Structural Reciprocity: Critical Overview
and Promising Research/Design Issues
Alberto Pugnale •Mario Sassone
Published online: 4 March 2014
ÓKim Williams Books, Turin 2014
Abstract Reciprocity is a structural principle that has fascinated designers and
builders throughout the world since ancient times. Despite the topic’s having been
studied by various academics, designers and researchers, a critical overview of the
references is still missing, as is an outline and discussion of the current and future
promising research/design issues. Further, no single text provides an exhaustive
deﬁnition of the principle of structural reciprocity and it must be critically recon-
structed from several different sources. This paper aims to ﬁll in these gaps, pro-
viding a complete and annotated list of references, in which historical examples, as
well as patents, research articles and terminological issues are discussed. A con-
sistent deﬁnition of structural reciprocity is also proposed, and the promising
developments of such a principle are outlined in order to guide designers and
researchers in the future.
Keywords Structural reciprocity Reciprocal frames (RFs) Nexorades
Mutually supported elements (MSE) Leonardo da Vinci Leonardo
grids Spatial structures Timber constructions Form-ﬁnding
Morphogenesis History of construction
A. Pugnale (&)
Faculty of Architecture, Building and Planning, University of Melbourne,
757 Swanston St., Melbourne, VIC 3010, Australia
Department of Architecture and Design (DAD), Politecnico di Torino,
Castello del Valentino, Viale Pier Andrea Mattioli, 39, 10125 Torino, TO, Italy
Nexus Netw J (2014) 16:9–35
The principle of reciprocity is based on the use of load-bearing elements which,
supporting one another along their spans and never at the extremities, compose a
spatial conﬁguration with no clear structural hierarchy.
An illustrative example will help the reader better understand the concept. Let us
consider three glasses, arranged on a table at the vertices of a hypothetical
equilateral triangle, and then imagine covering that area using just three kitchen
knives. Considering that the glasses are the only supports, the knife handles should
ﬁrst be placed over the glass openings, and the blades should be made to overlap one
another, like a fan. The resulting conﬁguration is the simplest reciprocal structure
made of three elements (Fig. 1).
Such a system has been used throughout the world since ancient times.
However, a comprehensive, annotated list of references regarding them has
not been developed yet. Furthermore, since designers and researchers works
frequently in independent ways, the current studies still need to be grouped
and compared systematically in order to outline the future promising design/
research activities. It is an explicit aim of this essay to respond to these needs
by proposing a critical overview of the topic supported by: (1) an annotated
list of references in both Western and Eastern cultures; (2) a complete lists of
patents on reciprocity; (3) a complete list of publications grouped by topic;
(4) a complete and annotated list of scientiﬁc terms coined and used by
different authors; (5) a consistent deﬁnition of structural reciprocity; (6)
several future research and design directions based on the current state of the
Differences and Similarities of a Fragmented History
The ﬁrst applications of structural reciprocity date back to Neolithic pit
dwellings, Eskimo tents and Indian tepees, as documented by Popovic Larsen
(2008). However, it was also present during the Roman Empire, when Julius
Caesar used it for the construction of a bridge over the Rhine: the structure was
made of interlocked timber elements, with the main purpose of simplifying the
joints. The project was described in Caesar’s ‘‘Commentaries on Gallic and
Civil Wars’’ and was later reconstructed by Palladio (Gros and Beltramini
2003). Apart from these ancient applications, independent from one another,
reciprocity has been independently studied and used in both Western and
Eastern cultures for centuries. The needs and purposes have been different, but
the structural outcomes have been similar. During the last few decades,
reciprocity has also become a research topic for academics, who have started to
study the mechanical, geometrical and construction aspects of reciprocal
10 A. Pugnale, M. Sassone
Reciprocity in Western Culture: Timber and Short-Beams
In Europe, structural reciprocity has mainly been used, at least until the twentieth
century, to span distances longer than the length of the available timber beams.
Several architects, builders, mathematicians and scientists all reasoned separately
about this problem, without evidence of being in contact with one another.
Between 1225 and 1250, Villard de Honnecourt drew some short-beam arrange-
ments in his sketchbook (Villard de Honnecourt 1959) for the construction of
reciprocal ﬂoors. Between 1220 and 1235, Alexander of Lincoln designed and built
Lincoln Cathedral using reciprocal supports, as reported by Hewett (1974).
Leonardo da Vinci explored at least six different spatial conﬁgurations based on the
principle of reciprocity, as can be seen from his sketches in the Codex Atlanticus,
folio 899v (Leonardo da Vinci 2000) (Fig. 2). To read the transcription of this page,
refer to Williams (2008).
After Leonardo, Sebastiano Serlio discussed the construction of planar ﬂoors
with short-beams in his ﬁrst book on architecture, but unfortunately proposed an
unbuildable structure (Serlio 1566; Yeomans 1997). John Wallis wrote about
different types of ﬂoors made of reciprocal elements in his ‘‘Opera Mathematica’’
(Wallis 1695), and this was the ﬁrst scientiﬁc text to be supported with structural
calculations, as it has been described in detail by Houlsby (2014).
