Proceedings of the 11th Space Syntax Symposium
#172
AN ARCHAEOLOGY OF THE PRESENT:
Topo-geometric Properties from the Invention of Geometrical Notation
to Non-Standard Variation in Architecture and Design
SOPHIA PSARRA
UCL, London, United Kingdom
s.psarra@ucl.ac.uk
ABSTRACT
As digital technologies revolutionise the ways in which buildings are produced there is a growing
risk for architecture to become a practice without a theory. Space syntax has contributed to
architectural research, through the description of systematic relationships between patterns
of use and spatial phenomena. Yet, in the last three decades it has primarily leaned towards a
theory of the city1. These are studied as the collective products of society that are either selforganising (cities), or operate independently of the agency of their architects (buildings). Yet,
from the viewpoint of architecture as a social discipline, there is a need to describe buildings and
their relationship to the city not simply as the emergent products of society but also as products
of design. This type of study requires theories and tools that describe topo-geometric properties,
or the interaction of spatial with geometrical patterns. It also needs to combine historical
research with morphological analysis. In this paper I explore the relationship between topology
and geometry through three key periods of Western architectural production: irst, the classical
invention of geometric notations in architectural drawings; second, the shift of emphasis by
modern architects to movement and visual information, freeing architecture from constraints
of axial geometrical planning; inally, the end of geometric and notational limitations on the
variability of forms with the rise of digital technology. Rather than providing a comprehensive
account of architectural design, this paper aims to understand the morphological traditions
from which contemporary architectural spaces and forms derive. I argue that as much as space
has been a silent instrument in architectural discourse, so has geometry been a silent conductor
in Hillier and Hanson’s theory of spatial coniguration. Aside to tools for topo-geometric
analysis, we need theoretical accounts of the ideas we ‘think with’, bringing space syntax and
contemporary architecture into the historical and morphological tradition.
KEYWORDS
Geometry, topology, movement, visibility, non-standard variation, algorithmic design
1 Buildings are also studied using space syntax theory and tools but no systemic understanding of buildings exists
across a wide range of building types. Further to this, the study of buildings has moved away from the early
attempts to build an internal theory of architecture through a clear understanding of the diference between
architecture and building. As such, seen from the perspective of architecture, most building studies using space
syntax fall into the realm of the normative, recycling old concepts and methods of analysing.
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1. INTRODUCTION: THE INVENTION OF ARCHITECTURAL NOTATION
Geometry was always present in architecture but the conscious employment of it goes back
to the Renaissance, where through intensive studies of ancient structures and inluential
patronage, architects, such as Leon Battista Alberti, Sebastiano Serlio and Andrea Palladio
established architecture as a discipline separate from artisanal inherited traditions. The purpose
was to raise it to an intellectual activity, conversing with learned men, poets, philosophers, and
literati. Alberti advises architects to conceive the design in the mind, and revise it many times
before building. Once revisions inish nothing should be altered for the better or the worse (1726).
Alberti produced the irst architectural treatise of the Renaissance, but it was un-illustrated and
written in Latin. It was Serlio (1475- c. 1554) who pioneered the use of high quality illustrations
to supplement the text, which was written in Italian (1611). Illustrations and the discovery of the
press spread the inluence of these books in the Western world. While previously architects had
to travel in order to study ancient ruins, books brought to them the treasures of antiquity in an
illustrated volume. Particularly pocket size books, like Palladio’s guide book to the ancient sites
of Rome, helped spread classical architecture and paved the road for its revival.
