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Ana-Maria RUSU
GEOMETRY AND COMPLEXITY
IN ARCHITECTURE
Abstract: As Constantin Brancuși (1876-1956) said „Simplicity is complexity itself“, simplicity and
regularity through the use of basic geometric forms has always played a central role in architectural
design, during the 20th century. A diachronic perspective, shows as the use of geometry and mathematics
to describe built form provided a common basis for communication between the processes of design,
fabrication and stability. Classic ways of representing geometry, based on descriptive methods, favor
precise language of bidimensionality easy to represent in a rectangular coordinate system. In recent years,
the importance of geometry has been re-emphasized by significant advances in the digital age, where
computers are increasingly used in design, fabrication and construction to explore the art of the possible.
Contemporary architecture transcend the limitations of Euclidean geometry and create new forms that are
emerging through the convergence of complex systems, computational design and robotic fabrication
devices, but which can also achieve higher levels of performance. Freeform architectural shapes and
structures play an increasingly important role in 21st century architectural design. Through a series of
examples, the paper relates to contemporary architectural explorations of complex, curvilinear surfaces in
the digital age and discusses how it has required rethinking the mode in which we traditionally operate as
architects. The analysis creates the possibility of comparisons between original and current design.
Key words: Geometry; Free-form; Architecture; Design process.
INTRODUCTION The first category of thin covers can in its own turn
be divided and classified based on geometrical form; the
As it is well-known, descriptive geometry is a branch resulting categories are:
of applied mathematics whose goal is to represent objects Surfaces of revolution;
bidimensionally, on one or more planes, allowing for a Surfaces of translation;
graphical solution to many problems that are connected Ruled surfaces.
to tridimensionality. Nowadays, in the digital era, when A second classification can be made based on the
computers play an ever increasing role in the projection Gauss curve:
and materialization of architectural designs, there have Singly curved shells. These shells have a zero
been profound changes in the evolution of the whole Gaussian curvature;
creation process which have led to the question whether Doubly-curved shells of positive Gaussian
geometry has not become solely a means of perception, curvature;
of building explanations, understanding, and representing Doubly curved shells of negative Gaussian
space; or does it still allow for the creation of new curvature. [1]
shapes? Although geometry has been the language of Classification based on geometrical developability:
choice between different design processes, production, Developable surfaces
and materialization of architecture, it receives new Non-developable surfaces
valences in the combination with digital technology. This The free forms cannot be defined analytically, only in
new technology transcends the limitations of the a more subtle way, through a network of coordinates, by
Euclidean geometry, generating free-form architectural employing functional calculations that are specific to the
volumes, a more organic aesthetics with an increased structurally conditioned optimization procedures.
level of structural performance and this makes the Up until now, when looking to create curved
answer to our previous question become more and more architectural shapes, one had to take into account:
confusing and distant. First of all must be possible to formulate
The complex architectural forms, which demonstrate mathematically;
a very free shape, have their beginnings set in the 1920’s, The second, easy to build;
when thin concrete shells started to be used as roof Thirdly, structure should have expressivity.
structures. Being able to make these structures in which Thus, thin continuous covers are known for their
curved architecture had to possess an adequate structural structural efficiency resulting from geometry, also called
strength, and an equally powerful elegance, represented a self-supporting structures, and for their reduced width
huge challenge given that knowledge, as well as compared to the openness. Besides being structurally
instruments, at that time was extremely limited. The efficient, these structures are also extremely simple and
classification of the thin covers presents two distinct elegant. As these architectural forms evolved, the
categories of shapes: analytical shapes and free shapes. structural analysis had a long and difficult history. As
they were developed and perfected sometime between
1950 and 1960, at a time when architects were using
JOURNAL OF INDUSTRIAL DESIGN AND ENGINEERING GRAPHICS 59
Geometry and complexity in architecture
them as a means of artistic expression, long before the
computer ever entered the architectural scene a
considerable amount of effort was required to check the
designs.