Amongst the treatises on carpentry, E
´my, who was professor of Fortiﬁcation at
the Royal Military School in Saint-Cyr, included some reciprocal examples in his
´de l’art de la charpenterie’’ (E
´my 1837). Reciprocity was also present in the
Fig. 1 An illustrative example of a simple reciprocal structure. Image:authors
This problem was also solved using an alternative construction technique, based on the overlapping of
layers of short load-bearing elements. This technique can be found, for instance, in the roof structure of
the Square Hall building in the Old Nisa, Mesopotamia, as described by Pizzetti and Zorgno (1980). It
was also proposed by Guarini for his Chapel of the Holy Shroud in Turin, Italy.
A more detailed description of historical works on reciprocity can be found in Popovic Larsen (2008).
Instead, a complete list of historical treatises, dealing with reciprocal structure, has been provided by Vito
Bertin on his website: http://www.vitobertin.hk/lw/10-reference/02-studies/index.html.
Structural Reciprocity 11
so-called ‘ﬂat-vaults’ described by Fre
´zier (1737), but which were invented by
Joseph Abeille and Se
´bastien Truchet. It was ﬁnally described by Rondelet (1810)
and Tredgold (1837).
As we mentioned before, reciprocal structures were used in Europe for
technological and construction reasons. Recent surveys and renovations of old
British buildings seem to conﬁrm this statement. The layout of the sub-ﬂoor
structure of the Wollaton Hall (Fig. 3) shows an irregular pattern of timber short-
Fig. 2 a Leonardo da Vinci, Codex Atlanticus, fol. 899v, with different concepts identiﬁed and
numbered. The reciprocal systems represented by Leonardo are redrawn in b–h.Continuous lines are
used to represent the working parts of the arrangements, which are clearly drawn and reproducible with
physical models. Unclear potions of Leonardo’s sketches, as well as construction lines are reproduced
with hidden lines. Hatches are then used to identify parts of the structures which were meant to be clad.
Image elaboration: authors. bTranscription of drawings A1 and A2. cTranscription of drawing B1.
dTranscription of drawing B2. eTranscription of drawing B3. fTranscription of drawing C2.
gTranscription of drawing E1. hTranscription of drawing E2
12 A. Pugnale, M. Sassone
beams which are arranged in a reciprocal way. They were clearly conceived not to
be seen as the photograph of the ﬁnished ceiling does not refer anyhow to its
reciprocal nature (Fig. 4). The same happens at the home of William Morris,
Kelmscott Manor, where the presence of reciprocal beams in the sub-ﬂoor structure
was only revealed during the most recent works of restoration (Insall 1972,2008)
Fig. 2 continued
Structural Reciprocity 13
(Fig. 5). A unique example that contradicts such logic can be found at Palazzo
Piccolomini in Pienza. The ceiling of its ‘‘Music room’’ presents a reciprocal
arrangement of timber beams which appears to be purely decorative: such a room is
one of the smallest of the Palazzo, but it is the only one using short-beams to span a
relatively short distance (Fig. 6).
Reciprocity in the East: Interwoven Structures and Symbolism
In Eastern culture, interest in reciprocal structures derives from two separate
concepts. On the one hand, especially in China, the use of interwoven strips of
bamboo for the realization of baskets is an old tradition that has been transferred to
objects of larger scale; the so-called ‘‘Rainbow Bridge’’ in Shandong is the main
example (Baverel 2000; Rizzuto et al. 2002; Di Carlo 2008). On the other hand, the
religious concept of Mandala, with its symbolic ‘magic circle’ shape (Gombrich
1979), has inspired the construction of circular reciprocal roofs in many Buddhist
temples all over Asia. Unfortunately, written documents concerning this practice are
only found after the year 1275, i.e., when Marco Polo arrived in China at the end of
the Song Dynasty.
A few contemporary Asian architects, such as Kazuhiro Ishii (Fig. 7), Yasufumi
Kijima and Yoichi Kan, still take advantage of structural reciprocity for their
constructions (Gutdeutsch 1996; Popovic Larsen 2008,2009).
The Basis of a Scientiﬁc Approach
Even though structural reciprocity has not had a linear history, and the evidence of
its use seems to be fragmented, a main point in common in both Western and
Eastern cultures is the use of timber as a construction material. In Europe, this
occurred for functional reasons, with the aim of developing ﬂat conﬁgurations. In
Fig. 2 continued
14 A. Pugnale, M. Sassone
Asia, designers were mainly guided by spiritual considerations in the construction of
3D structures, which recalled symbolic shapes.
