Reciprocal Tree-Like Fractal Structures

Abstract

Sometimes the complex structures of nature inspire human constructions. Gothic construction has shown that forces can cross space along intricate paths that may even be arbitrary if correctly dimensioned. In some way, ribbed structures are like trees where the branches conduct forces instead of sap; they operate as branches and trunks descending by fractal ways. Here we discuss reciprocal tree-like fractal structures and the difficulty in their design and erection and solutions for constructive details, as well as the possible analytical questions and automatic generation by means of proper software. The results are shown in the design of the Natural Interpretation Centre in Melilla where we have proposed two connected trees like shown at figures included below.

Architecture Imitates Nature

Sometimes we are surprised because nature appears to build its structures following our designs, but really we are the imitators. Whatever the sources of our inspiration, there is no doubt that we recall experiences of forests and jungles (Fig. 1). Gothic architecture was born in countries where forests were sacred and ribbed structures derived directly from trees and branches (Fig. 2).

Fig. 1

Lady Chapel in the Wells Cathedral according to Felix Escrig. Image: (Escrig 1998), reproduced by permission

Fig. 2

Westminster Abbey according to F. Escrig. Image: (Escrig and Valcarcel 2004), reproduced by permission

Gothic architects made their designs according to nature and, without knowing it, also made use of the theory of fractals. Their first designs were absolutely structural, as in the vaults of Notre Dame in Paris, where some aspects of classical composition were respected, in that ribs are well organized along longitudinal, transversal and diagonal edges. However, in that same structure there appears a new concept, more geometric than structural. The rose windows are illogical structures because they act as pre-stressed stone fabric to load transverse forces. Here the architects introduced an elementary fractal design, where a two-level design appears to “grow” from the centre (Fig. 3).

Fig. 3

South rose window, Notre Dame, Paris

It took some time to introduce the fractal concept into the design of vaults but then this was done with an unsurpassed mastery, first in wood, as in the vaults of Bath Abbey, and later in stone, as in the vault of the Chapel of Henry VII in Westminster Abbey, directly inspired by the geometry of rose windows.

The complete freedom and imitation of nature arrived later, with masterpieces never before seen. Examples include the vaults of the entrance and hall of Prague Castle, where the stone branches seem to grow and escape off the vault surface. Studies of the fractal components of Gothic architecture have been published by many important researchers and form part of the main studies in architecture design (Goldberger 1996; Bovill 1996).

In our own day there are a great number of proposals that develop fractal and tree-like growth forms, such as the Sagrada Familia by Antony Gaudí and Frei Otto’s designs for the tree-like supporting structure of Terminal 1 in the Stuttgart airport (Goldberger 1996). More recent designs include those for the Tote Banqueting Hall in Mumbai by Serie Architects and the Supertrees in Singapore. The Oriente train station in Lisbon by Santiago Calatrava (1998) and the Mercat Santa Caterina in Barcelona by Enric Miralles (2005) can be also considered as tree-like fractal structures. The most recent designs, such as the metal sculpture by Zaha Hadid (2012) and the design for the redevelopment of King’s Cross Station in London by John McAslan + Partners exhibited at the 2012 Venice Biennale explore other materials and geometries that make the field of this study more and more interesting.

Now we would like to introduce a new design to add to this extensive list, a design which we have called ‘reciprocal tree-like fractal structures’.

Reciprocal Fractal Tree-Like Structures

On fol. 899v of the Codex Atlanticus Leonardo sketched a few patterns now called ‘reciprocal frames’ that have been studied in depth. Elsewhere we have demonstrated the great capacity of these designs to act as real structures provided that the joints are properly connected (Sanchez and Escrig 2011). The main characteristic of these structures is that the diameter of bars makes the geometry very complex, with the final form rising out of the plane. The feasibility of these kinds of structures have been shown by Olga Popovic Larsen (2008). Our innovation is that while such designs are usually proposed as roofs supported at their external edges, we designed umbrella-like trees.

Our purpose was thus to study the possibility of combining the concepts of tree-like fractal growth and reciprocity (Fig. 4).

Fig. 4

Umbrella tree-like reciprocal structure model with several levels in two phases of assembling

We can increase the levels of growth by either maintaining the same number of branches at each level or by duplicating them each step or new level, as trees does. It may be that, in the strict sense of the term, systems that do not multiply their elements as they grow are not considered fractal, but this is in any case a way of extending their arms based in a defined mathematical form that can be automatized. Tree-form umbrellas that are reciprocal grids can be studied from the point of view of fractal theory.

