Three-dimensional reciprocal structures: morphology,
concepts, generative rules
Dario PARIGI1 and Alberto PUGNALE2
1 Assistant Prof., Dep. of Civil Engineering, Aalborg University, Aalborg, DK, email@example.com
2 Assistant Prof., Dep. of Civil Engineering, Aalborg University, Aalborg, DK, firstname.lastname@example.org
This paper present seven different three dimensional structures based on the principle of structural
reciprocity with superimposition joint and standardized un-notched elements. Such typology could
be regarded as being intrinsically three-dimensional because elements sit one of the top of the other,
causing every configuration to develop naturally out-of the plane.
The structures presented here were developed and built by the students of the Master of Science in
“Architectural Design” during a two week long workshop organized at Aalborg University in the
fall semester 2011.
Keywords: structural reciprocity; complex spatial structures; morphology; conceptual design.
The principle of structural reciprocity, 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.
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) .
According to these generative rules, reciprocal configurations that use a superimposition joint, i.e.
where un-notched bars sits on the top and in the bottom of each other could be regarded as being
intrinsically three-dimensional because they develop naturally out-of the plane, and their geometry
is extremely difficult to predict and control.
This paper present seven different three dimensional configurations developed and built by the
students of the Master of Science in “Architectural Design” during a two week long workshop
organized at Aalborg University in the fall semester 2011 . Since the use of physical models is
the most diffused tool to 'find' the form of reciprocal configuration due to the direct interaction with
the designer, the students were called to explore the 3D peculiarity of reciprocal structures by
building scale models and full-scale prototypes.
2. The concept of three dimensionality
In reciprocal configurations with superimposition joint elements sits on the top and in the bottom of
each other, and if they are un-notched their axis are not aligned and separated by a distance called
eccentricity. This causes such configurations to develop naturally out-of plane, and their geometry
becomes soon very difficult to predict and control because the position of each elements at the same
time determines, and is determined, by the position of adjacent ones.
An example of such configuration is the structures of Leonardo Da Vinci (Figure 1). Despite being
not evident from the original plan representation, once elements are placed one on the top of each
other the structure develops out-of-plane as a dome-like structure.
In order to avoid the out-of plane deviation notched elements can be introduced to re-align elements
axis and eliminate the eccentricity. This solution was adopted in the past to get configurations that
develop in plane as in the slab by Johannis Wallis, as shown in Figure 2. Another solution to reduce
geometric complexity is to align elements axis by not superimposing one on the top and in the
bottom of each other, as in the slab designed by Sebastiano Serlio (Figure 3).
However such intrinsic three dimensionality could be regarded as an opportunity if the overall
geometry of the reciprocal configuration can be predicted. Hybrid optimization strategies proved to
be an effective computational tool in controlling their geometry, such that complex three
dimensional configurations with potentially any shape can be obtained with the use of standardized
un-notched round elements .
In this framework, the use of physical models is a tool that triggers the exploration of new
typologies because of the direct interaction of the designer with the complex geometry of the model
itself. The structures presented here were developed using physical scale models and built in a full
scale prototype with the aim to explore and test how three dimensionality of superimposition joint
could lead to reciprocal configuration:
1. that expand in more than one direction or it has no direct reference to a surface;
2. where every joint is a reciprocal superimposition joint.
The aim is to stimulate design thinking in a more spatial way, with a typology that is intrinsically
three dimensional. This allow to understand the potential of such configurations to create
architectural spaces by excluding those typologies whose final result is predictable.
As an example of structures that can expand in one direction is Da Vinci's bridge represented in
Figure 4: such configuration can be expanded indefinitely along one direction by repeating the basic
fan. Placing two or more bridges side by side we can cover a larger span in the transversal direction,
however the direction of expandability remains the longitudinal one.
Surface structures could be easily obtained by creating regular or non regular patterns repeating
along two directions. Such typology was excluded because the results that could be achieved by
using physical models are predictable and lead to dome-like structures.
