The Costa surface is a minimal surface. It was first described in 1982 by the Brazilian mathematician Celso José da Costa.
A minimal surface is a surface where the area is everywhere locally minimal, which means that no small part of the surface can be changed so that the area becomes smaller. Some minimal surfaces, like the plane, the helicoid (a spiraling surface), and the catenoid (which is formed by rotating a catenary, the curve formed by a hanging chain), have been known a for long time. Minimal surfaces can be realized as soap films extended from wires. The surface tension contracts the surface in every point so that the area becomes minimal.
The Costa surface is the first discovered minimal surface after the three mentioned above and is remarkable in several ways. You can see it as two pants glued together in the crotch. It has also been said that it resembles a certain type of samba hat. Two tunnels lead downwards; two tunnels lead upward. The surface is infinite—here we only show the central portion around the tunnels.
Topologically (i.e., disregarding distances) the Costa surface is like a bicycle tube with three punctures. This should be compared with the plane, which is a punctured sphere, with the helicoid, which topologically is a plane, thus also a punctured sphere, and with the catenoid, which is a twice punctured sphere. It was believed for a long time that these three were the only embedded minimal surfaces that could be formed by puncturing a compact surface, but then Celso Costa discovered that he could puncture a torus (a bicycle tube) three times and get a samba hat.
This unique model of the Costa surface in stoneware clay has been realized after extensive experimenting by Jana Kasselbäck, Krukrike, at Ulva Kvarn north of Uppsala, in the summer of 1997. Its glazing consists of crushed minerals. It has been burnt in a furnace at 1,280 degrees Celsius (2,336 degrees Fahrenheit). (Jana Kasselbäck is nowaday active at Austergårds Stenkyrka, Tingstäde.)
This model was on exhibit at Gustavianum in Uppsala, but during a visit to this museum on 2019 June 23 I found that it was no longer there and nobody at the reception knew anything about it, not even that it had been there since 1997.
Costa, Celso, 1984. Example of a complete minimal immersion in R3 of genus one and three embedded ends. Bol. Soc. Bras. Mat. 15, 47–54.
Dirkes, Ulrich; Hildebrandt, Stefan; Küster, Albrecht; Wohlrab, Ortwin, 1992. Minimal surfaces I. Springer-Verlag. XII + 508 pp.