The principle of structural reciprocity: history, properties and design issues
Assistant Professor, PhD
Assistant Professor, PhD
Poul H. KIRKEGAARD
Assistant Professor, PhD
Politecnico di Torino
This paper deals with the principle of structural reciprocity, considering its origins in both
Occidental and Orient culture and aiming to highlight the definition, main peculiarities and
interesting aspects of such concept referring to its application to the world of construction. Issues
spanning from history, form-finding and morphology, structural behaviour and construction
techniques are discussed in the paper, which should be considered as a starting point to stimulate
future research and design directions/approaches.
Keywords: reciprocity, spatial structures, timber constructions, form-finding, morphogenesis,
history of construction.
The principle of reciprocity in structural design and construction, i.e. the use of load bearing
elements to compose a spatial configuration wherein they are mutually supported one another, has
been known since the antiquity. Evidence of its application can be initially found in Neolithic pit
dwellings, Eskimo tents and Indian tepees, as documented by Popovic Larsen , as well as
during the Roman Empire, when Julius Caesar used it for the construction of a bridge on the Rhine,
which was made of interlocked timber elements (described in his Commentaries on Gallic and Civil
Wars) with the purpose of simplifying the joints among elements, as also reconstructed and drawn
by Palladio . However, apart from these ancient applications, independent one another and
developed worldwide, the principle of reciprocity has been then autonomously studied and used, on
the basis of different needs and purposes, in both Occidental and Oriental culture for centuries, until
it became a research topic for academics. During the last decades, they started a new set of works
related to structural, geometrical and constructive issues of 'reciprocal' structures.
In this framework, the present paper aims to present a historical-critical overview on the principle
of reciprocity in the world of construction, highlighting its origins in different cultures with
differences and similarities, involved research fields and architectural design issues.
2. Different roots and curious similarities of a fragmented history
2.1 Reciprocity in the Occidental culture: timber and short-beams
In Europe, the concept of spanning distances longer than the length of the available timber beams
was the main reason for the use and development of the principle of reciprocity until the last
century. In spite of the fact that there was other techniques to deal with large span issues and short-
beams, such as that used in the roof structure of the 'Square Hall' building in Old Nisa in
Mesopotamia, dated around 2000BC and described by Pizzetti and Zorgno  (after also used by
Guarini for the design of his domes), different architects, scientists, mathematicians and
constructors reasoned about this problem without relations and documented continuity on one
another. Villard de Honnecourt drew in his sketchbooks some grillage assemblies for the
construction of floors inspired to the principle of reciprocity between 1225 and 1250 , then
Alexander designed and built the Lincoln cathedral using reciprocal supports between 1220 and
1235, as reported by Hewett . Leonardo Da Vinci explored at least five different spatial
configurations based on the principle of reciprocity, as shown his sketches inside his "Codex
Atlanticus"  , practically spanning through the main regular and non-regular 2D and 3D
geometrical configurations. Furthermore, Sebastiano Serlio discussed the construction of planar
floors with short beams in his first book on architecture (in Italian), dated 1566  , and John
Wallis wrote in his "Opera Mathematica"  about different kind of planar configurations of
structures with reciprocally supported elements, also providing examples of calculation worth to be
further investigated. Thus, Émy, who was professor of Fortification at the Royal Military School
Saint-Cyr, dedicated his efforts in developing a book about Carpentry, the "Traité de l'art de la
charpenterie" , in which some examples of reciprocal structures can be found, also for the
construction of the so-called 'flat-vaults' invented by Joseph Abeille and Sébastien Truchet in 1699
, and finally Rondelet  dealt with the principle of 'short beam' for the construction of floors
in his treatise dated 1810, in a similar way to what was presented in 1837 by Thomas Tredgold in
his "Elementary Principles of Carpentry" .
2.2 Reciprocity in Orient: interwoven structures and symbolism
In the Oriental culture, the interest in the use of reciprocal structures derived from different
inspiration concepts. On the one hand, especially in China, the use of interwoven strips of bamboo
for the construction of baskets is an old tradition that has been then transferred to objects of larger
scale, such as bridges of which the first example is the so-called 'Rainbow Bridge' in Shandong
. On the other hand, the religious concept of Mandala and the symbolic shape of the
'magic circle'  has been used as a direct reference for the development and construction of
simple circular 'fans' made of reciprocally supported elements, used in many Buddhist temples all
around Asia. Unfortunately, written documents concerning that period can only be found for the
period after 1275, in which year Marco Polo arrived in China and therefore at the end of the Song
Dynasty. However, in the last decades different Asian architects, such as Kazuhiro Ishii, Yasufumi
Kijima and Yoichi Kan, took advantage of this principle for their constructions, as described by
Popovic Larsen  and Gutdeutsch .
