## Surface

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

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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.