More importantly, it was orthographic projection, the use of techniques to survey existing
fragments and generate plans, sections and elevations that led to the design of classical
buildings. Until the emergence of digital architecture in the late 20th century, orthographic
projection has been a method of representation in drawings and books, a tool for collecting data
about buildings and inally a method of design. Raphael described the method of surveying and
designing through scaled drawings as follows: ‘…you should draw always measuring everything
with the scale, and use a line that equals the width of the base of the entire building. From the
central point along this line, draw another straight line that makes on either side two right angles;
this will be the centre line of the building. From the two extremities of the width line draw two
parallel lines, perpendicular with the base line; these two lines should be as tall as the building is
to be. Between these two lines, which make the height, you should then measure of the columns,
the pilasters, the windows and other ornaments drawn on the front part of the building. And do all
this always drawing the lines from every single extremity point of the columns, pilasters, openings,
or whatever else, such that these lines are parallel to the lines at the extremities’ (Hart and Hicks,
2009: 186).
Rafael further goes on to describe how the elevation (exterior wall) and the section (interior
wall) are derived from the plan. Corresponding parts are joined with parallel lines, which are
the conservers of true measure. These lines are considered to be representations of light paths
with the source set at ininite distance. ‘…the interior wall shows the inside of the building – half,
that is, if cut down the middle….In short, with these three orders or styles, it is possible to consider
in minute detail all the parts of any building, inside and out’ (ibid.). This method of drawing was
essential for the building to be constructed on true measures. But it was also the method that
created space. As Robin Evans explains: ’…architectural space would remain, one way or another,
limited by and bonded to the pictures that normally gave access to it…projection was an extra
ingredient grasping more or less cautiously at the imaginary space behind the three drawings…’.
Evans continues: ‘if the side you see is the mirror image of the side you do not see – if, that is,
the building is symmetrical about the sectional plane – you see it all through one cut…Vertical,
bilateral symmetry is economical within the conines of the technique…A centre line projected
through the cavity easily converts into a processional axis. Then the axial route will show up on
the principal elevation as a principal entrance, thereby converting the simple, binary equality of
left and right sides (a-a) into a tripartite, therefore hierarchical, centralised symmetry (a-b-a)…This
is why in most classical architecture design and building are in a near perfect accord’ (1995: 118119). Another way of saying this is that the building as three-dimensional physical space was an
identical scaled copy of the design (Carpo 2007).
2. IDENTITY: ISOMETRIC INVARIANCE BETWEEN DESIGN AND BUILDING
The invention of geometrical notations provided - like an analogue algorithm - instructions
for producing a building, as well as for experiencing it from the inside. We can illustrate this
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by looking at Palladio’s Villa Rotonda (igure 1). The most integrated positions in the Rotonda
are situated in the circular hall, drawing to themselves pathways and views through the entire
building. All systems of spatial relations, such as physical elements, lines of movement and
sight obey the same laws of invariance. All registers of symmetry correspond with each other
so that when viewers move in the villa, geometry conditions their vision, movement and
appreciation of the relationships between the parts and the whole. The transformations of
visual ields, expressed by the way in which the angles and radials of visual polygons (isovists)
change from space to space are symmetrical along the axes of movement. At the same time,
views are symmetrical from symmetrical positions (igure 1). Aligning the geometrical axes
with the processional axes, as Raphael advised, has the efect of geometrically controlling
the variability of views, so that the whole building can be experienced as a stable image.
Figure 1 - Villa Rotonda, graph (left) visual integration (right)
Group theory is the branch of mathematics that describes symmetry as the properties that
remain invariant under a transformation. In terms of geometry, the Rotonda has six symmetry
transformations: relection on two axes, and rotation on 90, 180, 270 and 360 degrees. In terms
of graph structure, the four entrances are symmetrical to each other with respect to the central
hall, and the outside, while each of them is asymmetrical in terms of their relationship to the
central space; the spaces next to the entrances are symmetrical to each other with respect
to each entrance, and so on. Geometrical symmetry and graph symmetry therefore, have the
same registers of invariance, creating an isometric correspondence between the building and
the design.
Palladio was aware of the diference between built forms and designs, manifested in his
exceptional capacity to respond to site and functional problems with elegant solutions
(Ackerman 1966). This is evident in his project for a pallazo in Venice in comparison with an
ancient house he published in his Four Books, or his Teatro Olimpico seen alongside the Roman
theatre by Vitruvius (Palladio 1570). Both designs were adjusted to the irregularity of the site
conditions. Yet, in Four Books Palladio presented an idealised view of architecture, eliminating
all adjustments of size and proportions necessary to address the realities of the physical fabric.