2.3 The bases of a scientific approach
It should be underlined that Leonardo Da Vinci was the only one in Occident who studied
reciprocal structures in their potential of being elaborated as complex geometrical configurations,
also in the three-dimensional space . John Wallis is also worth to be cited as the first author who
approached the principle of reciprocity with a modern scientific approach . He considered a
precise and well-defined problem related to covering a long span with a flat grillage of reciprocal
short-beams, and he proposed a rational and logical calculation of the forces acting on each element
of the proposed configurations. Therefore, he dealt with the problem in mathematical and structural
terms, but also giving inspiration and focusing on practical aspects of construction of timber frames.
In summary, the principle of reciprocity has not had a linear history and the evidences of its
knowledge and use around the world seems to be largely unrelated to one another. However, a
common point in the development and application of the principle of reciprocity is the strict
relationship with the use of timber as constructive material, in both Occidental and Oriental culture,
but it was more a practical and constructive issue in Europe, aimed to the development of flat
configurations, while it was also a spiritual reason in Asia, with the realization of three-dimensional
structures recalling magical and religious symbolic shapes.
2.4 The years of patents and the development of research
In the last century, the principle of reciprocity continued to stimulate the interest of designers, and
for the first time it became a topic of interest in the field of scientific research. Ten different patents
based on the principle of MSE was registered between 1965 and 1992, starting with the foldable
structure by Piñero (filed in 1961 in USA, published in USA and Canada in 1965) presenting a
MSE spatial configuration of bars . Thus, the reinforced concrete ceiling made of
prefabricated units hinged together by Walle and Prinz (filed in 1971 and published in 1975) ;
the MSE geodesic construction proposed by Bijnen in 1977 ; the building element for
construction of interlocking grids by Daniel Gat in 1981 , also described in a scientific article
previously appeared in 1978 in "Architectural Science Review" ; the interwoven construction
developed by Melvin Crooks in 1978 (and patented in 1980) made of flexible so-called 'A'-shaped
elements, intended to be used for three-dimensional structures such as domes ; the three-
dimensional structure of reciprocally supported timber beams connected with notched joints
registered by Graham Brown in UK, Canada and as international patent between 1989 and 1993
, and finally the Rotegrity system, conceived as a interwoven structure made of flexible
elements based on the principle of reciprocity, developed and patented in 1992 by the Simple
Science Toys based in Eugene, Oregon, USA.
Focusing on scientific publications in the last decades, an increasing number of papers started to
appear in conference proceedings and journals. As already mentioned, the first was written by
Daniel Gat referring to his SIGMA system . Thus, different researches based on the principle of
reciprocity have been developed and all of them are referring to the initial investigation by Chilton
et al., published in 1992, 1994 and 1995 . These authors were the first using the
term 'Reciprocal Frame' in a scientific paper (a first introduction to terminological heterogeneity in
the scientific research on reciprocity is provided by Popovic Larsen ) and they also provided a
precise definition and description of the principle restricted to closed circuits of elements. In fact, in
that period John Chilton started a collaboration with Graham Brown at Nottingham University, who
recently patented a reciprocal three-dimensional structure, previously mentioned, and decided to
increase his level of understanding of its structural behaviour. At the same university, the first PhD
thesis on that topic was published. It was written by Olga Popovic and completed in 1996 .
Chilton was the advisor of the research involving historical references and architectures based on
the RF principle, also focusing on geometrical, structural and construction considerations.
The following papers deal with different aspects of structures designed on the principle of
reciprocity, and can be roughly classified in: (1) related to morphology and geometry of spatial
structures based on reciprocity , also in their polyhedrical configurations ;
(2) dealing with form-finding and morphogenesis of such structures with computational tools
, recently also combined with the rapid prototyping of the timber elements; (3)
concerning joints and connections of reciprocal structures ; (4) focusing on the structural
behaviour ; (5) investigating the kinematic potential in order to generate adaptive
structures/architectures  mainly inspired from Leonardo Da Vinci's work 
and on the Piñero's patents ; (6) working on material and sections for their realization ,
(7) or presenting and discussing architectures, constructions and prototypes .
2.5 Reciprocity in contemporary art, architecture and industrial design
Considering realized architectures in the last century, some famous and unrelated examples
implementing the principle of reciprocity have been built in order to covering long spans, such as in
the Mill Creek Public Housing project by Kahn designed in 1952-53, in the Berlin Philarmonie by
Scharoun, built in 1960-63, and in a salt storage building in Lausanne by Atelier Gamme
Architecture, realized in 1989. Some structures have been also realized with the aim of giving shape
to spiritual philosophies, therefore involving symbolic meaning, such as in the Casa Negre by Josep
Maria Jojul (1915-26) and in all Graham Brown and previously cited Japanese constructions.
