F. Madeo and M. A. Schnabel (eds.), Across: Architectural Research through to Practice: 48th Interna-
tional Conference of the Architectural Science Association 2014, pp. 405–416. © 2014, The Architectural
Science Association & Genova University Press.
COMPUTATIONAL MORPHOGENESIS APPLIED TO
THE CHURCH OF LONGUELO
Reflections upon a possible parametric interpretation of the
original design concepts
ALESSANDRO LIUTI and ALBERTO PUGNALE
The University of Melbourne, Australia
Abstract. Before the introduction of NURBS-based CAD software
and optimisation, the design of form-resistant structures was based on
the use of either experimental tools (physical form-finding) or analyti-
cal surfaces, and architects were challenged in the articulation of
spaces from the intrinsic characteristics/rules of structural forms. An
outstanding example of this kind is provided by the Church of Lon-
guelo, which was built by architect Pino Pizzigoni in Italy, between
1961-1966. It was conceived as composed by two major elements – an
irregular frame and a set of shells suspended to it. The entire design
process was based on the calculation of the frame on which the shells
have been just added as a dead load. This paper presents one possible
way to redesign the church parametrically. Comparison with the orig-
inal design is not performed at the final formal level, which can logi-
cally differ, but around the concepts behind the project. The aim is to
show how current digital design and optimisation tools are affecting
the way architects design. But, at a higher level, the purpose is also to
highlight where conceptual design is now taking place in the process.
Keywords. Parametric design, optimisation, computational morpho-
genesis, Pino Pizzigoni, Church of Longuelo.
The term ‘parametric’ was born in the world of mathematics to define all
those “equations that express a set of quantities as explicit functions of a
number of independent variables, known as ‘parameters’” (Weisstein 2003).
Its architectural transposition was introduced by Luigi Moretti during the
Forties and indicates a consistent system of elements and relationships
406 A. LIUTI AND A. PUGNALE
among them, which is founded on the construction of topological rather than
The world of forms reveals itself to us through the differences be-
tween one form and another. […] Differences are the inevitable, in-
transgressible flashes of reality and of the forms; they are the forms.
[…] A non-elementary form is constituted by a group of differences
that are themselves connected by relationships that express and oblige
their order and sequence. The complex of these relationships is the
structure of a form, which can be expressed in the abstract as a com-
plex of pure relationships (Moretti, 1957).
The term ‘parametric’ identifies the only invariant of each and every de-
sign process with the potential of standardisation in digital design (Aish,
2005). Through parametrics, digital technology is changing into a resource
for conceptual design, which is used to formulate problems in a different
way and construct resolution tools and strategies in an interactive manner.
This paper aims to highlight this phenomenon by means of parametrically
redesigning the historical case study of the church of Longuelo, by architect
Pino Pizzigoni (1961-1966). Before the introduction of NURBS-based CAD
software and optimisation, the design of form-resistant structures was based
on the use of either experimental tools (physical form-finding) or analytical
surfaces, and architects were challenged in the articulation of spaces from
the intrinsic characteristics/rules of structural forms. The presence of an ex-
isting project permits a clear comparison between design processes and
shows where conceptual design is now taking place.
2. The church of Longuelo
Pino Pizzigoni designed the church of Longuelo, near Bergamo, in Italy, be-
tween 1961 and 1966. This is generally considered his most relevant project
based on shell structures. His interest for shell and spatial structures started
after the Second Post War. First, he built a few prototypes of hyperbolic pa-
raboloids and umbrellas in a field of his own property. Then, he applied that
experience to the roof design of some schools and factories around Bergamo
(Deregibus and Pugnale, 2010).
The church spans over 900 square meters, for 18 meters top height. It is
conceived as a double-reflectional symmetrical room made of four identical
quarters. Each quarter is made of 5 reinforced concrete hypar shells and a
beam frame. Figures 1 and 2 show, respectively, an external and an internal
view of the church.
COMPUTATIONAL MORPHOGENESIS APPLIED 407
Figure 1. The church of Longuelo, external view. Photograph by Carlo Deregibus.
Figure 2. Interior of the church. Photograph by Carlo Deregibus.
2.1 ORIGINAL DESIGN AND CONCEPTS
Pizzigoni’s archive is currently located in the public library of Bergamo
“Angelo Mai”. It can be assumed from the documents found that Pizzigoni
based the design of the church of Longuelo on three main propositions:
408 A. LIUTI AND A. PUGNALE
1. The use of hypar shells
2. The Möbius ring
3. The tent pitched by God (Figure 3).
Geometrically speaking, four hypar-surfaces are joined together in the
shape of a ring and then topologically transformed into a Möbius strip. A
fifth hypar is placed between the other ones in order to give strength to the
overall configuration. The use of hypar shells offers several spatial configu-
rations for the proposed frame. Moreover, the topology through which he ar-
ranged the hypars, allowed the architect to put to practice the concept of
Möbius ring as well. Another feature related to the hypar is that the internal
space results in a single, smooth surface that echoes the concept of the tent
pitched by God, metaphorical / biblical concept described by the gospel of
John (Deregibus and Pugnale, 2010).
