# Design and Fabrication of Free-Form Reciprocal Structures

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## Abstract

Due to their non-hierarchical nature, the geometry of reciprocal assemblies cannot be described conveniently with the available CAD modelling tools or by hierarchical, associative parametric modellers. The geometry of a network of reciprocally connected elements is a characteristic that emerges, bottom-up, from the complex interaction between all the elements’ shape, topology and position, and requires numerical solution of the elements’ geometric compatibility. A computational method, the “Reciprocalizer”, has been developed by the authors to predict and control the geometry of large networks of reciprocally connected elements, and it has been now perfected and included in a larger procedure that can be regarded as an extremely flexible and capable design tool for the generation of free-form reciprocal structures. The design tool has been applied for the design and realization of a free-form structure composed of 506 round, un-notched wooden elements with a diameter of 22 mm. This paper focuses on the geometry of reciprocal systems and the unique issues of fabrication posed by such assemblies.

## Keywords

Reciprocal frames Free-form Geometry Fabrication## Introduction

- i.
the presence of at least two elements allowing the generation of a certain forced interaction;

- ii.
that each element of the composition must support and be supported by another one;

- iii.
that every supported element must meet its support along the span and never in the vertices (Pugnale and Kirkegaard 2011).

When a superimposition joint is used (i.e., un-notched bars sit on the top and on the bottom of each other, as shown in Fig. 4a or 5a) reciprocal structures develop naturally out-of-plane because the elements’ axes are not aligned; for this reason they can be defined as intrinsically three-dimensional.

Due to their non-hierarchical nature, the geometry of a reciprocal assembly is extremely difficult to predict and control, and it cannot be described with available CAD software or by hierarchical, associative parametric modellers. The geometry of a network of reciprocally connected elements is a characteristic that emerges, bottom-up, from the complex interaction between all the elements shape, topology and position.

The three-dimensionality can be regarded as a design opportunity, for the possibility to create three-dimensional configurations with simple joints and standardized elements. A computational method called the “Reciprocalizer” has been developed by the authors to predict and control the geometry of large networks of reciprocally connected elements (Parigi and Sassone 2012; Parigi and Kirkegaard 2013, 2014).

## Geometry of Reciprocal Structures

*b*

_{ i }and

*b*

_{ j }.

- i.the engagement length
*l*_{ ij }, which measures the position where each element is supported along the supporting element (Fig. 3a) - ii.the eccentricity
*e*_{ ij }, which measures the distance between elements axes, directly dependent on the elements thickness and shape (Fig. 4a); - iii.the specification of whether element
*b*_{ i }sits on the top or on the bottom of element*b*_{ j }with respect to a reference vector*r*_{ j }whose tip indicates the top position (Fig. 5a).

It is possible to observe the effect of changing each of the geometric parameters in the morphology of a three-element fan (Figs. 3b, 4b, 5b).

The out-of-plane deviation increases with the increasing of the eccentricity value *e* _{ ij }; on the contrary, for large values of the engagement length *l* _{ ij }, the out-of-plane deviation is low, and it greatly increases if the engagement length value is close to zero.

The top/bottom position involves a topological change in the nature of the joint. In a three-element fan, four possible combinations of top/bottom positions determine four distinct characteristic geometries. When all elements are placed on the top or in the bottom of each other respectively a dome-like or reversed dome-like geometry is created. On the other side, when elements connect with mixed top/bottom positions, geometries that will prove to be fundamental for the creation of free-form shapes are created.

As a consequence of the non-hierarchical nature of reciprocal assemblies, those parameters are not independent, and any change in one parameter value should be followed by the simultaneous adjustment of all the other parameters values in the assembly in order to maintain the geometric compatibility. This is why the geometry of a reciprocal assembly cannot be conveniently described with available CAD modelling tools or by hierarchical, associative parametric modellers. The geometry of a network of reciprocally connected elements requires the use of iterative numerical methods to solve the physical interplay between all sticks in the assembly.