Another type of structure that was explicitly excluded by rule no. 2 is the one obtained by
superimposition of different fans - with this generative rule the joint between two different fans is
not reciprocal, being constituted by two bars that does not interlock with each other, instead they
merely touch and need to be fixed with other connecting methods.
Figure 4: Da Vinci's
Figure 5: reciprocal surface
Figure 6: fans
superimposition by Popovic
Figure1: Da Vinci L, un-notched
Figure 2: Wallis J, notched
Figure 3; Serlio S, aligned axis
3. The structures
The students at Aalborg University developed seven reciprocal structures, which interpret in
different declination the assigned task. Different strategies to generate three dimensional structures
that have no reference to a surface are analyzed below, highlighting:
1. the generative rule;
2. the three dimensional potential;
3. relation between the physical model and the created space.
The identification of those elements aims to generalize in a systematic way the results, also
suggesting the extension and application of the generative rules to this class of structures to obtain
potentially infinite number of variations and re-combinations.
3.1 Structure 1 - "Bugs"
3.1.1 The generative rule
This structure is generated starting from the classic typology of the Leonardo Da Vinci Bridge
(Figure 1) consisting of 6 longitudinal and 4 transversal interlocked bars, which can expand along
one direction only. The generative rule is the elimination of two transversal bars and the rotation of
four longitudinal bars in order to meet and to be supported directly by the two longitudinal bars
3.1.2 The three dimensional potential
With the rotation of the four longitudinal bars (the 'legs' of the 'bug') the elemental fan can expand
into four different directions. Each leg can be extended both upward and downward, enabling the
possibility to create three dimensional patterns of elements that develop in three dimensions at
3.1.3 Relation between the physical model and the created space
The realized model highlighted the difficulty in controlling the geometry of such configuration and
the need of an absolute precision in order for the bars of different bugs to meet at the desired
Figure 1: Leonardo
Figure 2: the generative
Figure 3: the elemental fan
Figure 4: three
Figure 5:array of bugs,
plan(top) and elevation
Figure 6: A three dimensioanl
array, plan(top) and elevation
Figure 7: the full scale prototype
3.2 Structure 2 - "Matrix"
3.2.1 The generative rule
The structure is based on the repetition of the reciprocal pattern represented in Figure 11 along two
orthogonal directions (Figure 8 and 9) at every node intersection. The structure is based on the node
of Figure 10 that allows to deal with three bars coming from three different orthogonal directions.
3.2.2 The three dimensional potential
The structure can be expanded indefinitely in the space along every direction.
3.2.3 Relation between the physical model and the created space
The prototype is a cube with extending legs elements that suggest the indefinite expandability of
such configuration along three directions (Figure 12). The final model appears as an orthogonal
'architectural frame' (Figure 13). It suggest also the possibility to generate different space frames
with different starting two dimensional patterns.
3.3 Structure 3 - "Neural network"
3.3.1 The generative rule
The generative rule is the combination of a positive (Figure 13) and a negative (Figure 14)
reciprocal fan by interlocking (Figure 15).
3.3.2 The three dimensional potential
The potential of such generative rule relies on the possibility to conceive it as a three dimensional
node that can expand along six different direction, three in the bottom (correspondent to the three
bars of the positive fan), and three in the top (corresponding to the three bars of the negative fan).
The two interlocked fans can also rotate relatively to the other, thus they can vary the direction they
are pointing at.
3.3.3 Relation between the physical model and the created space
The realized structure demonstrates that the node enable the possibility to freely design spatial
structures by combining the nodes at different heights and position, with almost unlimited
possibilities for expandability. This concept can be seen as an open-ended structure in which the
issue is related to the design of its ending points.
Figure 13:Positive fan
Figure 14: Negative
positive and negative fan
rotation of the two
3.4 Structure 4 - "Hypar"
3.4.1 The generative rule
Starting from a three bars fan, an outer layer of bars is added, each one interlocking with two bars
of the inner layer. Many layers could be added until physical limitations are reached (Figure 19).