Also some experimental pavilions have been constructed, such as the Forest Park structure by
Shigeru Ban and ARUP AGU, or the two interesting experimentations uniquely designed by Cecil
Balmond and ARUP AGU - the H-edge pavilion realized together with the students at Penn
University and the Serpentine Gallery 2005, designed in collaboration with Siza. In both examples,
the principle of reciprocity has been explored in its architectural potential of being used with
different materials, element sections, joints and planar or three-dimensional configurations,
providing a fundamental evidence of much this pseudo-typology should be still explored by
In the field of industrial design, Pino Pizzigoni realized during the Fifties timber and marble chairs
and tables made of reciprocally supported elements, called by the architect as structures with
'interlocked members' emphasizing the attention on the joints (See Pizzigoni Archive: PIZ N, 1948)
. Finally, the sculptors George Hart and Rinus Roefols  have been interested in the principle
of reciprocity, using it for the conception of some of their geodesic domes and complex three
3. Peculiarities and interesting aspects of reciprocity
The term reciprocity derives from the Latin 'reciprocus' which is composed by the two parts 'recus',
meaning backwards, and 'procus', translated as forwards. The etymological significance of
reciprocity is therefore 'moving backwards and forwards', implying the practice of exchanging
things with others for mutual benefits. Such definition stresses the obliged, forced return of a certain
action, slightly differing from 'mutuality', which refers to a voluntary exchange, or 'alternative',
which focuses more on the time aspect of acting at regular intervals.
In the world of construction, the application of the principle of reciprocity requires:
- the presence of at least two elements allowing the generation of a certain forced interaction;
- that each element of the composition must support and be supported by another one;
- that every supported element must meet its support along the span and never in the vertices (in
order to avoid the generation of a space grid with pin-joints).
A structure or spatial configuration respecting such conditions has some peculiar characteristics and
interesting aspects which have to be taken into account during design but also stimulate the
exploration of their expressive potential in architectural terms.
3.1 Reciprocity vs. hierarchy
According to the second requirement of a structure developed on the principle of reciprocity, each
element of the configuration must work, at the same time, as support and supporter of other
elements, generating, with the repetition of such scheme, a pattern wherein all the components of
the system play the same role, without differences in terms of structural hierarchy. This situation is
clearly opposite to that of other structures wherein the elements have distinct functions depending
on their respective positions and relations. For instance, a frame is 'read' as girders and columns, in
a truss the upper and lower chords transfer bending while the web elements transfer shear, even the
well-known theory of shells shaped as revolution surfaces refers to parallels and meridians to
interpret the tie effect of the formers on the latter, giving the possibility to the designer of 'reading'
the structural hierarchy of the composition, and thereby distinguishing between main and secondary
elements, vertical and lateral systems, bracing, slabs, etc. In structures based on the principle of
reciprocity the possibility of an intuitive understanding of the structural behaviour is intrinsically
lost. It was already clear in the past, when Wallis was studying reciprocal grillages , that
following the load paths inside the structure is a good starting point to evaluate the internal forces
acting in each element. However, it was equally clear that the path choice, i.e. the sequence of
elements to use for computing the internal reactions is totally arbitrary. Recently such issue has
been also investigated by Kohlhammer and Kotnik , confirming previous studies. Reciprocal
structures seem to have an intrinsic 'generative' property, being understandable more as a
reproduction of elementary/minimum patterns (fans) rather than instantiations of higher level
The prevalence of reciprocity on hierarchy has consequences both on the architectural conception
and structural design levels. The absence of a hierarchical logic in the composition of structural
elements is relatively far from the Occidental culture, in which from gothic cathedrals to
Calatrava’s bridges the discovery of load paths and the compression of the load bearing
mechanisms are an integral part of the architectural experience, and many aspects of the mechanical
behaviour, such as redundancy and robustness are related to this aspect, as explained in the paper by
Balfroid et al. .
3.2 Form-finding/morphogenesis techniques and design approaches
The definition of simple two- and three-dimensional configurations of reciprocal structures, as well
as of those based on regular polyhedra, can be found by means of different geometrical techniques,
as clearly described by Popovic Larsen , Baverel  and Stacchetti . However, the use of
physical models is probably the most diffused tool to 'find' the form of reciprocal configuration due
to its more direct interaction with the designer who can better see the assembly of elements in space.
However, the adaptation of the principle of reciprocity to more general reference shapes requires
the use of some computational aid in order to predict the geometrical configuration of the frame.
Furthermore, another degree of difficulty in the description of its final geometry could be given by
the use of non-regular frames, characterized by different unit cells in shape and dimensions, or non-
regular frames with non-regular topology, that also implies a variation in the dimensions of bars and
position of joints.