Physical models can be used at an exploratory level, to discover new configurations (Parigi and Pugnale 2014). However, the use and development of the typology requires a design tool that allows for predicting and controlling the geometry of large networks of reciprocally connected elements. The Reciprocalizer mentioned earlier is capable of handling different families of reciprocal systems based on the joint type.

### Classification and Morphology of Families of Reciprocal Structures Based on the Joint Type

In the simplest possible reciprocal configurations, each locally supported element sits on the top of the supporting element with a superimposition joint. When structures are composed with this rule, the friction of the joint is sufficient to keep them stable, and thus they can be erected by stacking one element on top of another. The arrangements sketched by Leonardo da Vinci in the *Codex Atlanticus* belong to this family (see Houlsby 2014). However, the rule places severe constraints on which geometries can be obtained; generally speaking, the structures obtained have positive Gaussian curvature.

When considering practical applications, a friction-only joint should be avoided in favour of a bilateral joint to improve the robustness of the assembly. When a bilateral joint is used instead of a friction-only joint, the structures continue to exhibit all of the properties that characterize reciprocal system defined in “Introduction”. When a bilateral joint is used, the supporting element can either sit on the top or on the bottom of the supported element; the joint will exhibit prevalent compression when sitting on the top and prevalent tension when sitting on the bottom of the supporting element. The use of both top/bottom support position activates one of the most intriguing characteristics of reciprocal systems: the possibility to create free-form shapes with standardized elements.

Additionally, when a bilateral joint is used elements could be arranged without the superimposition joint, with their axis aligned. Historically, the proposals for a reciprocal roof by Sebastiano Serlio belong to this family (see Houlsby 2014).

- i.
friction-only joint (top support position);

- ii.
bilateral superimposition joint/(top or bottom support position);

- iii.
bilateral joint with aligned elements axes.

Each of these typologies presents interesting and unique properties, in terms of both geometry and possible practical applications.

### Reciprocal Structures Families Obtained with the Reciprocalizer

The architecture of the “Reciprocalizer” algorithm was set up for the highest generality, to provide the possibility to generate configurations in any of the three families of reciprocal structures just listed.

The Reciprocalizer is thus capable of generating configurations in all three conditions: where each supported element meets the supporting element in the top position, where each supporting element is met indifferently in the top or in the bottom position, and where elements join with their axes aligned.

*e*

_{ ij }), engagement length (

*l*

_{ ij }), and top/bottom support position for each

*b*

_{ i }

*–b*

_{ j }connection—and adjusting the values according to the goal configuration, which may fall into one of the three families described in “Classification and morphology of families of reciprocal structures based on the joint type”.

In solution 1 (Fig. 6b), the support position is set for all connections *b* _{ i } *–b* _{ j } to be the top position. Therefore, each locally supported element meets the supporting element in the top position. The resulting configuration can be erected with friction-only joints, stacking one element on the top of each other.

In solution 2 (Fig. 6c), the algorithm ignores the specification of the top/bottom support position. Therefore, any top/bottom position is accepted, and the only condition set at each connection *b* _{ i } *–b* _{ j } is that the geometric compatibility is assured by the contact position. As a consequence, the algorithm computes the reciprocal configuration which is spatially the closest to the starting configuration geometry, and it will generate configurations with mixed top/bottom support positions. With those settings the algorithm exhibits the ability to adapt reciprocal assemblies to any free-form shape. A requirement for such configurations is that the joint must be bilateral if the supported elements connect the supporting element in the bottom position.

In solution 3 (Fig. 6d), elements join with their axes aligned. In this case, the value of eccentricity is set to 0, regardless the elements shape. In those configurations, the elements’ shape does not affect the eccentricity value, and the choice of the top/bottom position parameter does not apply. This solution has interesting practical applications for the possibility to create flat configurations, and for the independence of the elements’ shape from the overall geometry of the assembly.