In this historical framework, Leonardo da Vinci was the only one in the West to
study the potential of elaborating reciprocal structures as complex 3D geometries
Fig. 3 Wollaton Hall, sub-ﬂoor structure survey. This layout shows an irregular pattern of timber short-
beams which was clearly conceived not to be seen. Image: Ed Morton, The Morton Partnership LTD,
London, UK, reproduced by permission
Fig. 4 Wollaton Hall, ceiling. The reciprocal arrangement of sub-ﬂoor timber beams is not visible as it
was clearly conceived to solve a construction matter. Photo: Peter Langley, reproduced by permission
Structural Reciprocity 15
Fig. 5 Sub-ﬂoor structure of the William Morris house (Kelmscott Manor) during the most recent
renovation works. This reciprocal arrangement of timber beams was clearly due to structural and
construction reasons, as the carpentry was not designed to be exposed. Photo: Donald Insall, reproduced
Fig. 6 Palazzo Piccolomini, Ceiling of the ‘‘Music room’’. The reciprocal arrangement of timber beams
appears to be purely decorative and designed to be seen. Such a room is one of the smallest of the
Palazzo. However, it is the only one using short-beams to span a relatively short distance. Photo: Il
Cenacolo srl, Rome, reproduced by permission. Image elaboration:authors
16 A. Pugnale, M. Sassone
(Leonardo da Vinci 2000). It was John Wallis who ﬁrst approached structural
reciprocity scientiﬁcally, like a modern research topic (Wallis 1695; Houlsby 2014).
The Years of Patents and Scientiﬁc Research
Over the last century, the principle of reciprocity has continued to stimulate the
interest of designers and, for the ﬁrst time, has become a topic of interest in the ﬁeld
of scientiﬁc research. As shown in ‘‘Appendix 1’’, several patents were registered by
different authors between 1924 and 2012. The ﬁrst was the lamella construction
system by Zollinger with reciprocal joints, which has been described in detail by
Popovic Larsen (2014). A foldable structure by Emilio Perez Pin
proposed a kinematic roof structure with nodes based on the superimposition of
bars, was then registered in the USA and Canada in 1965 (Figs. 8,9). A detailed
description of this project can be found in the monograph about Pin
˜ero by Escrig
The German engineers Erwin Walle and Sigurd Prinz patented a reinforced
concrete ceiling made of prefabricated units hinged together, which curiously
recalls the reciprocal slabs designed by Kahn for the Mill Creek public housing
project in Philadelphia, dated 1952–1953. The patent was ﬁled in 1971 and then
published in three different parts. The building element for the construction of
interlocking grids patented by Gat is also worth mentioning. This was actually the
ﬁrst work to be also published as an article in a scientiﬁc journal (Gat 1978).
However, it was the 3D closed circuit of sticks patented by Graham Brown that
Fig. 7 Seiwa Bunraku-Kan, Puppet theatre complex designed by Kazuhiro Ishii. Photo: Kentaro
Tsukuba, reproduced by permission
Even though Gat’s system was the ﬁrst patent also published in a scientiﬁc journal, it was not the ﬁrst
article ever to discuss reciprocity. A paper by Donald Dean about a ‘new’ structural system called
‘Lamella Grid’ had in fact already been published in 1964 (Dean 1964).
Structural Reciprocity 17
deﬁnitively inspired and kick-started academic research on structural reciprocity.
Brown established contact with the University of Nottingham, with the intention of
increasing his understanding of the mechanical behaviour of such a system, which
he called ‘‘Reciprocal Frame’’. This led John Chilton’s group to make several
publications during the 1990s (Chilton and Choo 1992,1994; Choo et al. 1994;
Chilton et al. Chilton 1995a,b), including the Ph.D. thesis by (Popovic 1996), which
has recently been revised and printed as a book (Popovic Larsen 2008). A complete
list of patents involving structural reciprocity is given in ‘‘Appendix 1’’.
Fig. 8 Emilio Perez Pin
patent. Detail of the node
Images: United States Patent and
Trademark Ofﬁce, reproduced
Fig. 9 Emilio Perez Pin
˜ero’s patent. Superimposition of bars allows and controls the kinematic
movement of the assembly Images: United States Patent and Trademark Ofﬁce, reproduced by permission
18 A. Pugnale, M. Sassone
Even though John Chilton’s group was the ﬁrst to use the term ‘Reciprocal
Frame’ in a scientiﬁc paper (it is here that a deﬁnition of the concept of closed
circuits of elements is found), other names by other academics and research groups
also exist (see ‘‘Appendix 2’’ for further details). Such terminological heterogeneity
creates a certain level of confusion that has been avoided, in this text, by adopting
the most logical and generic term ‘structural reciprocity’, or simply ‘reciprocity’.
Structural reciprocity is a very extensive research topic, and has been approached
from very different points of view. Looking at the current publications, at least
fourteen different sub-topics can be deﬁned: (1) morphology and geometry of
reciprocal spatial structures, including (2) polyhedrical conﬁgurations; (3) form-
ﬁnding and morphogenesis of such structures with computational tools, recently
also combined with (4) the rapid prototyping of timber elements; (5) the design of
joints and connections; (6) analysis of the structural behaviour; (7) the study of the
kinematic potential; (8) investigations on materials and sections for construction; (9)
the use of planar panels; (10) discussion/criticisms of projects and prototypes; (11)
teaching structural reciprocity; (12) history of reciprocal structures; (13) art and
sculpture, (14) reciprocal structures based on Leonardo’s grids. ‘‘Appendix 3’’ lists
the main research works published in each category.
