The purpose of this essay is not to trace the origin and development of the discipline of geometry in history; it is rather to discuss its need and function in the past and its relevance in the present times. However, before we embark on this debate, it might be more helpful to first look into the definition, meanings and significance that the discipline of geometry has held in various cultures.
It has always been a human desire to comprehend existence and its very context, both at the physical and metaphysical level. The quest is to find an answer to the basic existential question: whether existence precedes essence or the essence precedes existence? Regardless of the circular nature of this inquiry, the ambit and context of this comprehension, from the very first, kept widening with advancements that helped in this understanding; starting with the immediate context of nature, extending to the regional, to global and further on to exploration and understanding of the universe. It is very clear in history that there were two contexts to this inquiry; one was more abstract, intangible and metaphysical, and the other one was tangible and apparently measurable because of the physical nature of the earth and universe that it is a part of. The discipline of geometry played a key role in this attempt to measure and quantify, though it is very obvious that any advancement in the measurable aspect informed the abstract one as well. Empirical inquiries into the shape and form of the universe led to rationalist abstractions in the fields of mathematics and geometry and vice versa as we witnessed developments in the field of theoretical physics. That is precisely why the disciplines of physics, mathematics and geometry are inseparable in forming a comprehensive understanding.
Since the Neolithic times, there has been an attempt to measure the earth. Tools and techniques were evolved to do this accurately and it gave birth to the discipline of geometry. The conception of the form of the Earth and the nature of the Universe played a significant role in the development of early principles of geometry. This quest led to a higher understanding of human existence and its relationship with the Universe and gave rise to a desire to strike a harmony between the two through an understanding of universal principles. Many older cultures have tried to reflect the cosmic order on earth in the form of layouts of their cities and important places. Babylonians, Ancient Egyptians and Pre Columbian Mesoamericans have followed the celestial order of heavenly bodies in the layouts of their cities, ziggurats and pyramid complexes. This was reflective of human consciousness and epistemological constructs of those times. The mythical aspect in this tradition was natural and quite understandable, as myth formed the bedrock of comprehension in these cultures.
The same tradition was present in Western civilizations as well, but more so in Ancient India where Tantric practices and art formed the basis of comprehending cosmology, cosmic energy and the ways of capturing and releasing this energy in spaces. The practice of Vastu Shastra was about the ordering of architectural spaces and formulated the principles for spatial layouts. In the Subcontinent, everyday life was structured in the context of mythology and was lived as a ritual. It is therefore very understandable that spaces and layouts were based on dialectical opposites such as good and evil and the energy associated with them. These principles were equally applied to places of religious importance like for example the Angkor Wat Hindu temple complex in Cambodia and in the layout of common houses. It is interesting to note that this consciousness was dependent more on region than faith, as we witness in India where references were drawn from Hinduism and Buddhism equally. This cultural hybridity created an expression that was very Indian in spirit.
If we take a broader view of these cultures in terms of their understanding of layouts, spaces, their significance and meanings, we realise that these were all inquiries into the nature of space. Space; as understood in the mythical context and as lived in the ritualistic context. Relationship between ritual and space and the abstract principles of geometry applied in the designing of these spaces created a deeper understanding of tradition in ancient and medieval societies in which there was no distinction between the sacred and the profane. Rather, there was no concept of the profane as such because faith and way of life were inseparable. This world view introduced the idea of ‘order’ as against disorder or ‘chaos’ which is a condition in which apparently there are no divine or universal principles guiding the structure of spaces and architecture. ‘Order’ found its physical manifestation in the shape of formal geometry that governed the structure of spaces and buildings. Belief and everything stemming from it served as the basic idea upon which all things were based: from the city form, to livelihood, to art and craft practices. It was something introspective that found its expression in abstraction.
The evolution of Cartesian geometry in Classical Greece was a major step forward in this understanding. It laid out the discipline of geometry on scientific and rational lines. The Islamic mathematicians and geographers of the Golden Age developed the discipline further and were able to establish its strong connection with other disciplines such as astronomy, mathematics and algebra. The observatory of Mirza UlughBaig in Samarkand is a surviving example of this advancement. The application of this development in the disciplines of art and architecture however restored its connection with the divine.
The use of geometry in various art forms practiced during the Islamic Period in Central Asia and Persia, and in the Subcontinent during the Sultanate and Mughal times achieved a high level of refinement. It was applied in calligraphy, miniature painting, architecture and urban planning and carried traditional meanings of geometric patterns and their significance.
