### Geometry

Many authors, with widely different professional backgrounds, believed in the existence of certain systems of proportion, scale and numbers.

Alberti wrote:

It is manifest that Nature delights in round figures, since we find that most things which are generated, made or directed by Nature are round ... We find too that Nature is sometimes delighted with figures of six sides; for bees, hornets, and all other kinds of wasps have learnt no other figure for building the cells in their hives, but the hexagon ... The polygons used by the Ancients were either of six, eight or sometimes ten sides. (Leone Battista Alberti, Ten Books on Architecture, Florence, 1485, cited in March and Steadman, The Geometry of Environment).

The various series of numbers (the Cantor set, the Fibonacci series), of curves (the Koch, the Minkowski and the Peano curves), the system of fractals, the golden section, Le Corbusier’s ‘modulor’ (based on repeated golden rectangle proportions) are but a few examples (Van Der Laan, 1983, Mandelbrot, 1983, Bovill, 1996). It has been assumed that certain such systems must be applied in architecture also (Padovan 1999, Salingaros, 2000). Geometric systems (as for example the module system) are transformed into number systems and vice versa (as for instance the Van Der Laan scale) (Van Der Laan, 1983). Certain styles and some architects did introduce various systems of proportions, scales, rhythms and measures. Palladio designed the plans of his villas on rectangles with whole number proportions: 1:1, 1:2, 2:3, 1:4, 3:8 (Elam, 2001, Padovan, 1999).

National Assembly, Dacca, Bangladesh, 1962–83, designer: Sher E. Banglaganar.

Geometric patterns (triangle, etc.) may be dominant on a façade.

Geometric patterns (triangle, etc.) may be dominant on a façade.

Architecture, after all, is a manifestation of geometry applied for the purpose of the design of buildings. Research attempted to create geometric systems for structural or architectural design, as has been seen already when discussing space frames, shells, domes and membranes. Some of the systems are of a pure mathematical or geometric character, in others structural or architectural design forms the basic background. There are attempts to develop fully automated structural design systems with geometric representation for structural domains, using automated techniques for finite element modelling, coupling self-adaptive integration of optimization techniques with geometry models (Kodiyalam, and Saxena, 1994). ‘Solid modelling’, meaning representation design, visualization, and analysis of three-dimensional computer models of real objects, finds application in the design of buildings but in other quite different fields also.

The attributes of symmetry and harmony gained favour in historical architecture: asymmetry, however, was appreciated only to the extent that it achieves harmony. On the other hand Viollet-le- Duc, a nineteenth-century architect, wrote: ‘Symmetry – an unhappy idea for which in our homes, we sacrifice our comfort, occasionally our common sense and always a lot of money’ (quoted by March and Steadman). Rhythm meant either repetition or variations with pleasing relationships. In modern and post-modern aesthetics, sometimes seemingly arbitrary deviations from repetition and disharmonic alterations became welcome. So, for instance, the memorial colonnade by Oscar Niemeyer was designed with variable column distances.

However, even the most sophisticated systems do not prevail forever, and invariably change over time. Styles and architectural design have to cope anew repeatedly with this transience and must devise their own solution for attaining pleasing appearances of buildings. What, however, is ‘pleasing’, is in itself a dynamic concept and the history of art and architecture continuously reports new design concepts that initially were judged to be ugly but as time went on were considered to be agreeable (Kroll, 1986).

The geometry of new architecture buildings may also display new features. Straight lines become curves, verticals and horizontals may be slanted and cut into each other at odd angles. Curves that traditionally featured in gothic, renaissance and baroque architecture are ignored; partly regular curves (circle, etc.) and individually designed curves take their place.

Naturally, the foregoing does not apply to all new buildings. Neo-classicist and late-modern buildings may adhere much more closely to the old rules.

Japanese architects have been ingenious in their application of geometric forms. Tadao Ando, for instance, favours a grid derived from traditional rice straw tatami mat with dimensions of 90 by 180 centimetres. Ando designs concrete walls with an exposed surface and each of his moulding boards (with the size of a tatami) has six holes through which the boards’ screws are driven. Arata Isozaki accords preference in his geometry to the square and the circle. In some designs he uses segments of curves and curved surfaces. On occasion grids are applied combined at slanted angles.

Size, scale and measure are changing. Large-size surfaces are articulated and contain uniformly spread identical small-scale elements or forms. In such cases a certain uniformity of the surface may be achieved and the contours of such surfaces can be selected almost at random.

It was pointed out that new architecture often extends components to the outside of buildings and sometimes into the air space. This is typical for suspended structures with external masts and cable systems but it can occur in other cases too, see, for example, Himmelblau’s office extension in Vienna. An innovative architectural component is the tall atrium often applied in large hotel buildings and office buildings. The internal height of such atria may reach up to 40 or more levels and poses a fresh challenge for their internal design (see the interiors of the hotels designed by John Portman).

Sebestyen, Gyula. 2003. New Architecture and Technology.

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