mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Costa's minimal surface or Costa's 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 that ...

discovered in 1982 by the Brazilian

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, mathematical structure, structure, space, Mathematica ...

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, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...

surface. Topologically, it is a thrice-punctured

torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...

. Until its discovery, the

plane Plane most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface * Plane (mathematics), generalizations of a geometrical plane Plane or planes may also refer to: Biology * Plane ...

helicoid The helicoid, also known as helical surface, is a smooth Surface (differential geometry), surface embedded in three-dimensional space. It is the surface traced by an infinite line that is simultaneously being rotated and lifted along its Rotation ...

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 describ ...

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 ENDS may refer to: End Mathematics *End (category theory) * End (topology) * End (graph theory) * End (group theory) (a subcase of the previous) * End (endomorphism) Sports and games *End (gridiron football) *End, a division ...

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 or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), hav ...

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

References

* * * Differential geometry Minimal surfaces Articles containing video clips {{topology-stub