This marked diference was in efect an outcome of the erudite climate of the period. Architects
had to demonstrate to their learned patrons that they practiced architecture as liberal art,
concerned with the abstract comparative understanding of architectural types, mathematics
and proportions, and not as mechanical art without erudition. The ideal geometries of Palladio
in his Four Books, demonstrate that he must have followed Alberti’s advice to architects, arguing
that the design, in essence an informational model, was the product of the author conceived
in the mind. The building on the other hand, was an identical copy of this product. In artisanal
practices it is the other way around. Artisans and craftsmen ‘inherit’ ‘designs’ from existing
building practices that survive the test of time and by word of mouth.
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If the Renaissance artefact was designed as a microcosm of the universe, and the universe had
mathematical origins (Wittkower 1971), the architectural creation had to provide the union
between mathematics and the world of the senses. Identity is a special property that deines a
symmetry group in mathematics, where a thing can be superimposed upon its image through an
isometric relationship of sameness. The classical system of notation established a relationship
of identity - or sameness - between geometry and space, or between the design and the
building. This type of relationship between the two ‘worlds’ caused the spatial to emerge from
the lat surface. Captured through geometry, the properties of space in architectural theory
have since remained an active but silent partner.
3. VARIABLY STANDARD: GEOMETRIC CONTROL VS. VARIABLE SPATIAL FORM
Palladio’s ideas travelled to the West in the 17th century, giving rise to English and American
Palladianism. In the 19th century studies of classical architecture using the generative potential
of taxonomy and classiication were produced by Jean-Nicolas-Louis Durand (1760-1834) in his
Cours’ d’ Architecture, a highly rationalised encyclopedic survey, consisting of formal schemata
that are literally empty of any speciic content (1805). The approach was rejected by the avantgarde architects in the 20th century, following ideas of organic evolutionary typology, based on
studies such as those by D’Arcy Thomson (1860-1948). Yet, as Frampton has recently argued,
there were three conlicting paradigms that shaped modernism: the technological, the classical,
and the vernacular (or the organic) (2016). The irst paradigm was about the impulse to use the
technological methods of the period. The second one was a normative standard embodying
a rational and international design culture. The vernacular model derived its strength from
organically grown built examples and from regional building culture.
The irst ‘modern’ architect before the modern movement was England’s John Soane (17531857). In Lincoln’s Inn Fields, Soane built incrementally a house that challenged the isometric
invariance between design and building. The distribution of visual integration captures the
grid-like geometry of Durand’s system, but the axes of symmetry are broken and the enilade
sequence of rooms, usually arranged in perspectival recession in Classicism, is distorted. The
axes of movement correspond neither with geometry nor with the axes of sight (Psarra 2009).
Compared to the villa Rotonda, visual ields here not only have more variation in terms of shape,
but also greater degrees of transformation along with movement (igure 2).It was Le Corbusier
Figure 2 - Soane’s Museum; Visual Integration (left); Isovist from points along axial line; Isovists and geometry of
room enclosures
who intensively engaged more than any other architect with the organic, modern and classical
models. In 1964 he published a site plan of his design of the Venice Hospital in his Oeuvre,
showing the hospital behind the train station together with a selection of other buildings in
Venice (igure 3). The drawing links the project with Palladio’s San Giorgio Maggiore, via the
Grand Canal dotted by Pallazzi, the Rialto, the Merceria and the Piazza San Marco. Le Corbusier
had always aspired to design a public building in Venice comparable in scale and impact to that
of Palladio’s convent, the grand piazza and the Ospedale Civile. In his Four Books, Palladio
described the convent as intended for the recreation of the ‘houses of the ancients’ (ibid.).