Reciprocity in Contemporary Architecture, Art and Industrial Design
Considering the works of architecture realized over the last century, we ﬁnd several
unrelated examples that have implemented structural reciprocity. Long spaces were
spanned with short beams in the Mill Creek Public Housing project, designed by
Louis Kahn in 1952–1953, but also in the Berlin Philarmonie by Hans Scharoun,
which was built in 1960–1963, and in a salt storage building in Lausanne, built by
Atelier Gamme Architecture in 1989 (Natterer et al. 1991). In order to introduce
spiritual philosophies into shapes, reciprocity was used in the Casa Negre by Josep
Maria Jujol in 1915–1926, (Fig. 10) (Ligtelijn and Saariste 1996), as well as in all
the structures by Graham Brown and the Japanese constructions mentioned earlier.
Some experimental pavilions, with the Arup Group’s Advanced Geometry Unit
group as the engineering consultant, have also been constructed: the Forest Park by
Shigeru Ban (McQuaid 2003), the H-edge pavilion by Cecil Balmond, realized with
the students at Penn University (Balmond 2007), and the Serpentine Gallery 2005
´lvaro Siza (Sakamoto et al. 2008). These examples clearly illustrate that the
architectural potential of structural reciprocity still needs to be explored—the use of
different materials, as well as the deﬁnition of element sections and joints, are some
of the possible research/design issues of the next few years.
In the ﬁeld of industrial design, Pino Pizzigoni realized reciprocal chairs and
tables made of timber and marble elements. Two examples can be seen in Figs. 11
and 12 (Pizzigoni Archive: PIZ N, 1948; Pizzigoni 1982; Deregibus and Pugnale
2010). Shifting to art, the sculptor George Hart used reciprocity to create simple
geodesic domes. Rinus Roelofs, instead, focused on complex three-dimensional
sculptures based on Leonardo’s grids (Roelofs 2005,2008). Some of his projects are
actually impracticable and are therefore just rendered as drawings or prototyped
though 3D printing techniques.
Structural Reciprocity 19
Characteristics and Interesting Aspects
The term ‘reciprocity’ comes from the Latin reciprocus, a composite of recus,
backwards, and procus, forwards. The etymological signiﬁcance of reciprocity is
therefore that of ‘back and forth’, evoking an exchange for mutual beneﬁt. In the
world of construction, situations in which structural systems imply an exchange of
actions is frequent—the voussoirs of an arch, for instance, achieve equilibrium
through mutual action/reaction. However, reciprocity differs from mutuality
because a transitive relation between at least two elements occurs only if such a
relation is perfectly symmetric.
The concept of structural reciprocity therefore refers to a speciﬁc subset of
structures, which are characterized by two main properties.
First, in each and every element the functions of supporting and being supported
by other elements must be separated instead of being overlapped. For instance, in
simply supported beams constraints are placed at the extremities and loads act at the
midpoints—the functions correspond to different position on the element and no
inversion is possible. In contrast, constraints and loads both occur at the ends of
truss bars—so the functions overlap here and the deﬁnition of supporting and
supported ends becomes merely conventional. This ﬁrst property also implies that
only beams and two-dimensional elements can form a reciprocal structure. The
forces are transferred through bending and shear with beams, both in-plane or out-
of-plane for 2D components.
Second, each and every element must be supported by the other one it supports.
As stated above, a perfectly symmetric relationship is needed to distinguish a
reciprocal structure from a simply mutual one.
Fig. 10 Casa Negre, by Josep Maria Jujol. One of the few European examples of a reciprocal structure
conceived for aesthetical purposes. Photo: Jaime Segura, reproduced by permission
20 A. Pugnale, M. Sassone
An example will be helpful here. Let us consider the beam arrangements shown in
Fig. 13. In Fig. 13a, each beam B is simultaneously supported by and supporting other
beams, while beam A directly takes the load and the beams C rest on the ground. This
system is not reciprocal because the supporters and those being supported are not the
same: beams B are supported by beams C, while they sustain, instead, beam A. In
Fig. 11 Chair for a Pedrini
collection, designed by Pino
Pizzigoni, 1948. Photo:
Pizzigoni Archive, PIZ N, card
3. Scanned by Carlo Deregibus
and Alberto Pugnale,
reproduced by permission
Fig. 12 Table prototype
designed by Pino Pizzigoni,
1948. Photo: Pizzigoni Archive,
PIZ N, card 4. Scanned by Carlo
Deregibus and Alberto Pugnale,
reproduced by permission
Structural Reciprocity 21
Fig. 13b, beam B is supported by beam A and supports beam C, while beam A is
supported by beam D, which is also supporting beam C. Such circularity allows us to
state that beam B is, at the same time, supported by and supporting beam A. This is also
visible from the actual arrangement, which does not organize the elements in a
sequence, but rather it forms a ‘loop’. In sequential systems, the positions and roles of
A→ B → CA→ B → C→ D→ A
Fig. 13 The relationship between elements in mutual and reciprocal structures: aleft antisymmetric,
which can be found in the Sindone chapel by Guarini; bright symmetric, which corresponds to the
arrangement of the Puppet Theatre by Ishii (see Fig. 7). The drawings are schematic and not to scale.