## Issues of Fabrication

Despite the apparent simplicity of the superimposition joint, when large, controlled networks of elements must be assembled, reciprocal structures require remarkable precision and uncommon expedients for their fabrication, because the overall geometry of the assemblage is the result of the complex and simultaneous interaction between all elements’ position and size. Therefore, the values of the geometric parameters at each connection are dependent on the values of all the others in the assembly. The practical consequence is that the elements must be assembled according to the exact parameters values, because if even a single connection sits outside a small tolerance, all the other elements’ positions are influenced and an overall modification of the geometry is determined. The extent of these modifications might require adjustment of all the other connections in the assembly in order to restore the geometric compatibility. Therefore, if one element is misplaced during assembly, it can cause a geometric incompatibility in the other elements of the assembly assembled according to the design parameters values.

This requirement for precision is challenged by the fact that each joint parameter is unique in free-form geometries, and in principle the elements’ spatial position requires six coordinates in order to be determined (either the position of the two end points with their coordinates *x*, *y*, *z*, or alternatively, the coordinates of one end point of the element and its rotation around the *x*-axis *φ* _{ x }, around the *y*-axis *φ* _{ y }, around the *z*-axis *φ* _{ z }).

For the realization of the reciprocal mesh dome of the Rokko Observatory, whose architectural concept by Architect Hiroshi Sambuichi had been constructively and structurally developed by ARUP (Kidokoro and Goto 2011), the complexity of assembling a three-dimensional reciprocal structure was immediately evident to the engineers and the contractors. The complexity was tackled by creating the geometry of the dome from the repetition in a symmetrical circular array of an identical “master” slice. This strategy enabled the possibility to create a single “gig” or temporary scaffolding for the master slice that supports the steel elements in their final position before they are fixed through soldering.

### The Scaffolding-Free Fabrication

We propose a method for fabrication of free-form reciprocal structures that does not require a temporary scaffolding to keep the elements in their final position before they can be fixed. The method is based on considerations regarding the kinematical determinacy of reciprocal systems, with a particular focus on networks composed of three-element fans.

The geometry of a network of fans can be understood as being formed at the local level of the rigid fan units, which, when connected among themselves into a network, determine the global geometry.

Therefore, if the network is assembled with the subsequent addition of individual elements, starting from an arbitrary position, each fan retains, upon completion, the final shape and position with the adjacent fans. The geometry is assembled bottom-up, by enabling, at the local level of the fan, the complex interaction between the elements shape, topology and position that is characteristic of reciprocal systems.

Our ability to build the overall geometry relies uniquely on our ability to precisely assemble each element with the adjacent one. In case circular elements are used, each connection between elements *b* _{ i }–*b* _{ j }, is uniquely determined by a single contact point for each of the two connecting elements, whose position should be identified along their surface, and that we call *P* _{ ij } on element *b* _{ i } and *P* _{ ji } on element *b* _{ j }.

*P*

_{ ij }and

*P*

_{ ji }are coincident. A statically determinate three-element fan is formed upon completion of the connections between elements

*b*

_{ i }–

*b*

_{ j },

*b*

_{ j }–

*b*

_{ k },

*b*

_{ k }–

*b*

_{ i }, or, respectively, the coupling of points

*P*

_{ ij }–

*P*

_{ ji },

*P*

_{ jk }–

*P*

_{ kj },

*P*

_{ ki }–

*P*

_{ ik }. When the three connections are completed, the fan is rigid, and each element retains its three-dimensional positioning with respect to the others in the assembly (Fig. 8).

