In mathematics Mathematics is the study of quantity, structure, space, and change. Mathematicians seek out patterns, formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions, specifically in topology Topology is a major area of mathematics concerned with spatial properties that are preserved under continuous deformations of objects, for example, deformations that involve stretching, but no tearing or gluing. It emerged through the development of concepts from geometry and set theory, such as space, dimension, and transformation, a surface is a two-dimensional In physics and mathematics, the dimension of a space is roughly defined as the minimum number of coordinates needed to specify every point within it. For example: a point on the unit circle in the plane can be specified by two Cartesian coordinates but one can make do with a single coordinate , so the circle is 1-dimensional even though it exists topological manifold In mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below. Topological manifolds form an important class of topological spaces with applications throughout mathematics. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions. The term “Euclidean” is used to distinguish these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity R³ — for example, the surface of a ball A ball is a round, usually spherical but sometimes ovoid, object with various uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used for simpler activities, such as catch, marbles and juggling. Balls made from hard-wearing materials are used in. On the other hand, there are surfaces, such as the Klein bottle The Klein bottle is a mathematical certain non-orientable surface, i.e., a surface with no distinct "inner" and "outer" sides. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a two-dimensional surface with boundary, a Klein bottle has no boundary. (For, that cannot be embedded In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup in three-dimensional Euclidean space without introducing singularities In mathematics, singularity theory is the study of the failure of manifold structure. A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width. What is meant by a singularity can be seen by dropping it on the floor. Probably there will appear a number of double points, at which the string crosses itself in or self-intersections.

To say that a surface is "two-dimensional" means that, about each point, there is a coordinate patch on which a two-dimensional coordinate system In geometry, a coordinate system is a system which uses a set of numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in 'the x-coordinate'. In elementary is defined. For example, the surface of the Earth Earth is the third planet from the Sun, and the densest and fifth-largest of the eight planets in the Solar System. It is also the largest of the Solar System's four terrestrial planets. It is sometimes referred to as the World, the Blue Planet,[note 6] or by its Latin name, Terra.[note 7] is (ideally) a two-dimensional sphere A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point. This distance r is known as the radius of the sphere. The, and latitude Latitude, usually denoted by the Greek letter phi gives the location of a place on Earth (or other planetary body) north or south of the equator. Lines of Latitude are the imaginary horizontal lines shown running east-to-west (or west to east) on maps (particularly so in the Mercator projection) that run either north or south of the equator and longitude Longitude , identified by the Greek letter lambda (λ), is the geographic coordinate most commonly used in cartography and global navigation for east-west measurement. Constant longitude is represented by lines running from north to south. The line of longitude (meridian) that passes through the Royal Observatory, Greenwich, in England, provide coordinates on it.

Surfaces find application in physics Physics is a natural science that involves the study of matter and its motion through space-time, as well as all applicable concepts, such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves, engineering Engineering is the discipline, art and profession of acquiring and applying technical, scientific, and mathematical knowledge to design and implement materials, structures, machines, devices, systems, and processes that safely realize a desired objective or invention, computer graphics Computer graphics are graphics created using computers and, more generally, the representation and manipulation of image data by a computer, and many other disciplines, primarily when they represent the surfaces of physical objects. For example, in analyzing the aerodynamic Aerodynamics is a branch of dynamics concerned with studying the motion of air, particularly when it interacts with a moving object. Aerodynamics is a subfield of fluid dynamics and gas dynamics, with much theory shared between them. Aerodynamics is often used synonymously with gas dynamics, with the difference being that gas dynamics applies to properties of an airplane A fixed-wing aircraft, typically called an aeroplane, airplane or just plane, is an aircraft capable of flight using forward motion that generates lift as the wing moves through the air. Planes include jet engine and propeller driven vehicles propelled forward by thrust, as well as unpowered aircraft , which use thermals, or warm-air pockets to, the central consideration is the flow of air along its surface.

Definitions and first examples

A (topological) surface is a Hausdorff In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently used and discussed. It implies the topological space Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology on which every point has an open neighbourhood homeomorphic In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function (from the Greek words ὅμοιος = similar and μορφή (morphē) = shape, form) is a continuous function between two topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of to some open subset The concept of an open set is fundamental to many areas of mathematics, especially including point-set topology and metric topology. Intuitively speaking , a set U is open if any point x in U can be moved by a small amount in any direction and still be in the set U. The notion of an open set provides a fundamental way to speak of nearness of of the Euclidean plane E². Such a neighborhood, together with the corresponding homeomorphism, is known as a (coordinate) chart. It is through this chart that the neighborhood inherits the standard coordinates on the Euclidean plane. This coordinates are known as local coordinates and these homeomorphisms lead us to describe surfaces as being locally Euclidean.