Thus, the march from the service yard to the embellished front of the city and to San Giorgio
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Maggiore in this map expresses Le Corbusier’s heroic entrance to the legacy of architecture
since Roman times. However, the link with Palladio’s convent and church is not simply because
Le Corbusier measured himself against the classical architect. It is also because he was aware of
the disciplinary roots of architecture with its emphasis on mathematical literacy and geometry.
It is in Venice that the irst translation of Vitruvius was published, and it is in Venice and the
Veneto where Palladio had practiced. Le Corbusier positions the hospital not only in the urban
context, but also within the disciplinary tradition of structured architectural knowledge, rooted
on classical humanism and against the organic context of Venice.
Figure 3 - Le Corbusier Site Plan of Venice
same time, he had absorbed other inluences for the project: irst, ideas of organic growth
and evolutionary design that were prevalent in the 60s. Second, the urban structure of Venice
based on interconnected squares, the separation and intersection of the two networks of
movement, that is, canals and pedestrian pathways (Psarra 2011, 2013.) Third, the pin-wheel
pattern, a schema that preoccupied him throughout his career. This pattern goes back to
the Villa La Roche (1925) built for a wealthy client to house his painting collection. This is the
house in which Le Corbusier invented the architectural promenade, guiding the visitor through
changes of direction along ramps, stairs and raised pathways. In contrast to the axial structuring
of movement through similar rooms in classical architecture, he used a twisting course of
movement, covering heterogeneous elements. As opposed to geometry shaping human
movement, he employed human empirical movement to shape the building.
But it was at the time he was designing the Villa Savoie (1928-1931) that the combination of
a simple Platonic volume with a turning path shows up as the irst instance of a career long
paradigm of designing (igure 4).. At that time he was collaborating with Paul Otlet (1868-1944)
on the Mundeneum (1929). Otlet was a signiicant igure in the history of information society
and the networked knowledge base of the future The Mundeneum was intended as a place
that would provide access to the world’s knowledge. Otlet envisioned a ‘city of knowledge’ that
would serve as a central repository for the world’s information. The World City was a utopian
vision, which like a universal exhibition would bring together all the leading institutions of the
world. It was formed as a giant circulation ramp into a ziggurat-shape to test the spiral idea at
a monumental scale. The two schemes, one domestic (Savoie), the other public (Mundeneum)
were worked in parallel, combining the simple volumetric form with the pattern of twisting
movement.
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Figure 4 - Le Corbusier, Villa Savoie.
In the design for the Museum of Contemporary Art (1931) Le Corbusier shaped the whole building
as a continuously unfolding wall. The museum is entered through an underground passage and
is without façade, absorbing the exterior into the interior. A few cuts are made in its surface to
allow the visitor to step outside the ixed itinerary and circulate in diferent ways. In 1936 he
used the pinwheel pattern again in the Centre for Contemporary Aesthetics. In 1939 he used the
same design theme in his designs for the French Pavilion in San Francisco and the exposition
in Liege. The same year marks the Museum of Unlimited Growth in Algeria, in which the future
expansion of the museum is based on a spiralling pattern indicated on the ground around the
simple box-like form of the building. The rotating path is combined with a central void and four
spaces, each on a diferent side of the volume, deining the pinwheel schema of composition.
Le Corbusier used this schema in 23 designs of diferent scale and social programme, from
museums and exhibition spaces to villas, including the monastery of La Tourette (igure 5)2.
Why was he preoccupied with this schema so consistently and what impact does it make.
2 It turns out that with the exception of the period of the Second World-War, there is no more than a 3-year gap to
the start of a new project that involves the spiral-swastika pattern. Total of 23 projects between 1923 and 1965. 1.
Villa La Roch-Jeanneret 1923 2. Villa Meyer 1925; 3. Mundaneum 1928; 4. Museum of Contemporary Art 1931; 5.
