Osama Usama

MSC. (Computing and design) Post-Graduate student

School of Architecture and Visual Art

University of East London, UK





Fractal Geometry in Architecture/Urban Design implies the presence of a dimension, which exists between line and area


This paper will attempt to pose four interrelated questions and their possible answers and the first one will be:


1) What is the importance of Conceptual design for the architect?

A study definition of Conceptual Design.


Conceptual Design is the part of form-making in which designers formulate the initial parameters for an artefact. At this stage, design ideas are set but not finalized. It can be argued that conceptual design representations should facilitate multiple interpretations of design elements, while at the same time, and allow these elements to be modified in a variety of ways.






Conceptual Design in architecture/urban design is the earliest stage where the Architect/Urban Designer finds shapes and gives birth to the initial physical manifestation of any forms, and therefore attempts to satisfy the occupational conflicts that result from Human Needs and the Built Environment; essentially this method is the result of a mathematical and geometric language (my study definition).


At the beginning of any project an architect / urban designer normally establishes the parameters generated by specific physical context. This is then fused with a more generic abstract idea of the building required. An example of this is a parking garage on a city site. This procedure can be held to be normal practise in architectural design and is self evidentially of fundamental importance in realising an architectural project.                 


In early stages of design, especially of Conceptual Design (CD), the designer has to make many important decisions which seek to balance the aesthetics of design with engineering precision, but the underlying reason for this is that form itself is really governed by geometry. Another fundamental question is how many reiterations in this process are required for a Conceptual Design to achieve practical fruition.

Until the advent of Modernism in architectural design the use of geometry was universal in virtually all traditional architecture especially in conceptual stages. Generally speaking architects thought out the major spaces, structure and elements of their proposed physical manifestation via geometry such as example shown which diagrams with the potential are for generating Islamic Domes



           ( Figure 1 )










Figure 1

Perhaps the dismissal of classical geometry from Conceptual Design has been a case of the Modernist architect arbitrarily rejecting the lessons of history. However with the discovery of NEW MATHS in the early 1970’s or greater potential was unlocked for the architect.


2) How does conceptual design move from two to three dimensions?


To derive area activity from linear activity we go through what I have termed Intellectual Compositional Architecture Activity (ICAA). There are several factors which affect the ICAA, the first factor being the line’s direction and the second is its proportion, the third, and most important, is human occupation or human intellectual activity. This has the potential to take fractal geometry into three dimensions and that particular formulation of the area could generate fractals in architecture, building or urban design. A good example of these is shown by Geoffrey Broadbent in his book DESIGN IN ARCHITECTURE, the illustration shows Euler’s method of analysing the systematic usage of the seven bridges in Konigsberg  


           (Figure 2, A view of showing the seven bridges over the River Pregel)


However when we are in the process of ICAA, we could  also be focussed and autonomous in our mathematical deliberations and thus directly generate fractals in architecture or we could be naive, unintentional or artificial in the use of fractals in architecture such the example shown below and this I maintain has generally been the situation. By naive I would suggest that Eiffel Tower in Paris. My contention is that the tower is made of many small trusses which have been progressively scaled down to build the whole structure. But that this is essentially two dimensional   (Figures a, b)


          (Figure 3-a)                                       (Figure 3-b)



3) Is Fractal Geometry being used in conceptual design?


The Mathematical definition of a Fractal is “an object   whose Hausdorff dimension is not an integer” (

A spiral is progressively Fractal in the way that the semi circle is scaled up or down to create the form. Through except in very few cases e.g.



(Figure 4) an example of the Fractal Tree


Frank Lloyd Wright in his later work (Palmer house, in Michigan) where he used self-similarity form of triangles in the plan (Figure A Plan of Palmer House, B self-similar triangles).  Here fractal geometry was only used in two dimensions as planning device. Where he used self-similarity form of triangles in the plan, but that was not part of the consistent design methodology.


