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.

The sculptor Helaman Ferguson built some large snow (1999) and stone (2008) Costa surfaces. He first used the first renderings of the surface to make beautiful 3d ephemeral models out of snow. His work is inspired by mathematical shapes and curvatures.

First snow sculpture by H. Ferguson, D. Schwalbe, T. Nemeth & S. Wagon, 1999 (credit: Stan Wagon)

Invisible handshake by H.Ferguson, at Macalester College, St. Paul, MN, 2008 (credit: Stan Wagon)

To the best of our knowledge the only Costa surface used as an architectural form maker is found in the Australian Wildlife Health Centre at the Healesville Wildlife Sanctuary, east of Melbourne, Australia. The fabric surface roof designed by MvS Architects and Michael Lester covers the central space of the center. It also increases the natural ventilation of the building and this way allows the visitors to enjoy a more comfortable visit of the center.

Elevation of the Wildlife center (credit: Peter Bennetts)

The natural ventilation happens through the openings of the surface. See the support system and the patterning (credit: Peter Bennetts)

The central space of the center is a welcoming space for the visitors. See the membrane patterning (credit: Peter Bennetts)

In the next posts of the “A physical Costa surface” series the form finding and fabrication of the Costa surface will be presented. Stay tuned!

EDIT 2016-05-05: the name of the all people involved in sculpting the snow Costa surface were added as well as the location of residency of the Invisible handshake sculpture 

Author: Victor Charpentier

References:
Krantz, S. G., & Parks, H. R. (2014). A Mathematical Odyssey. Springer.
Bruter, C. (Ed.). (2002). Mathematics and art: mathematical visualization in art and education. Springer Science & Business Media.
Weisstein, E. W. (2003). Costa Minimal Surface.
Meeks III, W., & Pérez, J. (2011). The classical theory of minimal surfaces. Bulletin of the American Mathematical Society, 48(3), 325-407.