Bata Boutique, 1935;
6. University Campus Rio de Janeiro 1936; 7. Centre of Contemporary Aesthetics 1936; 8. Pavilion des Temps
Nouveaux, 1937 9. Museum of Unlimited Growth , 1939; 10.French Pavilion in San Francisco, 1939; 11.Exposition
Habitat 45, 1945; 12.Urban Development, Saint-Die, 1946; 13.Exposition Synthese Des Arts, Porte Maillot, 1949;
14.Cultural Centre of Ahmedabad, 1951; 15.Tokyo Museum, 1959; 16.Etude d’urbanisation, Meaux, 1957; 17.Le
Couvent de la Tourette, 1950; 18.Museum at Chandigard, 1959; 19.Cultural Centre Chad , 1960; 20.Museum of
the Twentieth Century, Eisenbach, 1963; 21.Museum of the Twentieth Century, Nantere, 1965; 22.Musee de
lotissement undated; 23.Venice Hospital, Venice, 1965
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Figure 5 - Le Corbusier’s Assembly at Chandigarh (top row left); La Tourette, (irst row second left), Tokyo Museum (top
row second right); Venice Hospital visibility integration (top row right); Cultural Centre at Ahmedabad, permeability
Integration (second row left); Venice Hospital, permeability integration (second row right); La Tourette, ground loor
(third row left); La Tourette, irst loor (third row second left); La Tourette second loor (third row second right);
Chapel at Ronchamp with furniture (bottom row left); Chapel at Ronchamp without furniture (bottom row right).
We can explore answers to this question in three projects, the only ones which were constructed
based on the pinwheel scheme: the Tokyo Museum (1959), the Museum of the Cultural Centre
at Ahmedabad (1951) and the Chandigard Museum (1959). As with the Rotonda, in the Tokyo
Museum there are four such axial connections, linking the central space with the exterior (igure
6).Contrary to the Rotonda, these axial elements travel along the perimeter of the central hall
rather than traversing the hall, which is placed at the geometrical centre. In addition, seeing
the central hall is diferent from accessing this space, since there is a break in the classical link
between visibility and movement. We see here the clear impact of inserting an object at the
centre of a layout and pushing the values of integration to the corners of the space (Hillier
2003). Looking at the graph of the main spaces, we see that there is graph symmetry only
with respect to space 1 – at the end of the ramp – which is of the main axis. In the classical
model of composition there are usually more spaces that have graph symmetry in relation to
other spaces, while at the same time the space with the highest value of integration is at the
geometrical axis of the building (igure 1). Compared with the Rotonda, where there is isometric
correspondence between geometry and space, the Tokyo Museum is a clear case of invariance
between the two kinds of properties.
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Figure 6 - Tokyo Museum. Axial lines of visibility and permeability (top left); Permeability visual integration (top
middle); Visibility visual integration (top right); Cultural Centre at Ahmedabad; Permeability visual integration with
pavilions (bottom left); Visibility visual integration (bottom middle); Permeability visual integration (bottom right)
Developed over thirty years of architectural activity the pinwheel pattern became for the Swiss
architect a standard independent form, producing diferent variations on a theme, based on the
invariance between geometry and the topological graph of a building3. There are three main
mechanisms through which invariance between these two systems is generated: irst, twofold or four-fold symmetry in relation to the outer envelope and the central space; secondly,
rotational symmetry articulating the relationship between the central space and the adjoining
galleries, pathways or openings (the pinwheel plan); thirdly, placing a void at the centre and
screening it from the rest of the layout, so as to disassociate the structure of views from the
structure of movement.
Le Corbusier dismissed the Beaux-Arts approach to composition, which is appreciated on axis
as anachronistic and dogmatic. Yet, he had a clear understanding of the strategies by which the
close association of moving and viewing was achieved in classical buildings. He used long axes
to organise a plan but in a way, which created variable rules, in the sense that what is invariant
in one system is not the same with that which is invariant in the other. The spectator produced
by this architecture shifts course with movement while exploring vistas that develop along
diferent directions, but is always in reference to the classical logic of geometrical composition.