In order to better understand Pizzigoni’s design, a three-dimensional
model in Rhinoceros® of the church has been developed. Free bars of the
frame have been explicitly avoided in order to focus on the spatial complexi-
ty provided by the shells (Figure 4). Now, by focusing on a single quarter, it
is possible to understand how the geometry of the shells works (Figure 5).
Figure 3. The three concepts behind the project of the church: hyperbolic paraboloids,
Moebius ring, the God tent (Pizzigoni Archive, scanned by C. Deregibus and A. Pugnale).
COMPUTATIONAL MORPHOGENESIS APPLIED 409
Figure 4. The 3D model of the Church of Longuelo.
Figure 5. The church was designed working on a single quarter of the structure. Four hypar
shells are joined together and form a Möbius ring. A fifth shell is then added.
3. Computational morphogenesis of the church
According to a classical engineering approach, the church would now be op-
timised to verify the accuracy of the original structural design and, therefore,
the extent of Pizzigoni’s expertise and intuition. Such a process generally
starts from a parametric definition of the existing geometry, as it was con-
ceived by the architect - no design is involved, but just a form-improvement
of a well-defined structural layout is performed.
In this Section, a different approach is shown. The aim is to use paramet-
ric design and optimisation to define a computational process of morphogen-
esis, in which conceptual design is highly involved.
410 A. LIUTI AND A. PUGNALE
3.1 DESIGN OF THE PARAMETRIC MODEL AND OPTIMISATION
As mentioned before, three main concepts led Pizzigoni towards the final
design of the church: (1) the use of hyperbolic paraboloids, (2) the Möbius
ring and (3) the God Tent.
The parametric model of the church is here defined from such concepts
rather than referring to the built geometry. This permits the optimisation al-
gorithm to generate new spatial configurations of the church, and therefore it
allows morphogenesis to take place.
The model is generated by means of Grasshopper®, a Rhinoceros® plug-
in that allows the definition of parametric systems and their visualisation in
3.1.1. Boundary conditions, design variables and solution domain
First, the symmetrical and centrical layout of the original church is pre-
served. The parametric model is therefore based on a single quarter of the
church, which is subsequently mirrored twice.
Second, each quarter of the original church is composed by five hypars,
four of which define a Möbius ring. The topology of such a ring is preserved
by the parametric model, but the position of the fifth hypar becomes a design
variable. Three different edge conditions are possible for this hypar, as
shown in Figure 6.
Figure 6. Four hypars define a Möbius ring and their topology is preserved. A fifth hypar can
be placed in three different positions and is a design variable of the optimisation problem.
COMPUTATIONAL MORPHOGENESIS APPLIED 411
Figure 7. Eight points define the geometry of a quarter of the church. The coordinates of the
points are design variables and their solution domain is shown here.
Third, the structural frame is determined by the edges of the hypars, and
the geometry of the edges is controlled by acting on the coordinates of eight
nodes. Defining the solution domain of such coordinates is a key point (Fig-
ure 7). A bounding box that embraces the volume of a church quarter is de-
fined and acts as the overall domain for positioning the eight nodes. Four
nodes are further constrained to the upper part of the box to generate a
closed roof (R1, R2, R3 and R4); two points are retained to the lower part of
the quarter to define the connections with the ground (B1 and B2); the last
two points are kept into the middle part of the quarter (M1 and M2).
Defining a parametric model of the church in this manner involves de-
sign. Therefore, this is only one of the possible ways to interpret the con-
cepts by Pizzigoni, and has a purely illustrative scope of what ‘parametric
thinking’ stands for.
3.1.2. Fitness function and penalty functions
Optimisation is here used as a source of inspiration. The aim is to develop
the church design through the generation of a wide spectrum of geometries,
which perform efficiently in structural, spatial and functional terms. In order
to do so, multi-objective optimisation is used.
Structural behaviour is the first performance. Nodal vertical displace-
ments of the church frame are calculated by means of the FE solver Karam-
ba, a Grasshopper® plug-in which was developed by Clemens Preisinger
with Bollinger+Grohmann (Preisinger, 2013). Karamba is chosen because
permits to rapidly connect geometric and structural models within a single
parametric environment, i.e. Grasshopper®. Applied loading refers to the
original calculations provided by Pizzigoni (Deregibus and Pugnale, 2010).
412 A. LIUTI AND A. PUGNALE
Spatiality and functionality are then considered together as the second
performance criterion. The idea is to calculate the maximum amount of
space covered by the structure with no intermediate supports. Without this
function, the algorithm would tend to reduce the structural frame size to-
wards unusable dimensions.