More generally, a (topological) surface with boundary is a Hausdorff In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently used and discussed. It implies the topological space Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology in which every point has an open neighbourhood homeomorphic In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function (from the Greek words ὅμοιος = similar and μορφή (morphē) = shape, form) is a continuous function between two topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of to some open subset The concept of an open set is fundamental to many areas of mathematics, especially including point-set topology and metric topology. Intuitively speaking , a set U is open if any point x in U can be moved by a small amount in any direction and still be in the set U. The notion of an open set provides a fundamental way to speak of nearness of of the upper half-plane H². These homeomorphisms are also known as (coordinate) charts. The boundary of the upper half-plane is the x-axis. A point on the surface mapped via a chart to the x-axis is termed a boundary point. The collection of such points is known as the boundary of the surface which is necessarily a one-manifold, that is, the union of closed curves. On the other hand, a point mapped to above the x-axis is an interior point. The collection of interior points is the interior of the surface which is always non-empty In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced. Many possible properties of sets are trivially true for the empty set. The closed disk In geometry, a disk is the region in a plane bounded by a circle is a simple example of a surface with boundary. The boundary of the disc is a circle.

The term surface used without qualification refers to surfaces without boundary. In particular, a surface with empty boundary is a surface in the usual sense. A surface with empty boundary and is compact is known as a 'closed' surface. The two-dimensional sphere, the two-dimensional torus In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle. In most contexts it is assumed that the axis does not touch the circle (in this case the surface has a ring shape and is called a ring torus or simply torus if the ring shape is implicit). Other types, and the real projective plane In mathematics, the real projective plane is a non-orientable two-dimensional manifold, that is, a surface, that has basic applications to geometry, but which cannot be embedded in our usual three-dimensional space without intersecting itself. It has Euler characteristic 1, hence a demigenus of 1 are examples of closed surfaces.

The Möbius strip The Möbius strip or Möbius band (alternatively written Mobius or Moebius in English) is a surface with only one side and only one boundary component. The Möbius strip has the mathematical property of being non-orientable. It can be realized as a ruled surface. It was discovered independently by the German mathematicians August Ferdinand Möbius is a surface with only one "side". In general, a surface is said to be orientable if it does not contain a homeomorphic copy of the Möbius strip; intuitively, it has two distinct "sides". For example, the sphere and torus are orientable, while the real projective plane is not (because deleting a point or disk from the real projective plane produces the Möbius strip).

In differential Differential geometry is a mathematical discipline that uses the methods of differential and integral calculus, as well as linear and multilinear algebra, to study problems in geometry. The theory of plane and space curves and of surfaces in the three-dimensional Euclidean space formed the basis for its initial development in the eighteenth and and algebraic geometry Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis, topology and number theory. Initially a, extra structure is added upon the topology of the surface. This added structures detects singularities, such as self-intersections and cusps, that cannot be described solely in terms of the underlying topology.

Extrinsically defined surfaces and embeddings

Historically, surfaces were initially defined as subspaces of Euclidean spaces. Often, these surfaces were the locus In mathematics, a locus is a collection of points which share a property. The term locus is usually used of a condition which defines a continuous figure or figures, that is, a curve. For example, in two-dimensional space a line is the locus of points equidistant from two fixed points or from two parallel lines of zeros If the function is mapping from real numbers to real numbers, its zeros are the points where its graph meets the x-axis. An alternative name for the root in this context is the x-intercept of certain functions, usually polynomial functions. Such a definition considered the surface as part of a larger (Euclidean) spaces, and as such was termed extrinsic.

In the previous section, a surface is defined as a topological space with certain property, namely Hausdorff and locally Euclidean. This topological space is not considered as being a subspace of another space. In this sense, the definition given above, which is the definition that mathematicians use at present, is intrinsic.

A surface defined as intrinsic is not required to satisfy the added constraint of being a subspace of Euclidean space. It seems possible at first glance that there are surfaces defined intrinsically that are not surfaces in the extrinsic sense. However, the Whitney embedding theorem asserts that every surface can in fact be embedded homeomorphically into Euclidean space, in fact into E⁴. Therefore the extrinsic and intrinsic approaches turn out to be equivalent.

