An efficient means to counterbalance order is rhythm, the micro-fractality of texture, colours, light and shadow, and the fractally structured environs (cultural landscapes, gardens). Extreme order leads to monotony and depression, whereas extreme diversity without a unifying structure creates disorder, confusion, and chaos. The Human mind rejects such unstructured diversity and the result may be disgust and dereliction. Ancient music, architecture, and landscapes generally achieved this overall balance between order and diversity. Yet, modern architecture and environs do not. Like Baroque and classical music, old artefacts are rich in variety, but also possess a strong, logical order that makes them comprehensible and legible. The result of this balance is the elevating feeling that only beauty can impart.
There are four main functions of proportion in relation to perception: (1) information reduction, (2) unity between the whole and its parts in regard to structural compatibility between the whole and the diversity of parts, (3) characteristics of grace and elegance (in high and traditional architecture), and (4) an ensemble or complementary relationships between architectural objects and the natural environment, e.g. regularity and irregularity. The sub-functions of proportion reinforce the order-diversity balance in very complex ways. Rhythm introduces directly stimulating counterbalances to proportional order. Further examples include the micro-fractal patterns of texture, colour and patina, light and shadow play, ornaments, symbolism, and the Gestalt laws, (which have much in common with symmetric and topological patterns).
The patterns of proportion, as the hidden extensional orders of shape, vary in practice when applied to 2D, 3D and 4D objects. (The human eye level also creates variations). Although architecture is a 3D art, 2D and perspective effects have a strong bearing on the aesthetic effect of its gestalt pragnanz. As a whole, the 3D art of architecture is primarily subjected to the most simple Euclidean geometry, the geometry or proportion systems of regular polygons of 3 (equilateral triangle), 4 (square), and 5 (pentagon, Golden Section). Although the shape algorithms constitute only a small part of mathematics, approximately 300 proportion codes can be developed from them, particularly by the help of grid and circle networks.
Pre-industrial architects applied the basic limits of simple Euclidean geometry on exterior (fašades, roofs) and interior surfaces. In interior spaces there was an expanded, but fitting, range of patterns. These included geometric patterns and the overlapping of different affine proportion (particularly fractal), symmetry and topology patterns. These patterns are embodied in the exterior fašade and interior structures, ornaments, textures etc. Gothic cathedrals may have reached the highest form of mastery: this master work can be found in stonecutting, tracery, glass and rose windows, paintings etc. The author has conducted substantial research on the use of proportion in worldwide vernacular and/or traditional architecture and has established a literature database devoted to proportion mentioned beyond.
I have continued to prepare a bibliographic database for 20 years, which now [A.D. 2007] has some 54,000 records of specialized and qualified literature on proportion and geometry in architecture, art, and other fields, regardless of their civilization, language, period and source of origin. Keyword lists are in English and German.
After many years, I have recently restarted my research work on the proportional analyses of traditional architecture extant in many European countries, New England, and Japan, with the assistance of CAD software. This software quickly uncovers the inaccuracies of elevation drawings published in the literature of traditional architecture. I own several thousand books with such drawings, and many others are available in the Heidelberg libraries, to be used as source material for proportion analyses; it is often possible to conduct several analyses of the same building.
A few US architectural journals which I contacted in autumn are also interested in publishing my Berkeley paper. I will publish several articles on the different aspects of aesthetics and proportion, demonstrating these properties with analyses prepared within CAD software and perhaps some hand-drawn proportion analyses soon. An essay will appear in essay volume edited by the Institute of Traditional Architecture (ITA), London, founded by HRH The Prince of Wales.