2. SHELL STRUCTURES
One of the best known architects who designed and
built a great number of thin covers from concrete shells
is Felix Candela. Although an architect by formation,
Candela remains known as one of the most important
builders and structure engineers. Almost all his works, Fig. 2 The hyperbolic paraboloid.
created between 1950-1960 around Ciudad de Mexico,
are still in great shape, after years of usage and after the
devastating consequences of some strong earthquakes in
the area. He used the hyperbolic paraboloid for a number
of these structures, whose shape allowed for a
remarkably small width in relation to their aperture.
Candela posited that „of all the shapes we can give to the
shell, the easiest and most practical to build is the
hyperbolic paraboloid.“ In order to better understand the
creation of a thin structure in the shape of a hyperbolic
paraboloid, we must first understand the geometrical
characteristics of this geometrical surface.
2.1 THE HYPERBOLIC PARABOLOID
The hyperbolic paraboloid is a quadric ruled surface Fig. 3 The hyperbolic paraboloid.
generated by a straight line that lies on two straight
directrices and is at all times parallel to a director plane. A hyperbolic paraboloid can be also defined by
It is built by tracing one generator at a time, as a distinct means of a skew quadrilateral ABCD, Fig.3.
variation of the general hyperboloid. The third straight A skew quadrilateral determines one hyperbolic
directrix opens upward to the infinite and is replaced by a paraboloid and only one. The axis of the hyperbolic
director plane parallel to the surface’s generators. paraboloid is the straight line parallel to the intersection
The hyperbolic paraboloid is a doubly ruled skew line of the two director planes; it may be determined by
surface. It contains two families of mutually skew lines joining the middle sections of the skew quadrilateral’s
that can generate the same hyperbolic paraboloid. The diagonals ABCD. The tip of the hyperbolic paraboloid is
first generator family is made of generators parallel to the the point on its surface where the tangent plane in that
first director plane, Fig.1. The second family is made of particular point is perpendicular on the axis of the
generators parallel to the second director plane. hyperbolic paraboloid. The two generators that pass
through the tip of the hyperbolic paraboloid are called
main generators. The main generators are the diagonals
of a parallelogram that can be obtained by joining the
middle of the skew quadrilateral’s sides AbCd. [3] The
surface of the hyperbolic paraboloid contains two series
of straight generators Fig.4.
Fig. 1 The hyperbolic paraboloid.
The second director plane is parallel to the two straight
directrices Γ1 Γ2, which support the generators in the
first family, Fig.2. [2] Thus the generators in the first
family may become directrices for those in the second
family and the other way round. The hyperbolic
paraboloid is the only ruled surface with two director
planes.
Fig. 4 The hyperbolic paraboloid.
60 VOLUME 10 SPECIAL ISSUE ICEGD JUNE 2015
Geometry and complexity in architecture
and the Y-axis divides the structure into two symmetrical
They allow the delineation of skew quadrilateral parts. To reduce the tension in the concrete shell at the
sections of equal measure. Or, in other words: any skew basis of the surface the foundation has been set and two
quadrilateral may be adjacent to a section of a surface of pairs of buttresses, in the front and in the back, sustain
a hyperbolic paraboloid. If two straight opposite sides are the charge. After almost 60 years since it was built, the
equally subdivided and the subdivision points are united chapel is still in exceptional condition.
by straight lines, they become generators of the doubly
curved surface by the hyperbolic paraboloid.
The paraboloid, as a translation surface, may be also
generated by a parabola which is moving parallel to its
axis, along with another parabola, having parallel axes
and pointed in different directions.
There are many examples of hyperbolic paraboloid in
constructions and architecture where it can be
encountered in the manufacturing of roof systems or in
other projects which require a large number of surfaces.
From a structural point of view, the double curvature of
opposite direction deals very well with the changing
game of the internal forces of tension and compression,
maintaining balance under any strain, if there is a
minimum manifestation of forces. The surface may bear
in any point or given direction the compression or
applied stress, tangential to its curvature. It may be rest
on the two lower points. [4] Fig. 6 Chapel Lomas de Cuernavaca
For transmitting the self-load which is the most
important load of a roof, the suspended parabolas with
the curvature downwards are preferred for the tension
efforts, and the parabolas with the curvature upwards are
preferred for the compression forces, Fig.5. It’s a positive
thing that the parabolas coincide perfectly with the
pressure lines, and that they are capable of supporting
their own weight. The deviation tendencies of the forces
from the parabolic curvature of the thin surface are
therefore very reduced from the very beginning.