At present, three main form-finding techniques has been studied in order to deal with these
problems. The first is known as Iterative Additions and is based on the addition of set of bars to the
frame, starting from the basic configurations of three or four elements ; the second is related to
the use of optimization strategies, such as Genetic Algorithms  or Relaxation , which
iteratively find the spatial definition of the frame (the solution) by reducing a measure of
geometrical errors (fitness function) from a non-compatible configurations (tentative individuals,
population); the third technique implies the modelling of the behaviour of kinematically
undetermined configurations in order to define their final shape.
It should be underlined that the definition of reciprocal spatial configurations on the basis of
physical models is always based on taking advantage of the main characteristics of reciprocity,
allowing the designer to follow a so-called 'bottom-up' approach bringing to spatial complexity
from the composition and modification/deformation of elementary/minimum configurations. On the
contrary, the use of computational techniques can only start from an over-imposed reference
geometry/shape, in a forced 'top-down' approach in which the final configuration is not a result of
the peculiar properties of reciprocity and could then, from the ontological point of view, by another
structural typology such as grid-shell. However, such discussion about design approaches should be
developed in a more general way, comparing for instance the design of reciprocal configurations
with that of more consolidated typologies such as grid-shells with planar quadrilateral elements.
Also in that case, the use of geometric rules of translation and scaling of curves, defined by Schaich
and Shober, could be considered as a bottom-up approach, while the 'a priori' definition of a
reference surface, and then the application of an optimization procedure, is always a top-down
design process. In addition to this geometrical problems of form-finding of reciprocal structures, a
new set of questions and potential experimentation fields are arising from the development of
digital fabrication techniques and CNC machines, which require a completely different approach to
the design of the frame elements since the new way of component production .
3.3 The use of timber: connections and joints
A structural system where all the elements work as beams subjected to shear and bending is
naturally related with timber constructions and technologies, which have been always preferred, in
the Occidental tradition, for the application in works where the functionality is the prevailing aspect,
such as simple bridges, cranes and machinery, slabs and roofs, instead of masonry, which has been
mainly used, on the contrary as reference material for significant buildings and infrastructures. In
this context, the principle of reciprocity has been mainly conceived and used to solve practical
construction problems, and by means of that, for instance, simple bridges can be rapidly built
without the need of iron keys or complex joints. In all cases, the superposition joint, with or without
a friction component, is an essential part of the timber system, and its design can be refined, as
when a small over-thickness is suggested to correct slab deflections with a suitable camber .
The strict relationship between timber elements and reciprocal joints deserves to be compared with
current glulam construction, where joints are frequently realized with steel plates, clamps, or pins,
connected to timber elements by bolts or needles. Such kind of constructions, efficient form the
point of view of constructability and structural behaviour, keep the joints separated from the base
material, suggesting that it could be replaced by other materials, as for instance the architect
Shigeru Ban does by using cardboard pipes. In reciprocal structures, connections are parts of the
members themselves and the design and detailing cannot be separated by the whole conception.
However, the mechanical behaviour of joints, in terms of degrees of freedom, presence of friction,
local deformability, displacement capability, etc, directly contribute to define the global behaviour
of the structure .
3.4 Static and kinematical determination
According to the previously stated third requirement of reciprocity, in this kind of structures
supported elements cannot meet their supports in the vertices. Such characteristic allows to define
the constraint between supporting and supported members in a wider way, considering sliding
hinges and prismatic joints as possible substitutes of rotation hinges, and offering to configurations
obtained with such a modelling of constraints unique kinematic properties which are difficult to
intuitively predict. A preliminary research, carried through the analysis of the kinematic matrix,
showed how different constraint patterns can lead to unexpected kinematic behaviour, which is
worth of interest in the context of this kind of structures . More recent developments 
propose a kinetic system based on the concept of reciprocity called Kinetic Reciprocal System
(KRS), i.e. a specific morphogenetic process, aimed to the generation of suitable KRFs with
assigned overall behaviour starting from sets of kinematic parameters, that involve the creation of
complex geometries of intersecting curves and lines which cannot be predicted directly from the
4. Conclusions and further developments
The principle of reciprocity has been studied and applied worldwide, starting from different
inspiration concepts and intended to achieve distinct purposes. However, some relevant historical
references still need to be further investigated for their scientific approach to the topic, such as the
"Opera Mathematica" by Wallis , and a set of architectural issues characterizing this principle
can offer new ideas for future research projects, such as the question of connections and joints in
timber reciprocal configurations, which stimulate the use of the material as a 'whole' in the
construction, in terms of defining elements and connections, and the possibility of studying
reciprocal structures as assemblies of elements with a high interest kinematic potential.
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