The Reciprocalizer: an Agile Design Tool for Reciprocal Structures
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Abstract
This paper presents the reciprocalizer, a grasshopper plugin developed to design reciprocal structures. The plugin can handle the full set of geometric parameters necessary to describe the geometry of reciprocally connected elements in realtime. This makes it an agile design tool for the exploration of the geometrical richness of reciprocal structures.
Keywords
Reciprocal frames Freeform Geometry FabricationIntroduction
The geometry of a reciprocal structure is extremely difficult to predict and control, and it cannot be conveniently described with standard tools offered by commercial CAD software. The geometry of a network of reciprocally connected elements emerges, bottomup, from the complex interaction between all the elements shape, topology and position, and requires numerical solution of the geometric compatibility.
The Geometric Parameters of Reciprocal Structures

the eccentricity e _{ ij } (p1 in the table shown in Fig. 1), which measures the distance between the axes of elements b _{ i } and b _{ j };

the engagement ratios l _{ ij } and l _{ ji } (p2 and p3 in the table shown in Fig. 1), which measure the contact position of elements b _{ i } and b _{ j } along their spans and with respect to one reference end;

the top/bottom position t _{ ij } (p4 in the table shown in Fig. 1), which specifies whether element b _{ i } sits on the top or in the bottom of element b _{ j } with respect to a reference vector r _{ ij }.
These four parameters constitute the complete set of values necessary to determine the geometry of a single connection between two elements. The calculation of those values is performed starting from the geometric entities measured at each b _{ i } b _{ j } connection (Fig. 2).
Nonhierarchical Geometry and the Need of a Geometry Solver
 1.
The position of b _{ j } must be adjusted to restore the geometric compatibility between b _{ j } b _{ k } (Fig. 4b);
 2.
the position of b _{ i } must be adjusted to restore the geometric compatibility between b _{ i } b _{ j } (Fig. 4c);
 3.
the position of b _{ k } must be adjusted to restore the geometric compatibility between b _{ k } b _{ i } (Fig. 4d).
The last step leads us back to the first step, because the position of b _{ j } must be readjusted to restore the geometric compatibility in connection b _{ j } b _{ k } (Fig. 4b), and so on to successive steps recursively.
This example shows how the modification of a single parameter affects the position of each and every element in the configuration. In both large or small configurations, the configuration should be followed by the simultaneous adjustment of the position of all elements in order to maintain geometric compatibility. In reciprocal configurations, the value of the four geometric parameters in each connection b _{ i } b _{ j } contributes to the spatial definition of the overall geometry. Therefore, the geometry of reciprocal structures is a result of the simultaneous complex interaction between shape, topology and position of all elements.
Given the interdependency of the positions of the elements, the geometry of a reciprocal structure can be described as nonhierarchical, since each element contributes equally to the generation of the geometry. This type of geometry cannot be described conveniently with standard tools offered by commercial CAD software including hierarchical, associative parametric modellers whose acyclic directed graphs can describe geometries whose parameters are independent of one another. In contrast, the geometry of reciprocal structures can be determined by setting up a solver that is able to find the geometric compatibility of elements using an iterative method.
The Reciprocalizer
Results
The speed at which the reciprocalizer solves the reciprocal geometry has been gradually improved since the previous implementations in Matlab (Parigi and Kirkegaard 2013a, b; Parigi et al. 2012). The reciprocalizer now allows fast explorations of multiple design alternatives, making it a suitable design tool for both initial and final design stages. For small configurations, the reciprocal geometry is computed in real time. For larger configurations, the time rises to a few seconds. At the same time, the calculation depth and speed can be adjusted according to the design needs.
Figure 6a shows the starting hexagonal mesh used for the generation of configurations in Fig. 6b, c. The starting mesh, together with the parameter values, is a required input for the reciprocalizer.
Fig. 6b shows a domelike reciprocal configuration obtained starting from the hexagonal mesh in Fig. 6a and using a variable engagement length value. The top/bottom position t _{ ij } is set to 1, meaning that each element sits on the top of the supporting one.
Figure 6c shows the adaptation of the hexagonal mesh to an arbitrary freeform surface. The starting mesh is first mapped on the surface. Then the geometric parameters are set, except for the top/bottom position t _{ ij }, which is left unconstrained (t _{ ij } = 0) and chosen by the reciprocalizer to best fit the target surface.
Figure 6d shows the adaptation of a Voronoi pattern to an arbitrary freeform surface. The top/bottom position t _{ ij } is left unconstrained (t _{ ij } = 0) and chosen by the reciprocalizer to best fit the target surface.

the data required to perform a FEM analysis by using FEM plugins available for Grasshopper, i.e., the new nodes’ coordinates and their connection topology;

the data required for the fabrication of reciprocal structures.
Several physical models were successfully built to validate the results from the reciprocalizer (Parigi and Kirkegaard 2013b, 2014).
Conclusions

exploring interactively the influence, often unpredictable, of the parameters’ values on the overall geometry, therefore triggering the exploration of the geometrical richness of reciprocal structures and the emergence of original designs;

producing a realtime feedback of the structural performance, therefore improving our understanding of the influence of the parameters’ values on the structural performance of reciprocal structures, and ultimately enabling structural improvement/optimization;

linking digital design with manufacturing by outputting the necessary data to produce and assemble the elements, enabling innovative CAD/CAM approaches in the digital crafting of reciprocal structures.
References
 Parigi, D. and Kirkegaard, P. H. 2013a. The reciprocalizer: A design tool for reciprocal structures. Paper 147. In Proceedings of Civil Comp 2013 conference (Cagliari, 2013), B.H.V. Topping and P. Iványi, ed. Stirlingshire, UK: CivilComp Press.Google Scholar
 Parigi, D. and Kirkegaard, P. H. 2013b. Efficient design and fabrication of freeform reciprocal structures. Pp 480–487 in Structure and Architecture: Concepts, Applications and Challenges, Paulo J. da Sousa Cruz, ed. London: Taylor & Francis.Google Scholar
 Parigi, D. and Kirkegaard, P. H. 2014. Design and Fabrication of FreeForm Reciprocal Structures. Nexus Network Journal 16, 1 (in this same issue).Google Scholar
 Parigi, D., Kirkegaard, P. H. and Sassone, M. 2012. Hybrid optimization in the design of reciprocal structures. In Proceedings of the IASS Symposium 2012: From Spatial Structures to Space Structures (Seoul, 21–24 May 2012).Google Scholar