Galapagos, a ‘black-box’ Genetic Algorithm implemented in Grasshop-
per®, is used to perform the optimisation. Genetic Algorithms (GAs) are
meta-heuristic search algorithms based on the mechanics of natural selection
and natural genetics (Coelho et al., 2014). They provide a robust and flexible
tool to solve complex problems and their meta-heuristic way of exploring
suitable solutions seems to be particularly helpful in architecture – designers
generally benefit from comparing several sub-optimal outcomes rather than
converging to a single optimal one.
Multi-objective is pursued through the aggregated fitness function F (1).
Such a function aims to minimise nodal displacements (2) while the area
covered by the shells is maximised (3). For both criteria, solution domains
F = c1 F’(dz) + c2 F’(A) + P (1)
F(dz) = ∑i (dzi) → F’(dz) [0; 1] (2)
F(A) = [A]-1 → F’(A) [0; 1] (3)
The optimisation process also includes a penalty function P, in order to
prevent self-intersections between shells. Two weighing coefficients, c1 and
c2, are finally applied to treat the performance criteria equally.
4. Design through optimisation: reflections
The GA was run multiple times to increase the amount of geometries gener-
ated through optimisation. Figure 8 shows a comparison among configura-
tions with similar performance but very different shape. In Figure 9, a fur-
ther reduction to five representative solutions is proposed to provide full
details for each of them. For these, views of the church interior are also ren-
dered (Figure 10) to understand the spatial quality they offer. The geomet-
rical variety given by the external axonometric views is not reflected by the
spatiality of the internal perspectives, which look very similar to one anoth-
er. Similarity is given by the position of the fifth shell edges, which is shared
by the five selected geometries. Structurally speaking, they are all very effi-
cient, but the quality of the interior spaces is much lower than in the case of
the original design, where light is brought in creating a kind of clerestory.
COMPUTATIONAL MORPHOGENESIS APPLIED 413
Figure 8. Variety of shapes generated by the optimisation process. Fitness decreases (im-
proves) from top to bottom but is comparable among geometries in the same row.
414 A. LIUTI AND A. PUGNALE
Figure 9. Five geometries with different shape but comparable and very low fitness.
COMPUTATIONAL MORPHOGENESIS APPLIED 415
Figure 10. Spatial quality of the interior, given by the five geometries in Figure 9.
When applied to historical case studies, optimisation is generally used to
verify the accuracy of the original structural design, and therefore the level
of the architect’s expertise and intuition. No design is involved, but eventual-
ly a form-improvement of a well-defined structural layout is performed.
In this paper, a different approach has been shown.
First, a parametric reconstruction of the church has been defined starting
from the original architect’s concepts rather than on the built form. This has
shown only one of the possible ways to translate Pizzigoni’s ideas into a sys-
tem of topological relationships.
Second, optimisation has been used as a source of inspiration. The defini-
tion of a suitable fitness function has been fundamental to list priorities for
the church design (such as structural performance, covered area, etc). A GA
has been run several times in order to explore a wide spectrum of possible
solutions, but mainly to appreciate how the initial settings (boundary condi-
tions, design variables, solution domain, etc) might affect the algorithm out-
Aish, R.: 2005, Extensible computational design tools for exploratory architecture, in Kolare-
vic, B. (ed.), Architecture in the Digital Age: Design and Manufacturing, Routledge,
London and New York, 17.
Coelho, R. F., Mendez Echenagucia, T., Pugnale, A., Richardson, J.: 2014, Genetic Algo-
rithms for structural design, in Adriaenssens, S., Block, P., Veenendaal, D., and Williams,
416 A. LIUTI AND A. PUGNALE
C. (eds.), Shell structures for architecture: Form Finding and Optimization, Routledge,
London and New York, 290-294.
Deregibus, C., and Pugnale, A.: 2010, The church of Longuelo by Pino Pizzigoni: design and
construction of an experimental structure, Construction History, 25, 115-140.
Moretti, L.: 2002, Form as structure, Spazio, in F. Bucci, and M. Mulazzani, Luigi Moretti:
Works and Writings, Princeton Architectural Press, 182-184.
Pizzigoni, A.: 1982, Pizzigoni: Invito allo spazio, progetti e architetture: 1923-1967, Electa,
Preisinger, C.: 2013, Linking Structure and Parametric Geometry, Architectural Design,
Pugnale, A., Méndez Echenagucia, T., and Sassone, M.: 2014, Computational morphogenesis:
Design of freeform surfaces, in Adriaenssens, S., Block, P., Veenendaal, D., and Wil-
liams, C. (eds.), Shell structures for architecture: Form Finding and Optimization,
Routledge, London and New York, 224-236.
Weisstein, E. W.: 2003, Concise Encyclopedia of Mathematics, Chapman & Hall/CRC, Boca