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.
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.