Costa’s Minimal Surface


Intuitively, a smooth surface in Euclidean space is locally area minimizing if any small deformation results in a surface of larger area. Soap films spanning a space curve, for example, satisfy this property. Costa’s surface is a minimal surface with certain interesting topological and geometric properties. The image shown here is a photograph of a solid model rendered with a 3D printer.

In [1], Palais describes a program for the visualization of mathematics through the use of computer graphics. He notes that visualization has been instrumental in some important mathematical discoveries and is also useful for educational purposes. With this in mind, he proposes the creation of an online interactive gallery of mathematical visualization, which he calls a “mathematical exploratorium.” As helpful as computer graphics are for visualization, it can be argued that there is an additional benefit to be had in viewing and handling an actual physical object. Indeed, there has been a long-standing tradition at German universities of producing plaster models of interesting geometric objects. Nowadays, physical models can be easily created by means of stereolithography, or “3D printing technology.”

3D printers produce solid objects from appropriate input data. They were originally created for rapid prototyping of new product designs but are increasingly being used for other purposes, such as highly customized manufacturing and scientific visualization. Their applications will continue to grow as the underlying technology improves and decreases in cost. One type of 3D printer uses a powder-binder technology to create objects via a layering technique: a thin layer of powder is spread across a planar surface, and then a print head applies a binder within the cross-sectional area of the object being created. This process is repeated, adding layer upon layer, until the object is complete.

More details about Costa’s surface and the method of 3D rendering can be found in my article “Visualizing Minimal Surfaces.”


  1. R. S. Palais, “The Visualization of Mathematics: Towards a Mathematical Exploratorium,” Notices of the American Mathematical Society, 46(6), 1999 pp. 647-658.