                (Figure 5-a)                                                      (Figure 5-b)                             

These attempts have also not been very systemic, for example; in the twentieth century some architects have researched or applied unconsciously the concept of self-similarity.

4) How might Fractal Geometry be used in conceptual   design?

This paper will have a philosophical context which is: how is the human intellect positioned within the living and working environment? The issue that I will address here is the generation of form in architecture design/urban design through mathematical rules in general and fractal geometry in particular. This would then be a result of an autonomous generation of form, as well as the pre-ordering of a creation by using deliberate mathematical notation rather than a naive and an unsystematic approach.


(Figure 6)  


(An elevation of staircase Source: J. F. Gabriel, Beyond the Cube page, 454) 


City Tower Project

Philadelphia, PA; 1952-1957; UNBUILT, by Louis Kahn


                           (Figure 7-a)                                    (Figure 7-b)


The connection between the above examples is that the staircase which is designed for the purpose of human movement is structurally formed on the transformation of triangles. Remarkably the unbuilt structure by Louis Kahn uses the same concept. A further example is shown in the appendix- see (Figures 10-a & b)    


The conscious and methodical use of Fractal Geometry in architecture can produce more than two dimensions, nevertheless to take it into three dimensions it will be necessary to engage in ICAA and this is a result of human occupation or human intellect as outlined above.  


Human activities, and the built environment can be expressed in terms of occupation and the conflicts of aesthetics and human needs, the designer must resolve the issue and satisfy these occupations. This in my opinion is a matter of Geometry.  An appropriate example would be design project set at the third year level for a law student residence in London. In (figure 8a) is shown my initial conceptual drawing in three dimensions. It represents a typical working design solution which most students would arrive at early in design process (Figure 8b) is the same solution which has been translated via the application of Fractal Geometry. This concept is now ready for a potentially huge range of manipulation


            (Figure 8a)                                                    (Figure 8b)                                                                                                                                                                                                                                                                                     




The Architect is concerned with the whole rather than with the parts that is a synthesis. Mathematics and in particular geometry is the language that can describe the interface between the human behaviour and the built environment




In conclusion the process of design in architecture is an equation, that has input and output, the input could be behaviour and environment and the output is the physical manifestation of many different forms, however the method is a mathematical and fractal geometric language


Selected references:


Alison and Peter Smithson, The Charged Void: Architecture, 2001, ( Monacelli Press, Inc.)

Batty Michael, and P. Longley, Fractals Cities, 1994, (Academic Press London)

Benoit B. Mandelbrot, The Fractal Geometry of Nature, nineteenth printing, 2000, ( W.H. Freeman and Co. )

Bovill, C., Fractal Geometry in Architecture and Design. 1996 (Birkhäuser, Boston)

Broadbent Geoffrey, Design in Architecture: Architecture and the human sciences, 1975, (John Wiley & sons, Inc.)

Christopher Alexander, Notes on The Synthesis of Form, 1964, (Harvard University Press)

Christopher Alexander r, The Nature of Order (2000, New York: Oxford University Press).

Grabrel J. Francois, Beyond the Cube, 1997, (John Wiley & sons, Inc.)


Issam el –Said, Islamic Art and Architecture, The  System of Geometric Design , Edited by Tarek, El-Bouri & Critchlow, 1993, (Garnet Publishing Ltd.)

March, L. & Steadman, P. The Geometry of Environment, 1971, (RIBA, London )

William I Mitchell, The Logic of Architecture, 1998, (MIT Press)

The Sri YantraUsing 9 triangles with transform and intersect four point the down centre with five point upper centre to from connection this is also Fractal 


The Second Arts building in University of Bath





                               (Figure 10-a)                                    (Figure 10-b)




Axonometric of second floor, shown connection to the way within the existing building, Luiz Neto brede, Amanda Marshall from the book about ( the charged Void: Architecture Alison and Peter Smithson)


This building shown above is I believe anther good example of architects using fractal geometry, in this case the concept is that of Geometric Progression internality and in the external envelope



09 November 2004