A Physical Costa Surface 3/3: Building the structure

The fabrication of a tensile structure is a complex design process. How can the mathematical shape and the form found geometry derived in the first and second parts of the series be used as the basis for a sculpture? In this final post of the “Physical Costa Surface” series, the Costa Surface sculpture takes shape.

The dimensions of the sculpture are 1.5m of height and 2m of diameter. In order to build the sculptural installation, four steps are necessary: patterning the surface, designing the interaction between compressive and tensile elements, cutting the fabric and assembling the pieces.


Thread direction in fabric – the warp direction is often prestressed during manufacturing

The first task to making this surface a physical reality is patterning. This operation is maybe the single most important in the design process. The success of the patterning will in part determine if the tensioned surface will wrinkle or not. Fabrics used in engineering projects have generally a high level of anisotropy with warp and weft directions of the weave determining the material properties. In loom manufacturing, the warp direction is generally pre-stressed while the weft is weaved. In our case we used a high quality nylon/spandex fabric presenting a four-way stretch (ideally equally stretchable in warp or weft). The fabric can accommodate large strains so the risk of wrinkling is minimized.

We performed the patterning on the initial mesh geometry of the form finding procedure (details can be found here). In this process three distinct patterns are produced. The figure below shows how the patterns are distributed over the surface. The patterns are shrunk to compensate for the pre-stress and large strains in the membrane.

Patterning of the 6-hole Costa surface

Interaction tensile / compressive elements

The visuals of the structure have been so far limited to the surface itself. The constraints of the mathematics are fixed boundary conditions. The constraints of the fabric impose the application of the tensile stresses. These will in turn modify the position of the boundaries.

In order to create rigid circular boundaries, 3/8in. (9.5mm) glass fiber reinforced plastic rods were used. They were bent into 1.5m  (top and bottom) and 2m (center)  diameter circular hoops and connected by aluminum sleeves (ferrules).

The top and bottom rings are equilibrated by bending active GFRP rods. As seen in the figure below, by being bent, the rods push the two rings apart. The actions of the rods are equivalent to the thrust of an arch, providing the necessary force to achieve a height of 1.5m as specified in the computational model.

Building the sculpture

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A Physical Costa Surface 2/3: Form Finding Process

How can we algorithmically approximate the form of the mathematically defined Costa surface? This question is at the center of this second blog post of the “physical Costa surface”series. The form finding approach introduces a physical dimension to the equation generating the minimal surface. Finding the shape can be done in several ways. However, whether it is physical form finding or numerical form finding, the … Continue reading A Physical Costa Surface 2/3: Form Finding Process

A Physical Costa Surface 1/3 : History of The Surface

In the spring of 2016, Ruy Marcelo Pauletti, associate professor at University of São Paulo, taught the graduate class “Non-linear analysis of prestressed structural systems” at Princeton University. For the final project students applied techniques of design, numerical analysis and fabrication to a tensile sculpture that is on display in Princeton.


The student’s animated rendering of a modified Costa surface with six holes

Tension structures embody purity of forms. There is little hidden in a pre-stressed membrane that serves both  as a form definer and a structure. Membrane structures are structurally efficient, span large distances and can express elegance. They are also very relevant for sculptures since they are lightweight and compelling to engineers, architects,  mathematicians and visual artists. The students in Prof. Pauletti’s course use dynamic relaxation and natural density as form finding methods for the design of the tensile sculpture and use patterning methods for its fabrication.

For their sculptural qualities, minimal surfaces (embodied by the form of soap films) are the focus of this design process. Those surfaces are characterized by the minimization of total area for a given boundary. As such, the student’s project is the design and fabrication of a Costa surface (see rendering above). The history behind this surface is remarkable. In 1982 Celso Costa, then a mathematics graduate student in Rio de Janeiro, disproved a longstanding conjecture. This conjecture said that the plane, the catenoid and the helicoid were the only finite topological types of complete embedded minimal surfaces in R3 (surfaces with no boundary and no intersection with themselves). He intuited that a three-time punctured torus with two catenoidal extreme ends and a planar middle end also satisfies the criteria of such a surface. For the mathematically inclined reader, his original work can be found here.

The first rendering of the surface was done by David Hoffman at University of Massachusetts at Amherst with help of programmer James Hoffman in 1985. You can watch David Hoffman talk about minimal surfaces and the history of the Costa surface in the video below.

Only at a few occasions has the Costa surface been materialized at an architectural form before. It is relatively simple to create the form from the mathematical equations but to build a tensile Costa surface sculpture is more complex. The challenges are producing the pattern layout, sewing and working with a flexible material.

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