In fact, any compact surface that is either orientable or has a boundary can be embedded in E³; on the other hand, the real projective plane, which is compact, non-orientable and without boundary, cannot be embedded into E³ (see Gramain). Steiner surfaces, including Boy's surface In geometry, Boy's surface is an immersion of the real projective plane in 3-dimensional space found by Werner Boy in 1901 . Unlike the Roman surface and the cross-cap, it has no singularities (i.e. pinch-points), but it does self-intersect, the Roman surface and the cross-cap, are immersions In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup of the real projective plane into E³. These surfaces are singular where the immersions intersect themselves.

The Alexander horned sphere is a well-known pathological Often, when the usefulness of a theorem is challenged by counterexamples, defenders of the theorem argue that the exceptions are pathological. A famous case is the Alexander horned sphere, a counterexample showing that topologically embedding the sphere S2 in R3 may fail to "separate the space cleanly", unless an extra condition of embedding of the two-sphere into the three-sphere.

The chosen embedding (if any) of a surface into another space is regarded as extrinsic information; it is not essential to the surface itself. For example, a torus can be embedded into E³ in the "standard" manner (that looks like a bagel A bagel is a bread product, traditionally shaped by hand into the form of a ring from yeasted wheat dough, roughly hand-sized, which is first boiled for a short time in water and then baked. The result is a dense, chewy, doughy interior with a browned and sometimes crisp exterior. Bagels are often topped with seeds baked on the outer crust, with) or in a knotted In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations . A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a mathematical knot. Physical properties such as manner (see figure). The two embedded tori are homeomorphic but not isotopic; they are topologically equivalent, but their embeddings are not.

The image In mathematics, the image of a subset of a function's domain under the function is the set of all outputs obtained when the function is evaluated at each element of the subset. The inverse image or preimage of a particular subset S of the codomain of a function is the set of all elements of the domain that map to the members of S of a continuous, injective function from R² to higher-dimensional Rⁿ is said to be a parametric surface. Such an image is so-called because the x- and y- directions of the domain R² are 2 variables that parametrize the image. Be careful that a parametric surface need not be a topological surface. A surface of revolution A surface of revolution is a surface in Euclidean space created by rotating a curve around a straight line in its plane (the axis) can be viewed as a special kind of parametric surface.

If f is a smooth function from R³ to R whose gradient In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change is nowhere zero, Then the locus In mathematics, a locus is a collection of points which share a property. The term locus is usually used of a condition which defines a continuous figure or figures, that is, a curve. For example, in two-dimensional space a line is the locus of points equidistant from two fixed points or from two parallel lines of zeros If the function is mapping from real numbers to real numbers, its zeros are the points where its graph meets the x-axis. An alternative name for the root in this context is the x-intercept of f does define a surface, known as an implicit surface. If the condition of non-vanishing gradient is dropped then the zero locus may develop singularities.

Construction from polygons

Each closed surface can be constructed from an oriented polygon with an even number of sides, called a fundamental polygon In mathematics, each closed surface in the sense of geometric topology can be constructed from an even-sided oriented polygon, called a fundamental polygon, by pairwise identification of its edges of the surface, by pairwise identification of its edges. For example, in each polygon below, attaching the sides with matching labels (A with A, B with B), so that the arrows point in the same direction, yields the indicated surface.

Any fundamental polygon can be written symbolically as follows. Begin at any vertex, and proceed around the perimeter of the polygon in either direction until returning to the starting vertex. During this traversal, record the label on each edge in order, with an exponent of -1 if the edge points opposite to the direction of traversal. The four models above, when traversed clockwise starting at the upper left, yield

• sphere: ABB ^{− 1}A ^{− 1}
• real projective plane: ABAB
• torus: ABA ^{− 1}B ^{− 1}
• Klein bottle: ABAB ^{− 1}.

Note that the sphere and the projective plane can both be realized as quotients of the 2-gon, while the torus and Klein bottle require a 4-gon (square).

The expression thus derived from a fundamental polygon of a surface turns out to be the sole relation in a presentation of the fundamental group of the surface with the polygon edge labels as generators. This is a consequence of the Seifert–van Kampen theorem.

Gluing edges of polygons is a special kind of quotient space process. The quotient concept can be applied in greater generality to produce new or alternative constructions of surfaces. For example, the real projective plane can be obtained as the quotient of the sphere by identifying all pairs of opposite points on the sphere. Another example of a quotient is the connected sum.

Connected sums

The connected sum of two surfaces M and N, denoted M # N, is obtained by removing a disk from each of them and gluing them along the boundary components that result. The boundary of a disk is a circle, so these boundary components are circles. The Euler characteristic χ of M # N is the sum of the Euler characteristics of the summands, minus two:

The sphere S is an identity element for the connected sum, meaning that S # M = M. This is because deleting a disk from the sphere leaves a disk, which simply replaces the disk deleted from M upon gluing.