The opportunity to build the assembly bottom-up is provided by the fact that the three-bar fan is kinematically determinate. When we consider on the other side a single four-element fan where elements are connected with internal hinges, it is a kinematically indeterminate assembly in both two- and three-dimensional configurations (i.e., it exhibits, besides the free body motions, one additional inextensional mechanism) (Fig. 7b). The assembly is a mechanism. However the contact point between bars does not change during the mechanism, therefore a similar assembly procedure might be adopted here as well. Further studies might suggest that it is convenient to suppress, at least temporarily, the additional inextensional mechanisms for the purpose of fabrication. A network of four-bar assemblies is kinematically indeterminate, but only if the assembly is formed by a regular repetition of fans, while the network is kinematically determinate if the assembly is formed by a non-regular repetition of fans (Parigi et al. 2014).

## The Prototype: From Digital Design to Manufacturing

- i.
the definition of the starting free-form surface;

- ii.
the definition of a network of elements connected at their ends with no axes eccentricity;

- iii.
the definition of the corresponding reciprocal configuration with the Reciprocalizer algorithm;

- iv.
the calculation and output of the fabrication data;

- v.
the fabrication.

### The Starting Free-Form Surface

### The Network of Elements

### The Reciprocalizer

### The Fabrication Data

*b*

_{ i }, the position of the contact point

*P*

_{ ij }with each of the

*b*

_{ j }connecting elements). Point

*P*

_{ ij }is located along element

*b*

_{ i }surface and its position can be described with two values: its distance from one reference end

*D*

_{ ij }, and the angle

*α*

_{ ij }that it creates with a reference line arbitrarily set on the side element

_{,}measured from the element axes and in a perpendicular plane (Fig. 14). Except for the elements on the boundaries, in general each element connects with four adjacent elements, and therefore four contact points must be described (Fig. 15).

In this case the table presents 2,024 lines, each one describing the position of one contact point. In principle, this table is the only document needed for the entire process of fabrication. However, the digital model of the structure was also supplied as a tool to visually find and double-check the position of the elements in the space.

### The Fabrication Process

*ID*and marking the position of the contact points

*P*

_{ ij }on each element (Fig. 16).

*D*

_{ ij }is marked. Then, starting from this distance, the angle

*α*

_{ ij }identifying the final position of

*D*

_{ ij }, is marked. In order to mark both

*D*

_{ ij }and

*α*

_{ ij }with the same measuring tool, the value of the angle

*α*

_{ ij }is converted into a distance

*A*

_{ ij }along the element circumference (Fig. 17) with the formula:

*D*

_{ ij }, and curved paths, the case of measuring the distance

*A*

_{ ij }along the element circumference (Fig. 18).

For the coupling of the elements, two solutions of the joint were developed and tested, each requiring a different drill hole on the markings of the points *P* _{ ij }.

## Conclusions

Our experience has proved that the structural typology of reciprocally connected elements with superimposition joints, if supported by a custom developed design and fabrication process, is well-suited to build low-cost, free-form shapes with standardized elements. The design and fabrication of the prototype had two main objectives: (1) to test the results of the Reciprocalizer algorithm, and (2) to test a fast, efficient construction method, scaffolding-free, suited for any irregular free-form geometries. The experiment was successful in confirming and validating the output obtained from the Reciprocalizer algorithm and the ability to match the shape of the digital model into the prototype by connecting the elements according to the precise position indicated in the fabrication table. It was also successful in demonstrating the possibility to build the structure without scaffolding, by starting from a central point and gradually extending to the sides, taking advantage of the kinematical determinacy of the three-element reciprocal unit.

The construction of the dome also proved to be an engaging and almost magical experience when the expected shape emerged from the complex interaction of hundreds of precisely assembled elements with a low-tech fabrication technique.

## Notes

### Acknowledgments

The authors would like to thank the Department of Architecture, Design and Media Technology and the Department of Civil Engineering, Aalborg University, for providing working spaces and funds for the construction workshop, the students of the first semester of the Master of Science program in “Architectural Design” at Aalborg University for their intense participation and Eng. Ryota Kidokoro, invited guest of the workshop. All photos are by Dario Parigi except Fig. 23 by Mathias Sønderskov.

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