ThreeDimensionality in Reciprocal Structures: Concepts and Generative Rules
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Abstract
Reciprocal systems based on superimposition joints, i.e. where unnotched bars sits on the top or in the bottom of each other, could be regarded as being intrinsically threedimensional because of their natural outof plane development. This paper presents seven of these threedimensional configurations, conceived and built by the students of the Master of Science in “Architectural Design” at Aalborg University. They have been developed as an integral part of a 2week workshop, organized and run by the authors during the fall semester 2011. Since physical models are instruments that trigger the exploration of new typologies because of the direct interaction they provide with the designer the students were called to deal with the issue of threedimensionality in reciprocal systems through scale models and actual scale prototypes.
Keywords
Structural reciprocity Spatial structures Morphology Conceptual designThreeDimensionality as a Design Opportunity
This paper deals with a morphological aspect of reciprocal structures based on superimposition joints, that is, their natural outofplane development due to unnotched bars sitting on the top or on the bottom of each other. Such configurations present elements with unaligned axes, and can be regarded as intrinsically threedimensional.
The resulting geometry is not straightforward to predict and control. This is a consequence of the nonhierarchical nature of reciprocal assemblies: the position of each and every element determines, and is at the same time determined not only by the position of the elements immediately adjacent, but also by the position of all the entire set of elements in the assembly. The geometry of a network of reciprocallyconnected elements can therefore be understood as a characteristic that emerges from the complex interaction between the elements’ shape, topology and position (Parigi and Kirkegaard 2014b).
On the other hand, the intrinsic threedimensionality of reciprocal structures can be considered as a design opportunity, and can be approached by using both numerical tools and physical models.
 the eccentricity e _{ ij }, which measures the distance between elements axes, directly dependent on the elements thickness and shape (Fig. 4);
 the engagement length l _{ ij }, that measure the position where each element is supported along the supporting element (Fig. 5);
 the specification of whether element b _{ i } sits on the top or in the bottom of element b _{ j } with respect to a reference vector r _{ j } whose tip indicates the top position (Fig. 6):

the possibility to solve and therefore predict the geometry of nonplanar reciprocal configurations;

the possibility to explore the immense morphological richness of reciprocal structures, by generating multiple configurations in a short time, and their variations based on the changes in the values of geometric parameters.
A different and complementary approach comes from the use of physical models. Physical models are instruments that trigger the exploration of new typologies because of the direct interaction they provide with the designer: complex geometry can be easily conceived and built. One characteristic of the use of physical model is the freedom of the designer with respect to the topology of the configuration; another is the detachment from any codified, structured framework. For this reason, physical models can be regarded as the preferred tool during conceptual design.
In this paper we highlight the potential in the use of physical models by presenting seven configurations conceived and built by the students of the Master of Science program in ‘Architectural Design’ at Aalborg University.
A Framework for a University Design Workshop

identifying a research and design topic (in this case, structural reciprocity);

studying its background, relevance and current development in structural and architectural design (stateoftheart in relation to basic reciprocal configurations, built projects, design tools and strategies using physical models and numerical tools);

highlighting a current design issue, in order to define potential developments for the generation of innovative concepts of structures and prototypes (Pugnale and Parigi 2012).
Development of 3D Reciprocal Systems: Workshop Rules

the structures must be extendable in more than one direction according to the Cartesian coordinate system;

the assemblies must not present any direct reference to surfaces;

the joints must be created through superimposition and unnotched bars.
Such limitations were aimed at concentrating the design efforts towards those spatial configurations which show the intrinsic threedimensionality of reciprocal systems.
Such a system is extendable in one direction by repeating the base fan. Even when we place two or more bridges side by side, in order to cover a larger span in the transversal direction, the system presents a unique longitudinal direction of expandability.
They were excluded from the workshop as the results are often quite predictable, and because they have already been explored by several designers and researchers.
The three rules refer implicitly to reciprocal systems based on the use of elongated/linear elements. The students were provided with timber sticks with a square cross section of 50 mm, the only material allowed for the realization of the prototypes. The maximum length of the elements was fixed at 3 m for reasons of transporting them. However, the construction of longer beams by means of conjoined elements was permitted.
All the structures were first developed through physical scale models, and then converted to actual scale prototypes.
Convergence to Seven Concepts of 3D Reciprocal Structures

the generative rules;

the threedimensionality and the potential for spatial growth;

the relation between structure and sequence of created spaces;

