The word ``surface'' is an important term in mathematics and is used in many ways. The most common and
straightforward use of the word is to denote a 2-D Submanifold of 3-D Euclidean Space. Surfaces can range
from the very complicated (e.g., Fractals such as the Mandelbrot Set) to the very simple (such as
the Plane). More generally, the word ``surface'' can be used to denote an -D Submanifold of an -D
Manifold, or in general, any co-dimension 1 subobject in an object (like a Banach Space or an
infinite-dimensional Manifold).

Even simple surfaces can display surprisingly counterintuitive properties. For example, the Surface of Revolution
of around the *x*-Axis for (called Gabriel's Horn) has Finite Volume but
Infinite Surface Area.

*See also *Algebraic Surface, Barth Decic, Barth Sextic, Bernstein Minimal Surface Theorem, Bohemian
Dome, Boy Surface, Catalan's Surface, Cayley's Ruled Surface, Chair, Clebsch Diagonal
Cubic, Compact Surface, Cone, Conical Wedge, Conocuneus of Wallis, Cork Plug,
Corkscrew Surface, Cornucopia, Costa Minimal Surface, Cross-Cap, Crossed Trough, Cubic
Surface, Cyclide, Cylinder, Cylindroid, Darwin-de Sitter Spheroid, Decic Surface, Del
Pezzo Surface, Dervish, Desmic Surface, Developable Surface, Dini's Surface, Eight Surface,
Ellipsoid, Elliptic Cone, Elliptic Cylinder, Elliptic Helicoid, Elliptic Hyperboloid,
Elliptic Paraboloid, Elliptic Torus, Enneper's Minimal Surface,
Enneper's Negative Curvature Surfaces, Enriques Surfaces, Etruscan Venus
Surface, Flat Surface, Fresnel's Elasticity Surface, Gabriel's Horn, Handkerchief Surface,
Helicoid, Henneberg's Minimal Surface, Hoffman's Minimal Surface, Horn Cyclide, Horn Torus,
Hunt's Surface, Hyperbolic Cylinder, Hyperbolic Paraboloid, Hyperboloid, Ida Surface,
Immersed Minimal Surface, Kiss Surface, Klein Bottle, Kuen Surface, Kummer Surface,
Lichtenfels Surface, Maeder's Owl Minimal Surface, Manifold, Menn's Surface, Minimal Surface,
Miter Surface, Möbius Strip, Monge's Form, Monkey Saddle, Nonorientable
Surface, Nordstrand's Weird Surface, NURBS Surface, Oblate Spheroid, Octic Surface, Orientable
Surface, Parabolic Cylinder, Parabolic Horn Cyclide, Parabolic Ring Cyclide, Parabolic Spindle
Cyclide, Paraboloid, Peano Surface, Piriform, Plane, Plücker's Conoid,
Polyhedron, Prism, Prismatoid, Prolate Spheroid, Pseudocrosscap, Quadratic Surface,
Quartic Surface, Quintic Surface, Regular Surface, Rembs' Surfaces, Riemann Surface, Ring
Cyclide, Ring Torus, Roman Surface, Ruled Surface, Scherk's Minimal Surfaces, Seifert
Surface, Sextic Surface, Shoe Surface, Sievert's Surface, Smooth Surface, Solid,
Sphere, Spheroid, Spindle Cyclide, Spindle Torus, Steinbach Screw, Steiner Surface,
Swallowtail Catastrophe, Symmetroid, Tanglecube, Tetrahedral Surface, Togliatti Surface,
Tooth Surface, Trinoid, Unduloid, Veronese Surface, Veronese Variety, Wallis's Conical Edge,
Wave Surface, Wedge, Whitney Umbrella

**References**

Endraß, S. ``Home Page of S. Endraß.'' http://www.mathematik.uni-mainz.de/~endrass/.

Fischer, G. (Ed.). *Mathematical Models from the Collections of Universities and Museums.*
Braunschweig, Germany: Vieweg, 1986.

Francis, G. K. *A Topological Picturebook.* New York: Springer-Verlag, 1987.

Geometry Center. ``The Topological Zoo.'' http://www.geom.umn.edu/zoo/.

Gray, A. *Modern Differential Geometry of Curves and Surfaces.* Boca Raton, FL: CRC Press, 1993.

Hunt, B. ``Algebraic Surfaces.'' http://www.mathematik.uni-kl.de/~wwwagag/Galerie.html.

Morgan, F. ``What is a Surface?'' *Amer. Math. Monthly* **103**, 369-376, 1996.

Nordstrand, T. ``Gallery.''
http://www.uib.no/people/nfytn/mathgal.htm.

Nordstrand, T. ``Surfaces.''
http://www.uib.no/people/nfytn/surfaces.htm.

von Seggern, D. *CRC Standard Curves and Surfaces.* Boca Raton, FL: CRC Press, 1993.

Wagon, S. ``Surfaces.'' Ch. 3 in *Mathematica in Action.* New York: W. H. Freeman, pp. 67-91, 1991.

Yamaguchi, F. *Curves and Surfaces in Computer Aided Geometric Design.* New York: Springer-Verlag, 1988.

© 1996-9 *Eric W. Weisstein *

1999-05-26