Fig. 14 Circularity of force ﬂows in a reciprocal beam. Image: authors
22 A. Pugnale, M. Sassone
the members are related, while in loops the concepts of beginning and end do not exist,
and the positions of the elements are totally interchangeable.
Loops are not only related to the geometrical conﬁgurations, but are also
intrinsically present in the equilibrated force ﬂows. This can demonstrated in the
simplest reciprocal system, i.e., the beam made of two aligned elements as shown in
Fig. 14. Let us assume that the structure is subject to a load condition of two forces
directed downward. A circular ﬂow can be veriﬁed by following how shear (S) and
axial (N) forces act in the different parts of the system.
The same circularity occurs in the conﬁguration shown in Fig. 15 (or in
Fig. 13b). Let us assume a generic load condition applied downward. The four
vertical links only transfer compression, while the inner parts of the elements
transmit shear with the same sign. This produces a circular force ﬂow, or loop.
Neither force values nor the application points affect the presence of a force loop
in any of the examples mentioned above. However, the distribution of external loads
is closely related to it. The scheme shown in Fig. 16 demonstrates that at least one
exception exists, and that exception implies an antisymmetric load condition.
Reciprocity vs. Hierarchy
According to the second property mentioned above, each and every element of a
reciprocal structure must be simultaneously supported by and supporting other
components. It must generate, with the repetition of such a scheme, a pattern in
which all the members of the system play the same role, without differences in
terms of structural behaviour. This situation is clearly in contrast to that of other
structures in which the elements have distinct functions depending on their
respective positions and relationships. The difference between reciprocity and
hierarchy is exempliﬁed by the schemes shown in Fig. 13.
In reciprocal structures, an intuitive understanding of the structural behaviour is
intrinsically lost. This was already clear in the seventeenth century, when, for the ﬁrst
time, John Wallis attempted to calculate reciprocal grillages by following the load
paths within the structure in order to evaluate the internal forces acting in each element
(Wallis 1695; Houlsby 2014). However, the choice of a speciﬁc path (i.e., the sequence
of elements to use to compute the internal reactions) was totally arbitrary. Recent
research by (Kohlhammer and Kotnik 2011) also seems to conﬁrm this.
Reciprocal structures have an intrinsic generative property, which can be
considered more as a reproduction of elementary/minimum patterns, or fans, rather
than instantiations of higher level abstract schemes. The absence of hierarchy
affects the architectural as well as the structural design of such systems. In the
Vito Bertin proposes an alternative description of how structural reciprocity works mechanically. He
states that a reciprocal conﬁguration should present three characteristics: (1) a set of levers that form a
static system, (2) a set of elements that mutually support each other and (3) self-connected components,
i.e. which behave at the same time as elements, nodes and connectors. Visit Vito Bertin’s website for
further details: www.vitobertin.hk.
Structural Reciprocity 23
Western culture in particular, the presence of a hierarchical logic in composition has
always been an integral part of the building experience. Furthermore, many aspects
of the mechanical behaviour of a structure, such as redundancy and robustness, are
also related to hierarchy.
Form-Finding/Morphogenesis Techniques and Design Approaches
The form-ﬁnding of 2D and 3D conﬁgurations of reciprocal structures, as well as of
those based on regular polyhedra, can be performed through different computational
techniques. Illustrative case studies have been provided by Baverel (2004), Popovic
Larsen (2008) and Stacchetti (2005).
At present, three main families of such techniques can be identiﬁed. The ﬁrst is
known as ‘iterative additions’. Starting from the basic conﬁgurations of three or four
elements, these techniques work by adding sets of new bars to the elementary frame
(Proll et al. 2010). The second is related to the use of optimization strategies, such as
genetic algorithms (Baverel et al. 2004) or relaxation (Douthe and Baverel 2009).
Such techniques ﬁnd the frame geometry (the solution) by iteratively reducing a
measured value of the geometrical errors (the ﬁtness function) from a set of non-
compatible conﬁgurations (the tentative solutions). The third models the behaviour
of kinematically undetermined conﬁgurations in order to deﬁne their ﬁnal shapes.
However, in architecture, the use of physical models is probably still the most
diffused way of exploring reciprocal conﬁgurations, as designers can intrinsically
Fig. 15 Circularity of force
ﬂows in a reciprocal grillage.