Freeing the various systems of properties from each other, or breaking the isometric invariance
between geometry and space was a strategy that was widely adopted by modern architects.
Many buildings seen in igure 7 use geometry simply as a supporting armature, rather than as
a generator of the design. Invariance across systems is nonetheless, present in buildings by
‘classical modernists’, such as Terragni’ Casa del Fascio, Mies’ Crown Hall, Tugedthat House
and Farnsworth House; Aldo van Eyck’s church and Asplund’s library in Stockholm. The nordic
classicism of Asplund’s library (igure 8) is particularly interesting when seen in relation to
Palladio’s Rotonda (igure 1) and Le Corbusier’s Assembly in Chandigarh (igure 8), where a
similar U-shaped geometry surrounds a main space with a circular chamber. However, although
3 The scheme of rotational symmetry through the swastika plan inds realisation even in his religious architecture, if
for example we look at La Tourette and the three entrances in Ronchamp.
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the three projects have a similar parti, their spatial systems are entirely diferent from each
other.
Figure 7 - Visual integration in a collection of 20th century modern buildings; Asplund Library, Stockholm,
(top right - in circle); Terragni Casa di Fascio (third row - in circle); Mies van der Rohe Farnsworth House
(fourth row left - in circle); Mies van der Rohe Crown Hall (fourth row middle – in circle); Aldo van Eyck’s
Church (bottom row – in circle)
As with Palladio who inluenced modern architecture, the use of geometry and space by these
early modern architects afected contemporary architectural practice. A key reference is Mario
Botta’s domestic architecture, using isometric invariance between geometry and space within
the principles of tripartite composition, front and back distinction and the conines of the
Platonic solid. Another reference is Rem Koolhaas’ Kunsthal as a square volume perforated by
two intersections, a road and a pedestrian ramp, and a continuous spiralling circuit of interior
space that covers diferent spaces and programmes4. Another clear case is Herzog and De
Meuron’s De Young Museum in San Francisco (igure 9). The architects have used the corporeal
geometry to inform the incorporeal geometries of moving and viewing. The building seems to
gather all the elements of a Beaux-Arts Museum: an open courtyard, a tower, a grand staircase,
a portico - and reassemble them in a new fashion. The reference of the building to the BauxArts is made evident by the analysis. In a manner that is reminiscent of Durand’s axial grid-like
composition, the pattern of integration picks up the axial lines of the building geometry, but
replaces orthogonal geometry with an oblique system of geometrical planning. Contemporary
architecture therefore, is still choreographed by the lines of sight and movement of the body.
Figure 8 - Asplund’s Library and Corbusier’s Palace of the
Assembly
4 The Kunsthal does not rely on isometric invariance between geometric and topological properties, but uses the
idea of the promenade inside an orthogonal geometry.
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4. NON STANDARD VARIATION
Traditional drawing was an additive process. The consistency and the essential associative
relations between plan, section and elevation, between one element and another, between
geometry and space were managed by the designer during the process of design. Raphael’s
method of orthographic projection guaranteed exactly that. Geometry was the practical,
conceptual and intellectual network of associations needed to establish internal coherence.
CAD software simply translated this additive logic within the digital realm. This means that even
though geometry and space in modern architecture were decoupled, there were by and large
no changes in terms of notational tools (plans, elevations and sections) or the strict repertory of
orthogonal geometrical forms until the rise of digital technology.
Figure 9 - Herjog and de’ Meuron, De Young Museum.
Over the last decades, digital architecture has led to interactive algorithmic models based on
associative logic, responding to variations in the design input by manipulating the entire system.