Connected summation with the torus T is also described as attaching a "handle" to the other summand M. If M is orientable, then so is T # M. The connected sum is associative so the connected sum of a finite number of surfaces is well-defined.

The connected sum of two real projective planes is the Klein bottle. The connected sum of the real projective plane and the Klein bottle is homeomorphic to the connected sum of the real projective plane with the torus. Thus, the connected sum of three real projective planes is homeomorphic to the connected sum of the real projective plane with the torus. Any connected sum involving a real projective plane is nonorientable.

Closed surfaces

A closed surface is surface that is compact and without boundary. Examples are spaces like the sphere, the torus and the Klein bottle. Examples of non-closed surfaces are: an open disk, which is a sphere with a puncture; a cylinder, which is a sphere with two punctures; and the Möbius strip.

Classification of closed surfaces

The classification theorem of closed surfaces states that any connected closed surface is homeomorphic to some member of one of these three families:

1. the sphere;
2. the connected sum of g tori, for ;
3. the connected sum of k real projective planes, for .

The surfaces in the first two families are orientable. It is convenient to combine the two families by regarding the sphere as the connected sum of 0 tori. The number g of tori involved is called the genus of the surface. Since the sphere and the torus have Euler characteristics 2 and 0, respectively, it follows that the Euler characteristic of the connected sum of g tori is 2 − 2g.

The surfaces in the third family are nonorientable. Since the Euler characteristic of the real projective plane is 1, the Euler characteristic of the connected sum of k of them is 2 − k.

It follows that a closed surface is determined, up to homeomorphism, by two pieces of information: its Euler characteristic, and whether it is orientable or not. In other words, Euler characteristic and orientability completely classify closed surfaces up to homeomorphism.

For closed surfaces with multiple connected components, they are classified by the class of each of their connected components, and thus one generally assumes that the surface is connected.

Monoid structure

Relating this classification to connected sums, the closed surfaces up to homeomorphism form a monoid with respect to the connected sum, as indeed do manifolds of any fixed dimension. The identity is the sphere, while the real projective plane and the torus generate this monoid, with a single relation P # P # P = P # T, which may also be written P # K = P # T, since K = P # P. This relation is sometimes known as Dyck's theorem after Walther von Dyck, who proved it in (Dyck 1888), and the triple cross surface P # P # P is accordingly called Dyck's surface.^[1]

Geometrically, connect sum with a torus (# T) adds a handle with both ends attached to the same side of the surface, while connect sum with a Klein bottle (# K) adds a handle with the two ends attached to opposite sides of the surface; in the presence of a projective plane (# P), the surface is not orientable (there is no notion of side), so there is no difference between attaching a torus and attaching a Klein bottle, which explains the relation.

Surfaces with boundary

Compact surfaces, possibly with boundary, are simply closed surfaces with a number of holes (open discs that have been removed); thus a connected compact surface is classified by the number of boundary components and the class of the corresponding closed surface – equivalently, by the number of boundary components, the orientability, and Euler characteristic. The genus of a compact surface is defined as the genus of the corresponding closed surface.

This follows quickly but not without note from the result for closed surfaces: removing an open disc from a closed surface yields a compact surface with a circle for boundary component, and removing k open discs yields a compact surface with k disjoint circles for boundary components; the locations of the holes is irrelevant because the homeomorphism group acts k-transitively on any connected manifold of dimension at least 2.

Conversely, the boundary of a compact surface is a closed 1-manifold, which, by classification of 1-manifolds (closed curves), is a disjoint union of a finite number of circles, and these circles can be filled in (formally, taking the cone), yielding a closed surface.

Compact orientable surfaces of genus g and with k boundary components are often denote as Σ_g,k, for example in the study of the mapping class group.

Riemann surfaces

A closely related example to the classification of compact 2-manifolds is the classification of compact Riemann surfaces, i.e., compact complex 1-manifolds. (Note that the 2-sphere, and the tori are all complex manifolds, in fact algebraic varieties.) Since every complex manifold is orientable, the connected sums of projective planes do not qualify. Thus compact Riemann surfaces are characterized topologically simply by their genus. The genus counts the number of holes in the manifold: the sphere has genus 0, the one-holed torus genus 1, etc.

