Surfaces (mathematics)

SUMMARY: Surfaces are two-dimensional manifolds, some of which have been studied for their special properties.

Living beings interact with much of the world through surfaces. Humans walk, eat, and sleep on surfaces. Surfaces like the one-sided Klein bottle, named for Felix Klein, stretch the imagination and are the subject of mathematical investigations. They are often represented using physical models as well as computer models, including sculptures and computer animations. In twenty-first-century classrooms, students investigate a variety of surfaces and their properties, including area and volume. The National Council of Teachers of Mathematics recommends an understanding of the area and volume of rectangular solids for primary school students; of prisms, pyramids, and cylinders for middle grades students; and of cones, spheres, and cylinders for high school students. The parametrization and volume of surfaces is further explored in a multivariable calculus course.

History of Study

Mathematicians have long developed the theory of surfaces, and they continue to investigate their properties. In addition to the plane, polyhedra, such as the surface of a cube or an icosahedron, are among the first surfaces studied by Euclid the ancient Greeks in geometry. Their view of surfaces was entirely different from the functional description used in investigations of surfaces in the twenty-first century. The Greeks also had a good knowledge of surfaces of revolution and pyramids. With the introduction of analytic geometry in the seventeenth century, the study of surfaces developed into one of the most studied branches of mathematics. Mathematicians like Carl Friedrich Gauss, Pierre Bonnet, Barnhard Riemann, Gaspard Monge, and their followers firmly established surfaces on a rigorous basis during the eighteenth and nineteenth centuries.

One of the greatest achievements of the theory of surfaces is the Gauss–Bonnet theorem. Versions of the theorem were explored by Gauss in the 1820s and Bonnet and Jacques Binet in the 1840s. The form of the theorem that is standard in undergraduate differential geometry courses is attributed to Walther von Dyck in 1888. Smooth surfaces are defined as those surfaces in which each point has a neighborhood diffeomorphic to some open set in the plane. This added structure allows the use of analytic tools. The parametric functions for a smooth surface define two quadratic differential forms: the first and second fundamental forms, which are local invariants defined as functions of the arc length. The Gaussian curvature of the surface is an isometric invariant; hence, an intrinsic property of a surface, which is known as the Theorema Egregium of Gauss. The Gaussian curvature measures the deviance of the surface from being flat at each point. The parametric equations of a surface determine all six coefficients of its first and second fundamental forms; conversely, the fundamental theorem of surfaces states that given six functions satisfying certain compatibility conditions, then there exists a unique surface (up to its location in space). A geodesic curve on a smooth surface is characterized as a locally minimizing path. In a sense, geodesics are the straight lines of surfaces, which are essential for defining the notions of distance, area, and angle on a surface. Bonnet investigated the geodesic curvature, which measures the deviance of a curve on the surface from being a geodesic. The Gauss–Bonnet theorem states that for an orientable compact surface, the total Gaussian curvature is 2π times the Euler characteristic of the surface, named for Leonhard Euler. A consequence of this theorem is that the sum of the interior angles of a geodesic triangle is greater than, less than, or equal to π, depending on if the Gaussian curvature of the surface is positive, like on a sphere; negative, like on a hyperboloid; or zero, like in the plane.

Types of Surfaces

The classification of surfaces is another topic that is explored in undergraduate geometry or topology classes. In 1890, Felix Klein asked what surfaces locally look like the plane. In Klein’s Erlangen Program, a space was understood by its transformations. Heinz Hopf published a rigorous solution in 1925 that arose from groups of isometries acting on the plane without fixed points. The surfaces are the plane; the cylinder; the infinite Möbius band, named for August Möbius; the flat Clifford torus or donut, named for William Clifford; and the flat Klein bottle. Intuitively, surfaces seem to always have two sides; however, the Möbius band and the Klein bottle have only one side. Other surfaces like the projective plane resemble the sphere, and many-holed donuts resemble hyberbolic space. The Euler characteristic is used to classify the topology of a surface.

Some important types of surfaces that are studied intensively in geometry and analysis are minimal surfaces with zero mean curvature, such as catenoids and helicoids; developable surfaces with zero Gaussian curvature, like the plane, the cylinder, the cone, or a tangent surface; and ruled surfaces that can be generated by the motion of a straight line, like the cylinder and the hyperboloid of one sheet. While some of these surfaces date to antiquity, others are more recent. In the eighteenth century, Euler described the catenoid and Jean Meusnier the helicoid. Discoveries in 1835 by Heinrich Scherk and in 1864 by Alfred Enneper included minimal surfaces that are now named for each of them. In the 1840s, Joseph Plateau’s experiments indicated that dipping a wire ring into soapy water will create a minimal surface. Jesse Douglas won a Field’s Medal in 1936 for his solution to Plateau’s problem in minimal surfaces. A minimal surface that originated at the end of the twentieth century is because of Celso Coasta in 1982.

Representations and Investigations

Algebraic geometers investigate algebraic surfaces, such as cubic or quartic surfaces that can be represented by polynomials. These led to rich mathematical investigations in the nineteenth and twentieth centuries. For instance, in 1849, Arthur Cayley and George Salmon showed that there were 27 lines on a smooth cubic surface. Quartic surfaces were of interest in optics, and mathematicians such as Ernst Kummer studied them.

While mathematicians had long built physical models of surfaces, which were typically housed in universities and museums, in the late twentieth and early twenty-first centuries, computer-generated surfaces revolutionized the visualization and construction of surfaces and led to many interesting mathematical questions. For instance, numerous mathematicians and computer scientists have explored the method of subdivision of surfaces, including Tony DeRose, and Jos Stam, who won a Technical Achievement Award in 2005 from the Academy of Motion Picture Arts and Sciences. This method often takes advantage of the similarity between the local structure of a surface and a small piece of a plane. For instance, surfaces may be represented using small flat triangle or quadrilateral mesh representations. These representations are easier to manipulate, but they can still appear smooth to the eye.

Bibliography

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