Introduction

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).

Fig. 1
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An illustrative example of a simple reciprocal structure. Image: authors

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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 art.

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 structures.

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.Footnote 1

Several architects, builders, mathematicians and scientists all reasoned separately about this problem, without evidence of being in contact with one another.Footnote 2 Between 1225 and 1250, Villard de Honnecourt drew some short-beam arrangements 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).

Fig. 2
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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. b Transcription of drawings A1 and A2. c Transcription of drawing B1. d Transcription of drawing B2. e Transcription of drawing B3. f Transcription of drawing C2. g Transcription of drawing E1. h Transcription of drawing E2

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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, Émy, who was professor of Fortification at the Royal Military School in Saint-Cyr, included some reciprocal examples in his “Traité de l’art de la charpenterie” (Émy 1837). Reciprocity was also present in the so-called ‘flat-vaults’ described by Frézier (1737), but which were invented by Joseph Abeille and Sébastien 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-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. 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).

Fig. 3
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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

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Fig. 4
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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

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Fig. 5
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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 by permission

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Fig. 6
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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

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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).

Fig. 7
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Seiwa Bunraku-Kan, Puppet theatre complex designed by Kazuhiro Ishii. Photo: Kentaro Tsukuba, reproduced by permission

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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 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 (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 Piñero, which 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 Piñero by Escrig (1993).

Fig. 8
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Emilio Perez Piñero’s patent. Detail of the node Images: United States Patent and Trademark Office, reproduced by permission

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Fig. 9
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Emilio Perez Piñero’s patent. Superimposition of bars allows and controls the kinematic movement of the assembly Images: United States Patent and Trademark Office, reproduced by permission

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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).Footnote 3 However, it was the 3D closed circuit of sticks patented by Graham Brown that 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”.

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.

Fig. 10
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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

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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 by Álvaro 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.

Fig. 11
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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

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Fig. 12
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Table prototype designed by Pino Pizzigoni, 1948. Photo: Pizzigoni Archive, PIZ N, card 4. Scanned by Carlo Deregibus and Alberto Pugnale, reproduced by permission

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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.

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. 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 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.Footnote 4

Fig. 13
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The relationship between elements in mutual and reciprocal structures: a left antisymmetric, which can be found in the Sindone chapel by Guarini; b right symmetric, which corresponds to the arrangement of the Puppet Theatre by Ishii (see Fig. 7). The drawings are schematic and not to scale. Images: authors

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Force Flows

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.

Fig. 14
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Circularity of force flows in a reciprocal beam. Image: authors

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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.

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Circularity of force flows in a reciprocal grillage. Image: authors

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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.

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Two directional force flows in a reciprocal beam under an antisymmetric load condition. Image: authors

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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 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 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 connections 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).

Fig. 17
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Scottish Parliament designed by EMBT, detail of the chamber roof structure. Photo: Yeh-Lun Chou, reproduced by permission

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Fig. 18
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Paper tower at the London Design Festival 09, designed by Shigeru Ban. Photo: Tom Parkin, reproduced by permission

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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.

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 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 Goto (2011).

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 Álvaro 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).

Conclusions

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.