‘They have already made it possible to envisage a continuous design and production process where
one or more designers can intervene, on a variety of two-dimensional visualisations and threedimensional representations (pint-outs) of the same object, and where all interventions or revisions
can be incorporated into the same master ile of the project’ (Carpo, ibid.). For Carpo, ‘under the
former dominion of geometry what was not measurable was not buildable. Now all that is digitally
designed is, by deinition and from the start, measured, hence geometrically deined and buildable…
today’s designers are not working on notations of objects but on interactive avatars of the objects
themselves’ (ibid.). Further Deleuze and Cache’s description of the ‘objectile’ deines design as
an algorithm rather than as an object, a parametric function which may determine an ininite
variety of objects, all diferent yet all similar as the underlying function is similar to all. Similar
to Hillier and Hanson’s notion of the genotype in the beady ring settlements, producing endless
phenotypical variations of the same model, the objectile can be collaboratively manipulated by
designers resulting in a series of non-identical elements. Carpo explains that together with the
demise of geometrical notation there is no longer the Albertian author of identical mechanical
copies (ibid.).
The invention of the digital not only enabled design to operate directly on three-dimensional
coordinates, but also provided a vast repertory of forms freed from constraints imposed by
buildable geometry. Yet, although orthogonal geometry, notations and the limitations they
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impose on formal variability have gone, geometry and its essential link with space are not
gone. Even when a building is not aesthetically revealed by spatial exploration, it still embodies
relationships of geometry and movement. Design software can produce diferent formal
outputs through inputs that afect the geometry of objects, but the impact of geometry on
space outputs is still in the blind spot, still in the shadows of these data, components and
node diagrams. It is here where the theory and method of space syntax can contribute, linking
geometry and space, generative and analytical approaches to design. Prior to discussing the
input of space syntax, it is important to explore what the appropriations of space and geometry
discussed here mean for architecture.
5. TOPO-GEOMETRIC PROPERTIES
Tracing morphological paradigms, I explored how geometry inluenced the development of
architecture as liberal art concerned with conscious design. By foregrounding a geometric
world of conceptual intelligibles, geometry in Classicism established an identity relationship
between design and building, geometrisising spatial structure (which in the three-dimensional
world is understood through moving and viewing) as a stable image. By externalising spatial
relationships, it made spatial and symbolic messages more pronounced. The technological
invention of the structural grid in the twentieth century lifted geometric constraints imposed
by load bearing partitions. Freed from geometric limitations, modern architecture established
variability in the relationship between geometry and movement.
Yet, in spite of diferent approaches, the complex relationship between geometry and space
facilitated translations from one programme to the other, including from cities to architecture
in both periods. Le Corbusier used the pinwheel scheme in public and private commissions, in
diferent sites, programmes and cultural contexts. Many of his museums were incubated in his
domestic architecture through the most private commissions. In Un Maison - Un Palais (1928),
in which he expressed the extension of his ideas from the private house to public buildings
and urban spaces, the villas became prototypes for a universal way of living, and the museum
a prototype for the city5. Inluenced by Alberti, Palladio also believed that a building is like a
city. His villas and churches were based on his studies of Roman baths, which he interpreted as
indoor miniaturized cities, theatrically framing space from the scale of the apse to the house
and the landscape. Is it the trans-nationality of these projects, the expansion from the house
to the scale of the museum, the hospital and the city as a whole, the intersection of social
programmes that are functionally very dissimilar peculiar to Palladio, Le Corbusier and Mies or
to the other contemporary architects?
For Beatrice Colomina domesticity has been the real source of modernity in Le Corbusier and
Mies’ museums (2009). I argue that the roots of modernity reach back into the villas of Palladio
in which the pattern of interconnected rooms and their lexible use difuse the boundaries
between the house as a space for private living and the house as art gallery, performance
space or theatre. In efect, Le Corbusier’s application of the pinwheel plan across diferent
building types and the translation of Venice’s spatial structure in the Hospital reveal that these
architects invested on generic properties of geometry and space over and above the functional
programmes of house, museum, hospital or urban space.