With the European enlightenment this discipline took a new turn. René Descartes introduced rationalism based upon deductive reasoning as a logical process governing the principles of geometry. Immanuel Kant introduced non-Euclidian geometry according to which the discipline was completely removed from empirical knowledge acquired through the senses; it was based on pure and internalised intuition of time and space.
This was basically a shift from tradition to modernity, from sacred to the profane. It created disconnect of the discipline of geometry with the divine and redirected it towards the physical world of appearances and matter.
The postmodern turn of events added further complexity and depth to the discipline after Einstein’s inquiries into the nature of space and time, establishing that the universe is curved and space is infinite. This discovery opened up a whole new debate on the relative nature of reality and maintainability of universal truths, especially in various fields of art that dealt directly with the representation of matter and space. The inquiry is therefore about the relative relationship of objects in space, and the relativity of space time itself, technically defined as differential geometry. Finally, recent development in computer and software technology has introduced fractal geometry that is based on the concept of evolving symmetry. In recent years, fractal geometry has served as a major inspiration in the organisation of space in architecture and urbanism.
We shall now look at the relevance and application of various concepts of geometry in the fields of architecture and urbanism in this region. This study might be helpful in shedding some light on the evolution of various epistemological paradigms of planning and design of spaces. When we look at the layout patterns of the earliest Neolithic settlements of Mohenjo-Daro and Harappa along the River Indus and River Ravi respectively, one is surprised to see the sense of order and planning in the form of grid-iron layouts of streets and blocks. It is certainly indicative of a high level of advancement and civilisation.
The same sense of order is seen in the slightly later Buddhist cities belonging to the Gandhara civilisation in which grid-iron planning and axial symmetry is maintained. This type of development is reflective of the connections and contacts this region had with other parts of the known world, as well as the foreign influences this region absorbed as part of its cultural evolution.
Though we do not have surviving examples of Hindu Shahi towns, we do have fairly intact examples of their architecture that displays a high level of understanding of geometry and proportioning system. The spatial layouts in this tradition were mainly ritualistic in nature. Few incomplete surviving examples of this period like the town of Malot in the eastern part of the Salt Range display a clear segregation of the sacred and the profane as discussed earlier in the essay. The temples are on the hilltop and form the acropolis, followed by the necropolis on the slopes and the town at the base.
Medieval cities of this region are rather confusing in their layouts and planning, firstly because these are living cities and secondly the medieval parts of these cities form a palimpsest that cannot be easily deciphered.
These older parts of the cities have absorbed various cultural influences and have transformed greatly over time. One can only rely on conjectures to complete a vague picture of how these cities might have looked originally. The walled city of Lahore is particularly a very interesting example. It presents its reader with a striking dichotomy. There are two distinct and contrasting planning types present side by side: the citadel containing the fort and the imperial mosque with a perfectly formal layout based upon sacred Islamic geometry and below it the rest of the Walled City with its almost organic labyrinthine layout patterns interspersed with examples of formal geometry like the Wazir Khan Mosque complex. This dichotomy raises certain very pertinent queries regarding the purpose and use of geometry as a tool for the structuring of the built environment, especially in the case of medieval Lahore. The question arises whether sacred geometry was used only in the designing of spaces and structures of the royalty, or whether the Walled City of Lahore, being a living city transformed over time and the spaces that were once laid out on formal geometry subdivided beyond recognition.
The British Colonial planning of Lahore was based upon formal baroque and neo-classical layouts for the fairly obvious purpose of initially intimidating the inhabitants of the city with the scale of urban spaces, and later for disciplining the society on modern lines. The formal planning and sense of zoning in the Colonial Period marks the beginning of the modern period in the Subcontinent that later culminated into less picturesque and more utilitarian spatial layouts.
The most important urban planning exercise undertaken after partition was the design and layout of the new capital city Islamabad for which a modern rationalist approach was preferred over the traditional or empirical. That might be partly because Doxiadis was a Greek planner and partly because the young country at the time was quite willing to embrace modernity and a modernist outlook.
Lastly, we shall look at the planning models adopted and followed in the past few decades. It is sad to note that the most noticeable, or rather, the only typology in planning and layout of new parts of cities and satellite towns is based on consumerism that follows the neo liberal model driven mainly by market forces. All the significance and purpose of geometry, traditional or otherwise appears to be irrelevant and meaningless when major planning decxisions are taken by developers instead of planners and architects. The only thing consistent is arbitrariness.