other considerations.
The identification of these features allowed a systematic generalization of the results. It can be helpful where the application and extension of these concepts is desired: an infinite number of variations and recombinations is theoretically possible.
Structure 1: ‘The Bug’
Generative Rules
The new base unit is obtained and consists of two longitudinal, two transversal and four diagonal bars.
ThreeDimensionality and Spatial Growth Potential
Thanks to the four diagonal bars, the base unit can expand in four different directions, and therefore they can be combined into twodimensional arrays. Furthermore, each diagonal bar can be extended both upward and downward, enabling the possibility to create threedimensional patterns of elements that develop in three dimensions at different heights.
Relation Between Structure and Sequence of Created Spaces
The space created by this system can be easily adapted to an architectural context. It can generate different levels, which are connected by sloping paths or ramps, and it can therefore be arranged to fit both related and independent spaces.
Other Considerations
Even though this concept has a great potential for infinite expansion in three spatial directions, the built model also highlighted some difficulties in the assembly: when a bar is placed at the wrong distance or angle with regard to the adjacent bars, any error in its placement not only affects the position of the bar itself, but also the overall geometry of the assembly. The prototype has therefore proved that a high level of construction precision is required.
Structure 2: ‘Matrix’
Generative Rules
The originality of the generative rule lies in the fact of transforming the wellknown twodimensional pattern into its corresponding threedimensional structure. In keeping with this rule, there are no continuous bars that span the entire structure in all three directions, and the threedimensional configuration extends according to the generative rule of the twodimensional pattern.
ThreeDimensionality and Spatial Growth Potential
The structure can be expanded indefinitely in space in every direction.
Relation Between Structure and the Sequence of Created Spaces
Other Considerations
The configuration is characterized by a joint type (Fig. 21) for which the transfer of forces in a reciprocal way can be considered as at the limits of the structural principle.
In fact, the engagement length, which is characteristic of reciprocal configurations, is equal to zero. However, the joint maintains the typical interlocking of reciprocal joints.
Structure 3: ‘Neural Network’
Generative Rules
ThreeDimensionality and Spatial Growth Potential
Relation Between Structure and the Sequence of Created Spaces
Other Considerations
The joint is extremely flexible and functional. However, in order to be rigid, all six composing elements must be present. This forces the use of unnecessary elements when less than six elements converge in the node, and might constitute an overall drawback or limit of the concept. In the built prototype, the problem was solved using very short elements to complete the joints, when less than six bars were converging.
Structure 4: ‘Monkey saddle’
Generative Rules
The bar length always needs to extend sufficiently over the meeting points to accommodate and support further additions.
ThreeDimensionality and Spatial Growth Potential
With a fan of four elements, the system already develops in eight directions. Starting fans made of five and more elements develop in ten or more directions according to the same logic.
The regularity of the final geometry is related to the dimensions of the starting fan––identical elements with equal engagement lengths and eccentricities grow maintaining the same proportions in every direction, while differences in these parameters result in nonregular or deformed structures.
Relation Between Structure and the Sequence of Created Spaces
Other Considerations
The application of this concept seems to be more appealing when geometrical regularity is maintained. In this way, the simplicity of the generative rules is highlighted, resulting in a sculptural structure. However, the control of precision throughout the construction is very difficult in practice because of the interdependence of the elements.
Structure 5: ‘Star Frame’
Generative Rules
ThreeDimensionality and Spatial Growth Potential
Relation Between Structure and the Sequence of Created Spaces
The integrity of the space it creates is a product of the fact that the top elements are interlocked and constitute a whole with the elements that go to the ground.
Other Considerations
Structure 6: ‘Flame’
Generative Rules
ThreeDimensionality and Spatial Growth Potential
Relation Between Structure and the Sequence of Created Spaces
The ‘flame’ configuration creates a conical or cylindrical internal space, simply by taking advantage of the reciprocal superimposition joint and variations in the connection angles.
In the built prototype, the tower converges in a single node after four iterations of bar additions; it does not offer the possibility to continue indefinitely, but further experimentations, together with more sophisticated construction techniques, could lead to the realization of a indefinitely expandable structure.
Other Considerations
The transformation of the scale models into a real prototype showed that the geometry is extremely difficult to control, because of the interdependence of the structural elements. Furthermore, the lower layers of bars are structurally more stressed than the higher ones. This can imply the use of elements with a variable cross section, adding a further degree of complexity to the development of this concept.
Structure 7: ‘Wave’
Generative Rules
ThreeDimensionality and Spatial Growth Potential
Even though the ‘wave’ derives from a surfacelike assembly, it can be considered as a 3D concept because quadrilateral and triangular fans develops, respectively on perpendicular planes.
The starting gridshell, made of fourelement fans, always guides the spatial growth of the ‘wave’. Theoretically, both long roofs, such as in the case of the new Milan Trade Fair by Studio Fuksa, and small pavilions can be developed by applying this concept.
Relation Between Structure and the Sequence of Created Spaces
Other Considerations
During the workshop, the presence of four bar fans generated stability problems in the initial gridshell structure. Even the addition of new threebar fans was not enough to reach a stable configuration, and the structure needs further bracing systems to be practical.
Conclusions
In this paper, seven concepts of reciprocal structures built using unnotched standardized elements have been presented. The different design proposals have been conceived and developed during a construction workshop at Aalborg University, with the aim of exploring the intrinsic threedimensionality given by the superimposition joint.
The generative rules of the built prototypes, as well as the spatial growth potential of the design concepts have been here described in order to provide a systematic overview of the workshop results.
The schemes proposed were developed in a researchbased environment, and their scope is intended for use as the starting point for the development of further variations of the same concepts or some real application in an architectural context.
Notes
Acknowledgments
The authors would like to thank the Department of Architecture, Design and Media Technology, Aalborg University for providing us working spaces and funds, the students of the 1st semester of the Master of Science in ‘Architectural Design’ at Aalborg University for their intense participation to the workshop, Prof. Poul Henning Kirkegaard for encouraging us in organizing this activity and Nathalie Balfroid and Marie Frier for their active collaboration during the whole process.
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