Fig. 5 The hyperbolic paraboloid
2.2 CHAPEL LOMAS DE CUERNAVACA
Made by Felix Candela in 1958, Chapel Lomas de
Cuernavaca, Fig.6 only needed one hypar to generate its
shape. The entry point was obtained by its intersection
with an inclined plane and the hyperbolic sides of the
church resulted from the intersection of the hypar with a
plane parallel to the ground.
Fig.7 represents the drawing of the Chapel, made by
Candela, with the two types of generators that form the
curved surface and Fig.8 offers a glimpse of the ongoing
building process, bringing to our attention the generators
that give the chapel its saddle-like shape, with a
remarkable width of only 4 centimeters. The grid of the Fig. 7 Chapel Lomas de Cuernavaca
generating lines is made of parallel and perpendicular
lines, refined at an optimum level for creating the shape,
JOURNAL OF INDUSTRIAL DESIGN AND ENGINEERING GRAPHICS 61
Geometry and complexity in architecture
and F respectively. An arbitrary end plane P’ generated
2
by end line D’ intersects with the two arcs in points (1,
1’) and (1, 1’). We obtain thus the line (110, 1’1 ’) as a
0
generator of the arrière-voussure surface. Similarly, we
can determine other generators, for example (22 , 2’2 ’),
0 0
(33 , 3’3 ’) etc. The crooked arrière-voussure surface
0 0
covers the opening only between the limit generators
1’1 ’ şi 5’5 ’. Between the level plane H’ and the end
0 0
plane P’ the opening is covered by by a general ruler
surface, whose generators’ two points of support are two
directing lines (the vertical in a, the end line D) and a
directing curve (the arc γ2, γ2’).
Fig. 8 Chapel Lomas de Cuernavaca
The calculation of the determined weight on a single
ruler surface, more precisely on a single parabolic
hyperboloid, was controlled through studies and
calculations without the aid of a computer. But this
process was not error-free because what we see today is
the second version, only 18 metres high; the first version
of the building, 24 metres high, had crumbled. This could
have been avoided if the builders and designers would
have had access to simulation programs that calculate the
degree to which a structure is pressed by its own weight.
Computer-assisted design developed in the 70’s and it
allowed architects to design more graphic, daring and
free shapes, bringing immense ease in their design,
construction and creation.
3 PARAMETRIC DESIGN
The new visions in architectural design are based on
more complex types of geometry, although on a basic
level they develop shapes and surfaces that are already Fig. 9 The arrière-voussure surface.
known in descriptive geometry; when multiplied, they
vary from one side to the other of the structure.
Designing these components by using only classical
instruments used to represent tridimensionality would
take a very long time and solving the structural problems
would represent an equally daunting task.
An interesting project that suggests this exact type of
relation between crooked geometrical surfaces is Norman
Foster’s Sage Gateshead Concert Hall. In order to gain a
better understanding of the complexity of this structure, it
is important that we notice the geometrical characteristics
of the arrière-voussure surface.
3.1 THE ARRIÈRE-VOUSSURE SURFACE
The arrière-voussure surface is a crooked surface
generated by a line that sits on two directing curved lines
situated in two parallel planes and on a line that is
perpendicular on the planes of the directing curved lines.
This directing line is sometimes called the surface axis.
A generator of the surface can be obtained by uniting the
intersection points between the two directing lines and an
arbitrary plane, starting in the right directing line. Thus,
we can choose the axis of the arrière-voussure surface Fig. 10 The arrière-voussure surface.
fig.9 the end line D (d, d’) and the directing lines can be
the arcs (γ , γ ’) and (γ , γ ’), situated in the planes F
1 1 2 2 1
62 VOLUME 10 SPECIAL ISSUE ICEGD JUNE 2015
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