In mathematics, Costa's minimal surface, is an embedded

minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...

discovered in 1982 by the

Brazil Brazil ( pt, Brasil; ), officially the Federative Republic of Brazil (Portuguese: ), is the largest country in both South America and Latin America. At and with over 217 million people, Brazil is the world's fifth-largest country by area ...

ian

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

Celso José da Costa. It is also a surface of finite topology, which means that it can be formed by puncturing a

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

surface. Topologically, it is a thrice-punctured

torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...

. Until its discovery, the

plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...

helicoid The helicoid, also known as helical surface, after the plane and the catenoid, is the third minimal surface to be known. Description It was described by Euler in 1774 and by Jean Baptiste Meusnier in 1776. Its name derives from its similarity ...

and the

catenoid In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis (a surface of revolution). It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. It was formally descri ...

were believed to be the only embedded minimal surfaces that could be formed by puncturing a compact surface. The Costa surface evolves from a torus, which is deformed until the planar

end End, END, Ending, or variation, may refer to: End *In mathematics: ** End (category theory) ** End (topology) **End (graph theory) ** End (group theory) (a subcase of the previous) **End (endomorphism) *In sports and games **End (gridiron footbal ...

becomes catenoidal. Defining these surfaces on rectangular tori of arbitrary dimensions yields the Costa surface. Its discovery triggered research and discovery into several new surfaces and open

conjectures In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in ...

in topology. The Costa surface can be described using the Weierstrass zeta and the Weierstrass elliptic functions.

References

* ''Ph.D. Thesis, IMPA, Rio de Janeiro, Brazil.'' * ''Bol. Soc. Bras. Mat. 15, 47–54.'' * Differential geometry Minimal surfaces Articles containing video clips {{topology-stub