Costa Minimal Surface

The Costa surface is a complete minimal embedded surface of finite topology (i.e., it has no boundary and does not intersect itself). It has genus 1 with three punctures (Schwalbe and Wagon 1999). Until this surface was discovered by Costa (1984), the only other known complete minimal embeddable surfaces in with no self-intersections were the plane (genus 0), catenoid (genus 0 with two punctures), and helicoid (genus 0 with two punctures), and it was conjectured that these were the only such surfaces.

Rather amazingly, the Costa surface belongs to the dihedral group of symmetries.

The Costa minimal surface appears on the cover of Osserman (1986; left figure) as well as on the cover of volume 2, number 2 of La Gaceta de la Real Sociedad Matemática Española (1999; right figure).

It has also been constructed as a snow sculpture (Ferguson et al. 1999, Wagon 1999).

On Feb. 20, 2008, a large stone sculpture by Helaman Ferguson was installed on the south deck of the Olin-Rice Science Center at Macalester College (photo courtesy of Stan Wagon).

As discovered by Gray (Ferguson et al. 1996, Gray 1997), the Costa surface can be represented parametrically explicitly by

			(1)
			(2)
			(3)

where is the Weierstrass zeta function, is the Weierstrass elliptic function with (OEIS A133747), the invariants correspond to the half-periods 1/2 and , and first root

(4)

(OEIS A133748), where is the Weierstrass elliptic function.