3.4.2 The three dimensional potential
The structure has three high points and three low points, and it develops in the space along these
directions. The geometry of such configuration turned out to be extremely difficult to control,
because of the interdependence of each element with the others.
3.4.3 Relation between the physical model and the created space
The scale model (Figure 20) and the final prototype define a space with sculptural qualities (Figure
22). Bars touches towards their ends and not along their span (Figure 21).
3.5 Structure 5 - "Star frame"
3.5.1 The generative rule
Starting from a six bars array where bars extend in both direction before and after the contact point
(Figure 23b/c), an outer array of bars is added (Figure 23e) where each bar supported by two bars
facing opposite in the first array (Figure 23d). A final array of six bars is added (Figure 23 g), where
each bar is supported by one bar belonging to the first array and one to the second array (Figure
3.5.2 The three dimensional potential
The configuration, despite the rigorous methods required to generate it, creates a seemingly random
''cloud' of bars (Figure 25). Each bar point into different direction, and it has no reference to
surfaces. The configuration seems to be a closed configuration, as it does not allow for further
3.5.3 Relation between the physical model and the created space
The 1:1 prototype was easily scaled up from the model. The integrity of the space it creates
descends from the fact that the top elements are interlocked and constitutes a whole with the
elements that goes to the ground.
3.6 Structure 6 - "Flame"
3.6.1 The generative rule
Starting from a three bars fan, a first array of three bars is added (Figure 27): each bar touch the
ground at one extreme. A second array is added where each bar being supported by the first array
and the three bars fan. A the third array is added, each bar being supported by the first and the
second array. The generative rule suggest that this operation can be repeated again starting now
from the third array, a three bars fan similar to the starting one.
3.6.2 The three dimensional potential
Depending on the angles created between bars, the model opens up the possibility to create a fully
reciprocal tower, indefinitely expandable. However, the realized prototype closes up representing a
3.6.3 Relation between the physical model and the created space
The configuration encloses a three dimensional space taking advantage of the three dimensionality
of the reciprocal superimposition joint. The step from the scale model to the prototype highlighted
that the geometry is extremely difficult to control, because of the interdependence of one element
with the other. In the realized prototype the tower closes up in the fourth array and does not offer
the possibility to continue indefinitely, however further experimentation could lead to a fully
3.7 Structure 7 – "Wave"
3.7.1 The generative rule
The wave combines four bars fan and three bars fan. The two typologies are assembled in an
orthogonal direction with respect to each other as shown in Figure 30.
3.7.2 The three dimensional potential
Despite the structure derives directly from a surface it is three dimensional because their
quadrilateral and triangular fans develops into two orthogonal directions.
3.7.3 Relation between the physical model and the created space
The structure encloses a space where the interaction between the two typologies of fans are
responsible of the overall stability and manifest itself in the different aesthetic qualities in the
exterior and the interior (Figure 31).
The structures presented here are based on the principle of reciprocity and are built using un-
notched standardized elements. Each structure interprets differently the intrinsic three
dimensionality of reciprocal structures with superimposition joint, and their generative rules are
reported in the paper as schemes whose aim is to generalize the concepts behind them.
It should be underlined that small variations on the position of elements and the variation of the
value of their thickness could influence dramatically the geometry of the configuration. However,
those elements were voluntarily omitted from our considerations in order to focus on the topology
of the connections, and not on the infinite variations that can be achieved by changing such
The authors would like to thank the Department of Architecture, Design & 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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history, properties and design issues", Proceeding of the IABSE-IASS symposium 2011
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 Pugnale A, Parigi D, “Approaching technical issues in architectural education”, Proceedings
of the IASS-APCS 2012, Seoul, Korea, 21-24 May 2012.
 Parigi D, Kirkegaard PH, Sassone M, “Hybrid optimization in the design of reciprocal
structures”, Proceedings of the IASS-APCS 2012, Seoul, Korea, 21-24 May 2012.
 Popovic Larsen O, Reciprocal Frame Architecture, Elsevier, Burlington, 2008.