Fig. 16 Two directional force ﬂows in a reciprocal beam under an antisymmetric load condition. Image:
24 A. Pugnale, M. Sassone
take advantage of the characteristics of reciprocity. With the aid of computation,
instead, a design process can only start from an over-arching reference geometry/
shape, a forced ‘top-down’ approach in which reciprocity does not inspire the
project but is just adapted to it.
In addition to these problems of structural and geometrical form-ﬁnding, a new set of
questions and ﬁelds of experimentation are emerging from the development of digital
fabrication techniques, mostly because the new way in which the components are
produced automatically requires a different approach to their design (Proll et al. 2010).
The Use of Timber: Members and Joints
In reciprocal systems made of elongated members, all the elements are beams
generally joined by simple overlapping. This naturally relates them to timber
construction, in which high ﬂexural strength members are usually available, while
the realization of joints is a complex problem. Furthermore, working with raw
timbers without the need of iron keys or complex connections made reciprocal
construction fast and reliable, as shown by the application to military bridges, such
as the one described by Caesar (Gros and Beltramini 2003).
The idea of joining members by overlapping has also a conceptual relevance: the
overlaps do not require any speciﬁc device, such as bolts and pins which are made
of timber or steel, but they inﬂuence the shape and the organization of the whole
structure. The curvature of a reciprocal bridge made of raw timbers is ruled by the
size of the timbers themselves and by the position of overlaps. Hence, there is an
intrinsic relationship between members and joints (joints are in fact parts of the
members) and the design and detailing is an integral part of the whole conception. In
terms of mechanical behaviour, such as for the number of degrees of freedom, the
effect of friction and of local deformability, and displacement capability, joints
directly contribute to deﬁne the global behaviour of the structure.
Let us compare reciprocal structures in wood with the current industrial detailing
of timber structures, such as glulam. Glulam structures generally present connec-
tions based on the use of additional steel devices, plates, clamps or pins, which are
bolted to timber. Even though they are efﬁcient in terms of constructability and
structural behaviour, steel joints do not appear to belong to the base material (see
Fig. 17). The conceptual and tectonic distance between the two materials is even
more emphasized when the timber members are replaced by composite or even
cardboard pipes, such as in the projects by Shigeru Ban (McQuaid 2003), and there
are no signiﬁcant changes in the steel joint system (see Fig. 18).
Static and Kinematical Determination
According to the ﬁrst property of reciprocity mentioned above, the supported
elements of reciprocal systems cannot come into contact with their supports at the
vertices. This allows the constraint between the supporting and supported members
to be deﬁned in an extensive manner: sliding hinges and prismatic joints become
possible substitutes of rotation hinges, and this offers unique kinematic properties to
the obtained conﬁgurations, which are difﬁcult to predict intuitively.
Structural Reciprocity 25
Preliminary research, carried out through the analysis of the kinematic matrix,
has shown how different constraint patterns can lead to unexpected kinematic
behaviour (Parigi et al. 2009; Parigi 2011; Parigi and Sassone 2011a). Recent
developments have proposed a kinetic system based on the concept of reciprocity,
which is called kinetic reciprocal system (KRS). Through a morphogenetic
procedure, suitable Kinetic Reciprocal Frames are generated with assigned overall
behaviour; the process starts from sets of kinematic parameters that create complex
Fig. 17 Scottish Parliament
designed by EMBT, detail of the
chamber roof structure. Photo:
Yeh-Lun Chou, reproduced by
Fig. 18 Paper tower at the London Design Festival 09, designed by Shigeru Ban. Photo: Tom Parkin,
reproduced by permission
26 A. Pugnale, M. Sassone
geometries of intersecting curves and lines, and which cannot be predicted directly
from the input data (Parigi 2011; Parigi and Sassone 2011b). The latest results of
such a research activity led to the development of a form-ﬁnding tool which is
called the ‘Reciprocalizer’ (Parigi and Kirkegaard 2014; Parigi et al. 2012,2014).
Other conceptually similar research is found in Goto et al. (2011) and Kidokoro and
Use of Planar Elements
When a reciprocal system is designed with elongated elements (i.e., with a set of
components that behave like beams) design efforts are mainly focused on three
aspects: (1) the deﬁnition of the elementary fans that have to be assembled, (2) the
study of their composition possibilities and (3) the selection of the jointing systems.
However, as suggested by the rare natural example of the cocolith, planar
components of different shapes can also be used to transfer forces in a reciprocal
way. The cocolith features circular tiles, but also squares, triangles and more
articulated, or irregular, geometries could also be considered.
