Mathematics and architecture

Mathematics and architecture have always enjoyed a close association with
each other, not only in the sense that the latter is informed by the former,
but also in that both share the search for order and beauty, the former in
nature and the latter in buildings. Mathematics is indispensable to the
understanding of structural concepts and calculations. It is also employed
as visual ordering element or as a means to achieve harmony with the
universe. Here geometry becomes the guiding principle.

In Greek architecture, the Golden mean or the Golden rectangle served as a
canon for planning. This corresponds to a proportion of 1: 1.618, considered
in Western architectural theory to be very pleasing. In Islamic
architecture, a proportion of 1: √2 was often used—the plan
would be a square and the elevation would be obtained by projecting from the
diagonal of the plan. The dimensions of the various horizontal components of
the elevation such as mouldings and cornices too were obtained from the
diagonals of the various projections and recesses in plan.

The optical illusions of the Parthenon at the Acropolis, Athens, could not
have been done without a thorough knowledge of geometry.

Ancient architecture such as that of the Egyptians and Indians employed
planning principles and proportions that rooted the buildings to the cosmos,
considering the movements of sun, stars, and other heavenly bodies. Vaastu
Shastra, the ancient Indian canons of architecture and town planning employs
mathematical drawings called mandalas. Extremely complex calculations are
used to arrive at the dimensions of a building and its components. Some of
these calculations form part of astrology and astronomy whereas others are
based on considerations of aesthetics such as rhythm.

Renaissance architecture used symmetry as a guiding principle. The works of
Andrea Palladio serve as good examples. Later High-Renaissance or Baroque
used curved and dramatically twisted shapes in as varied contexts such as
rooms, columns, staircases and squares. St.Peter's Square in Rome, fronting
the St. Peter's Basilica, is an approximately key-hole shaped (albeit with
non-parallel sides) exterior space bounded by columns giving a very dynamic
visual experience.

The term Cartesian planning given to the planning of cities in a grid-iron
fashion shows the close association between architecture and geometry.
Ancient Greek cities such as Olynthus had such a pattern superimposed on
rugged terrain giving rise to dramatic visual qualities, though proving
difficult to negotiate heights. Modern town planning used the grid-iron
pattern extensively, and according to some, resulting in monotony and
traffic problems.

The beginning of the twentieth century saw the heightened use of Euclidean
or Cartesian rectilinear geometry in Modern Architecture. In the De Stijl
movement specifically,the horizontal and the vertical were seen as
constituting the universal. The architectural form therefore is constituted
from the juxtaposition of these two directional tendencies, employing
elements such as roof planes,wall planes and balconies, either sliding past
or intersecting each other. The Schroeder House by Gerrit Rietveld is a good
example of this approach.

The most recent movement-Deconstructivism-employs non-Euclidean geometry to
achieve its complex objectives resulting in a chaotic order. Non-parallel
walls, superimposed grids and complex 2-D surfaces are some external
manifestations of this approach which is exemplified by the works of Peter
Eisenman and Zaha Hadid.

In recent times, the concept of fractals has been used to analyse many
historical or interesting buildings and demonstrate that such buildings have
universal appeal and are visually satisfying because they are able to
provide the viewer a sense of scale at different levels/ distances of
viewing. Fractals have been used to study Hindu temples where the part and
the whole have the same character.

As is apparent, architecture has always tried to achieve ends that not only
relate to function, but also to aesthetics, philosophy and meaning. And in
many a case, the means has been the beauty and structure of mathematics.