In a building that is like a house, a museum and a city, functional demands imposed by site
and social programme are just one ilter among others. Without circumventing functional
requirements, these architects were concerned with crafting geometry, space and exploring their
limits in diferent frameworks of functionality. The relationship between these programmes and
between the building and the city were based on topo-geometric properties that are common
to all. By interfacing generic relationships related to urban space and architectural space, these
architects extended over and above ontological and functional distinctions between functional
building types, architecture and urban contexts. Topo-geometric properties of moving and
viewing are shared among cities as multi-authored products of society and architecture as selfconscious product of design. If the former arise as the collective outcome of micro-economic
activity and the reproducibility of culture, the latter are the result of conscious intentionality
that recognises patterns common to all and translates them through creative invention.
5 Palladio A. (2002), The Four Books of Architecture, Cambridge Mass: The MIT Press.
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Describing space as a topological relational ield, Hillier’s analytic theory has in the last four
decades studied buildings and cities in relation to human activity and function. Hillier explains
that geometry gets into the topology of the urban network, afecting through the intersections
of street lines and the angles of their incidence their spatial structure (1997). However, in
spite of extensive application in real projects, the theory of coniguration treats architecture
and cities solely as empirical objects, based on properties of embodied movement and vision
from the ground. It does not take into account how architecture is conceived and produced,
including the ways in which generic properties of space and form get into the topology of
buildings and streets traversing these two worlds from within and without. If space has been
a silent partner in architectural discourse, geometry has been a silent conductor in the theory
of spatial coniguration. For Hillier, the reason for this deicit is that the relationship between
design and use passes not through geometry or form but the realm of space (1996). Yet, with
a clear focus on how cities and designs inluence one another as revealed by this analysis, the
picture is more complex than the clear-cut split into analysis and design, architecture and cities,
aesthetic and social practice. The study of topo-geometric properties should make it possible
to explore buildings and cities both as the non-authored products of society and the authored
products of design.
The examples studied here help reveal a genealogy of ideas around which the concerns of
architects converge and the architecture as a discipline is deined. Architecture concerns critical
commitment to comparative architectural knowledge on the part of an empirical historical
architect, that is, a person endowed with historical consciousness (and an unconscious).
Historical consciousness means that the fact that Palladio built before Le Corbusier, and Le
Corbusier operated before Rem Koolhaas is as signiicant as the morphological exploration of
their buildings. Comparative knowledge and historical consciousness establish an architect’s
place in history in relation to the available knowledge of ideas and tools that shape the discipline
up to one’s present, together with the possibilities and limitations for the future one’s historical
position enables and withholds.
If innovation and the creative imagination proceed from the intersection of possibility with
constraint, the intersection of comparative knowledge and dependence on historical sequence
brings us to the question of the imagination. At the beginning of this paper I argued that digital
technology might reduce architecture into a practice without a theory. If over the centuries
Eucleidian geometry was a platform of mediation through which space/building could be
visualised and ideas would be linked to the three-dimensional material world, in computational
design mediation between the thought process and the empirical object is abstracted through
algorithms and scripting. In the former the cognitive processes that underline design establishing
consistency relationships between parts are with the designer. In the latter, the designer
produces an interactive digital model responding to variations in the input by manipulating
the entire system, enabling to design a process rather than a single object. Lending ideas
syncretically to the eye all at once, geometry enabled architects to link form to space, idea to
building and intuition to logic.
Computational design can provide the scientiic and philosophical exploration of design
possibility and virtuality, extending and surpassing the designer’s intellect. However, it
needs to engage the relationship between geometry and topology rather than simply the
mathematically generated styles of software engagement. It also needs to bring the abstract
logic of computational design into the realm of principled understanding and the historical
sequence of ideas that inluence architecture and the imagination. The architectural imagination
transgresses functional constraints, social programmes, ontological and historical categories
by transferring generic properties across diferent domains, in ways, which enable one to make
innovations and overcome constraints in a work. The principles of space and geometry are not
just generic tools, but also the instruments of the critical faculties in architect’s imagination.
Abstract comparative knowledge and historical consciousness can raise space and geometry
from silent instruments to the level of abstract comparative thought, towards a unitary theory
of generation and explanation in architecture and the architectural imagination – towards an
archaeology of the present.
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