In order to guide future morphological research activities, we have distinguished
ﬁve main categories of reciprocal conﬁgurations based on planar components. The
ﬁrst type uses planar elements as ‘thick’ elongated elements. This category includes
all those reciprocal structures in which the planar elements are approached in the
same way as elongated ones. For instance, Werner Blaser designed tables and chairs
made of fans, in which planar timber panels were reciprocally interlocked. A few
buildings have also been designed with thick linear elements; the Serpentine Gallery
2005 by A
´lvaro Siza and Cecil Balmond is probably the most relevant of all. A
second category considers planar elements as groups of elongated elements. This
includes all those reciprocal structures in which the planar elements can be
substituted by fans made of sticks. A simple example is that of triangular tiles that
replace fans of three linear elements each. The tiles can have three different
engagement lengths, whereas only two are possible for elongated elements. A third
family includes all those conﬁgurations in which the planar elements are part of a
truss. A forth category groups the reciprocal systems in which the members can
transmit bending moment via the notches, but are assembled differently than in the
previous three categories. The ﬁfth and last category collects all the possible
remaining conﬁgurations (Baverel and Pugnale 2012,2013).
The research and experiments on reciprocal structures currently underway are the
heirs of an ancient tradition in both the East and West. We hope that the attempt at
synthesis of the state of the art (deﬁnition, terminology, methodologies of form-
ﬁnding and analysis) will allow those engaged in such research to view and present
their efforts in terms of a broad context, overcoming the fragmentation that has
prevented reciprocal structures from receiving the attention they deserve, and
fostering a fruitful exchange of information, resources and results.
Structural Reciprocity 27
See Table 1.
Table 1 List of patents involving the principle of structural reciprocity, arranged according to the release
Author Title Patent No. Date issued Country
Friedrich Zollinger Means of manufacturing space-
covering structures such as
roofs, frame walls, and the
US 1483037 A 5 February 1924
(ﬁled in 1922)
CA 707069 6 April 1965
(no ﬁling date)
Three dimensional reticular
3185164 25 May 1965
(ﬁled in 1961)
Guy Monteilhet Nouvelle structure pour la
couverture des ba
FR 1560454 21 March 1969
(ﬁled in 1968)
Vorrichtung und Verfahren zur
DE 2152580 A 3 May 1973
(ﬁled in 1971)
LU 72572 A1 9 May 1975
(ﬁled in 1975)
Draagconstructie voor huizen,
hallen, en dergelijke
NL 7507054 A 13 June 1975
(ﬁled in 1975)
carrier-grid beams as
prefabricated units hinged
together with zones covered
by prefabricated slabs
DE 2152580 B2 17 July 1975
(ﬁled in 1971)
Charpente pour maisons, halles
BE 835087 A1 16 February 1976
(ﬁled in 1975)
Perfeccionamientos relativos a
estructuras de soporte para
casas, cobertizos y ediﬁcios
ES 438831 A1 17 September
1976 (ﬁled in
DE 2514036 A1 30 September
1976 (ﬁled in
Charpente pour maisons, halles
FR 2306309 A1 29 October 1976
(ﬁled in 1975)
Suhlbetondecke mit als
DE 2152580 C3 27 June 1977
(ﬁled in 1971)
NL 7603046 A 27 September
1977 (ﬁled in
CH 594114 A5 30 December
1977 (ﬁled in
28 A. Pugnale, M. Sassone
See Table 2.
Table 1 continued
Author Title Patent No. Date issued Country
Melvin Crooks Building construction of
4182086 8 January 1980
(ﬁled in 1978)
A building Element for
construction of interlocking
IL 55404 A 27 February 1981
(ﬁled in 1978)
Graham Brown Three-dimensional structures WO 89/08172 8 September
Graham Brown Three-dimensional structures GB 2235479 A 6 March 1991
Graham Brown Three-dimensional structures CA 1320812 C 3 August 1993
(ﬁled in 1989)
Stadium building US 5351449 A 4 October 1994
(ﬁled in 1993)
Geodesic structure US 7389612 B1 24 June 2008
(ﬁled in 2001)
Improvements relating to roof
truss structures (A bracket for
connecting roof joists in a
GB 2483263 A 7
(ﬁled in 2010)
Table 2 List of terms used in scientiﬁc literature to indicate structural reciprocity or those somehow
related to such a principle, sorted according to publication date (historical books, treatises, projects and
patents are not included)
Term and related terms Variants and related
First use in
Lamella beams 1D circular arrays of
lamella beams in a
Dean et al.
In 1997, Dean (1997)
published an article on
lamella beams arranged
in circular patterns
explicitly referring to the
RF studied in
Nottingham by Chilton
Selfsupporting interlocking grids
for multiple applications
Gat at (1978) Coined by Daniel Gat at
the suggestion of A. Ben-
Arroyo, his colleague at
Technion, Israel Institute
Structural Reciprocity 29
Table 2 continued
Term and related terms Variants and related
First use in
Mandala-Dach Mandala roof Chilton and
Probably coined by
Graham Brown, together
with a German
colleague, and ﬁrst cited
by Chilton and Choo in
1992. Later cited by
Olga Popovic Larsen and
Reciprocal frame (RF) Reciprocal structure,
Coined by Graham Brown,
even though the term is
not used in the
descriptive text of his
Serlio ﬂoor Serlio’s device,
Used only by David
Yeomans in his article
Multi-reciprocal grid (MRG) Saidani et al.