Non-compact surfaces

Non-compact surfaces are more difficult to classify; one simple case is puncturing (removing a point, or multiple points) from a closed manifold, but even a question such as the open subsets of the plane is very complicated — for example, the complement of the Cantor set is a non-compact surface. Another example of a non-compact surface is Jacob's ladder, which looks like a ladder made of handlebodies.

Proof

The classification of closed surfaces has been known since the 1860s,^[1] and today a number of proofs exist.

Topological and combinatorial proofs in general rely on the difficult result that every compact 2-manifold is homeomorphic to a simplicial complex, which is of interest in its own right. The most common proof is (Seifert & Threlfall 1934),^[1] which brings every triangulated surface to a standard form. A simplified proof, which avoids a standard form, was discovered by John H. Conway circa 1992, which he called the "Zero Irrelevancy Proof" or "ZIP proof"^{[note 1]} and is presented in (Francis & Weeks 1999).

A geometric proof, which yields a stronger geometric result, is the uniformization theorem. This was originally proven only for Riemann surfaces in the 1880s and 1900s by Felix Klein, Paul Koebe, and Henri Poincaré.

Surfaces in geometry

Polyhedra, such as the boundary of a cube, are among the first surfaces encountered in geometry. It is also possible to define smooth surfaces, in which each point has a neighborhood diffeomorphic to some open set in E². This elaboration allows calculus to be applied to surfaces to prove many results.

Two smooth surfaces are diffeomorphic if and only if they are homeomorphic. (The analogous result does not hold for higher-dimensional manifolds.) Thus closed surfaces are classified up to diffeomorphism by their Euler characteristic and orientability.

Smooth surfaces equipped with Riemannian metrics are of fundational importance in differential geometry. A Riemannian metric endows a surface with notions of geodesic, distance, angle, and area. It also gives rise to Gaussian curvature, which describes how curved or bent the surface is at each point. Curvature is a rigid, geometric property, in that it is not preserved by general diffeomorphisms of the surface. However, the famous Gauss-Bonnet theorem for closed surfaces states that the integral of the Gaussian curvature K over the entire surface S is determined by the Euler characteristic:

This result exemplifies the deep relationship between the geometry and topology of surfaces (and, to a lesser extent, higher-dimensional manifolds).

Another way in which surfaces arise in geometry is by passing into the complex domain. A complex one-manifold is a smooth oriented surface, also called a Riemann surface. Any complex nonsingular algebraic curve viewed as a real manifold is a Riemann surface.

Every closed orientable surface admits a complex structure. Complex structures on a closed oriented surface correspond to conformal equivalence classes of Riemannian metrics on the surface. One version of the uniformization theorem (due to Poincaré) states that any Riemannian metric on an oriented, closed surface is conformally equivalent to an essentially unique metric of constant curvature. This provides a starting point for one of the approaches to Teichmüller theory, which provides a finer classification of Riemann surfaces than the topological one by Euler characteristic alone.

A complex surface is a complex two-manifold and thus a real four-manifold; it is not a surface in the sense of this article. Neither are algebraic curves or surfaces defined over fields other than the complex numbers.

See also

• Volume form, for volumes of surfaces in Eⁿ
• Poincaré metric, for metric properties of Riemann surfaces
• Area element, the area of a differential element of a surface
• Roman surface
• Boy's surface
• Tetrahemihexahedron

Notes

1. ^ "ZIP proof" is an example of RAS syndrome.

References

1. ^ ^{a b c} (Francis & Weeks 1999)

• Dyck, Walther (1888), "Beiträge zur Analysis situs I", Math. Ann. 32: 459–512
• Gramain, André (1984). Topology of Surfaces. BCS Associates. ISBN 091435101X. (Original 1969-70 Orsay course notes in French for "Topologie des Surfaces")
• Bredon, Glen E. (1993). Topology and Geometry. Springer-Verlag. ISBN 0-387-97926-3.
• Massey, William S. (1991). A Basic Course in Algebraic Topology. Springer-Verlag. ISBN 0-387-97430-X.
• Francis, George K.; Weeks, Jeffrey R. (May 1999), "Conway's ZIP Proof", American Mathematical Monthly 106 (5), http://new.math.uiuc.edu/zipproof/zipproof.pdf, page discussing the paper: On Conway's ZIP Proof

External links

• Math Surfaces Gallery, with 60 ~surfaces and Java Applet for live rotation viewing
• Math Surfaces Animation, with JavaScript (Canvas HTML) for tens surfaces rotation viewing
• The Classification of Surfaces
• History and Art of Surfaces and their Mathematical Models

Categories: Surfaces | Geometric topology | Differential geometry of surfaces | Analytic geometry

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