Similar works have been investigated by others, but such studies are not abundant and have been undertaken from other point of view (Sieder et al. 2012).

Deployable Reciprocal Tree-Like Fractal Structures

Another aspect that we have introduced in our studies is the mobility. This is one of our main objectives in the design of meshes composed with bars. This is because not only can they change in form over time, but also because they greatly facilitate assembly, since each component is carried from the fabrication plant to the worksite in a compact parcel (Escrig 2012).

Figure 5 shows our first attempts by means of a model that simultaneously fulfills the conditions of reciprocal quality, fractal growth and deployability by twisting and sliding joints. Solving all requirements at the same time is very complicated, not only because of the need for the design of a proper joint but because of the simultaneous movement that is required. Our proposal consists in twisting the main supports around a hyperbolic surface by means of two rings placed at equal distances from the hyperboloid centre.

Fig. 5

The same two-level umbrella with sliding joints to make it deployable

Figure 6 shows the growth process, as well as the addition of other levels of bars in the deploying process.

Fig. 6

The three-level umbrella with sliding joints to make it deployable

If we build this system by computer methods it is possible to check the model in real time, as seen in Escrig (2013) and Fig. 7. Figure 8 shows details of joints that have to twist and slide during the deployment process. The design of these joints in a built example will be seen below.

Fig. 7

The four-level umbrellas obtained by a computer program. Image: authors

Fig. 8

Details of joints for the sliding and twisting bars to make possible the deployment of reciprocal structure umbrella

If we superpose several steps of deployment in the same graphic we can obtain a figure that relatively simple but appears complex. We can profit from this kind of image to design a real structure that will be shown in the next section (Fig. 9).

Fig. 9

The four level umbrella deploying obtained by a computer program

An Application for a Reciprocal Frame Fractal Tree Like Structure

Melilla is located in the north of Africa and it has a dry climate with almost desert vegetation. With the project proposed in 2008 we intended to adapt a form inspired by both Muslim tents and local trees. We profited from our previous studies to define a complex bunch of branches connected in such way that every bar is linked to others by means of only two points and each supports the origin of another. The initial site was a desolate place that would be revitalized (Fig. 10), but in the end the structure was built in an urban park with many other buildings whose designs were also interpretations of nature. To complete the design we checked some models, such as those shown in Fig. 11, and put together a global proposal (Figs. 12, 13).

Fig. 10

Initial design for a Centre for Nature Interpretation in Melilla, North Africa

Fig. 11

Preliminary model to define a tree-like structure for the Melilla design

Fig. 12

Front (below) and rear (above) elevations for the final design proposal

Fig. 13

General view of the final proposal for the Melilla Visitors Centre for Nature Interpretation

If we consider the size of bars and the support system that reciprocity imposes, the solution increases in complexity. It then becomes necessary to draw each bar with the correct diameter and to define its position in a precise way (Fig. 14).

Fig. 14

The design considering the size of pipes

Another problem was the design of a kind of joint capable of rotating, sliding and being fixed when arriving at the correct position. For this we needed to invent a special solution that permits us to orient the bars in every direction in space (Figs. 15, 16). In Fig. 17 we show the sequence of assembly and the angles to complete the construction.

Fig. 15

Preliminary sketch of the joint design

Fig. 16

Joints can twist and slide in every position in space to orientate bars

Fig. 17

Sequence of mounting bars with predefined angles

Figure 18 shows the actual construction, where we can see that the solution proposed is correct. To facilitate assembly we placed a ring around the pipes to situate the geometry correctly (Fig. 19).

Fig. 18

Group of joints solved with disc plates to connect bars

Fig. 19

With the correct position of bars indicated by means of a ring around the pipes

The Building in Progress

After having checked the proposed solution and completed the main building, we proceeded to install our structure, which consisted in two symmetrical tree-like structures (Fig. 20). To cover the enceinte we decided to use a tensile fabric roof supported by the extremities of the highest branches, as shown in Fig. 21. Figure 22 shows the similarity of this structure with natural trees, and in Figs. 23, 24 and 25 we show the final built structure.

Fig. 20

The installation of tree structures

Fig. 21

Mounting the fabric roof

Fig. 22

Artificial tree-like structures in front of and behind natural trees

Fig. 23

Mounting the fabric roof

Fig. 24

The finished roof in January 2011

Fig. 25

Lateral view of the finished roof