Proposed by John Chilton
to Olivier Baverel when
he showed him his early
research on reciprocal
Nexorade Nexor, fan Baverel
et al. (2000)
Coined by Hoshyar
Nooshin to identify the
structures studied by
Olivier Baverel after he
moved to the University
Leverworks Mutually supported
Coined by Vito Bertin to
explain his research on
reciprocal systems. Such
a name emphasizes the
lever principle involved
in structural reciprocity
Multi-reciprocal element (MRE) Rizzuto et al.
Used by Joe Rizzuto to
differentiate his research
from that on MRG
published by Messaoud
Saidani and colleagues
Mutually supported element
Used by Joe Rizzuto in an
attempt to generalize the
principle of reciprocity
Leonardo’s lattice Leonardo grid Roelofs
First coined by Rinus
Roelofs. Also Used by
Duvernoy (2008) and
other authors in the
special issue of the
Nexus Network Journal
on Leonardo’s reciprocal
Selfsupporting framework Proll et al.
Only used by Mischa Proll
and Andreas Gu
their thesis and related
30 A. Pugnale, M. Sassone
See Table 3.
Table 3 The main publications on reciprocity, classiﬁed according to the topic
Morphology and geometry Baverel and Saidani (1999), Baverel et al. (2000,2006), Baverel
and Popovic (2011), Se
´chal et al. (2011)
Polyhedrical reciprocal structures Rizzuto et al. (2001b), Baverel (2007), Rizzuto and Hulse (2007),
Di Carlo (2008), Rizzuto (2009)
Form-ﬁnding and morphogenesis
with computational tools
Baverel et al. (2004), Douthe and Baverel (2009), Brocato and
Morandini (2010), Goto et al. (2011), Parigi and Kirkegaard
(2012), Parigi et al. (2012), Parigi et al. (2014), Peng et al.
Rapid prototyping Parigi and Kirkegaard (2014), Gu
¨nther and Proll (2013), Song-
Ching Tai (2012)
Joints and connections Rizzuto et al. (2001a), Rizzuto and Popovic Larsen (2010)
Structural behaviour Brocato (2011), Douthe and Baverel (2014), Gelez et al. (2011),
Kohlhammer and Kotnik (2011), Saidani and Rizzuto (2004),
´nchez and Escrig (2011)
Parigi et al. (2009), Sassone and Parigi (2010), Goto et al. (2011),
Kidokoro and Goto (2011), Parigi (2011), Parigi and Sassone
(2011a,b), Parigi and Kirkegaard (2012), Parigi et al. (2012)
Materials Pizzigoni (2009), Brocato and Morandini (2010)
Use of planar panels Blaser (1992), Baverel and Pugnale (2012,2013)
Table 2 continued
Term and related terms Variants and related
First use in
Lamella ﬂock Tamke and
Used as the title of a
research project by
CITA, based in
Copenhagen, in which
structures, inspired by
Zollinger’s work, have
Interlocking frame Song-Ching
Used by Alan Song-Ching
Tai to identify short
members that span
across large surfaces and
lock to each other with
Reciprocal Frames are
considered as a structural
sub-family of IFs, which
are not necessarily
Structural Reciprocity 31
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Teaching and didactics Gelez and Saby (2011), Tho
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Pugnale and Parigi (2012)
History Popovic (1996), Di Carlo (2008), Duvernoy (2008), Popovic
Larsen (2008), Williams (2008), Pugnale et al. (2011), Houlsby
Art and sculpture Roelofs (2008)
Leonardo’s grids Popovic (1996), Sa
´nchez and Escrig (2006), Duvernoy (2008), Di
Carlo (2008), Popovic Larsen (2008), Roelofs (2005,2008),
Williams (2008), Sa
´nchez and Escrig (2011)
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Alberto Pugnale is a lecturer in Architectural design at the University of Melbourne, Australia. In 2007,
he won the ﬁfth edition of the IASS HANGAI Prize, related to the study of complex architectural/
structural bodies. He has been a assistant professor at Aalborg University, Denmark (2010–2012), and an
invited lecturer in France and Italy. At present, he is member of the International Association for Shell
and Spatial Structures (IASS) and is a licensed architect in Europe. His research interests are in the
computational morphogenesis of free-form structures, reciprocal structures and history of construction.
Mario Sassone Ph.D. and Assistant Professor at Politecnico di Torino, is member of the 209 Committee
of American Concrete Institute and of the International Association of Spatial and Shell Structures
(IASS). His research ﬁeld mainly concerns the computational analysis of time dependent effects
behaviour in concrete and steel concrete composite structures, and the structural and architectural
optimization of RC shells by means of artiﬁcial intelligence techniques. He is currently working on
bridging the gap between architectural issues and computational approaches to structural and physical
problems. Several members of his research group have been awarded with the IASS HANGAI prize.
Structural Reciprocity 35