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A bubble is a minimal-energy surface of the type that is formed by soap film. The simplest bubble is a single sphere, illustrated above (courtesy of J. M. Sullivan). More complicated forms occur when multiple bubbles are joined together. The simplest example is the double bubble, and beautiful configurations can form when three or more bubbles are conjoined (Sullivan).An outstanding problem involving bubbles is the determination of the arrangements of bubbles with the smallest surface area which enclose and separate given volumes in space.

Let a vault consist of two equal half-cylinders of radius which intersect at right angles so that the lines of their intersections (the "groins") terminate in the polyhedron vertices of a square. Two vaults placed bottom-to-top form a Steinmetz solid on two cylinders.Solving the equations(1)(2)simultaneously gives(3)(4)One quarter of the vault can therefore be described by the parametricequations(5)(6)(7)The surface area of the vault is therefore givenby(8)where is the length of a cross section at height and is the angle a point on the center of this line makes with the origin. But , so(9)and(10)(11)(12)The volume of the vault is(13)(14)The geometric centroid is(15)

A sphericon is the solid formed from a bicone with opening angle of (and therefore with ) obtained by slicing the solid with a plane containing the rotational axes resulting in a square cross section, then rotating the two pieces by and reconnecting them. It was constructed by Israeli game and toy inventor David Hirsch who patented the shape in Israel in 1984. It was given the name "sphericon" by Colin Roberts, who independently discovered the solid in the 1960s while attempting to carve a Möbius strip without a hole in the middle out of a block of wood.The solid is not as widely known as it should be.The above net shows another way the sphericon can be constructed. In this figure radians . A sphericon has a single continuous face and rolls by wobbling along that face, resulting in straight-line motion. In addition, one sphericon can roll around another.The sphericon with radius has surface area and volume(1)(2)The centroid is at the..

A tube of radius of a set is the set of points at a distance from . In particular, if is a regular space curve whose curvature does not vanish, then the normal vector and binormal vector are always perpendicular to , and the circle is perpendicular to at . So as the circle moves around , it traces out a tube, provided the tube radius is small enough so that the tube is not self-intersecting. A formula for the tube around a curve is therefore given byfor over the range of the curve and . The illustrations above show tubes corresponding to a circle, helix, and two torus knots.The surface generated by constructing a tube around a circleis known as a torus.

A double bubble is pair of bubbles which intersect and are separated by a membrane bounded by the intersection. The usual double bubble is illustrated in the left figure above. A more exotic configuration in which one bubble is torus-shaped and the other is shaped like a dumbbell is illustrated at right (illustrations courtesy of J. M. Sullivan).In the plane, the analog of the double bubble consists of three circular arcs meeting in two points. It has been proved that the configuration of arcs meeting at equal angles) has the minimum perimeter for enclosing two equal areas (Alfaro et al. 1993, Morgan 1995).It had been conjectured that two equal partial spheres sharing a boundary of a flat disk separate two volumes of air using a total surface area that is less than any other boundary. This equal-volume case was proved by Hass et al. (1995), who reduced the problem to a set of integrals which they carried out on an ordinary PC. Frank Morgan,..

A projective plane, sometimes called a twisted sphere (Henle 1994, p. 110), is a surface without boundary derived from a usual plane by addition of a line at infinity. Just as a straight line in projective geometry contains a single point at infinity at which the endpoints meet, a plane in projective geometry contains a single line at infinity at which the edges of the plane meet. A projective plane can be constructed by gluing both pairs of opposite edges of a rectangle together giving both pairs a half-twist. It is a one-sided surface, but cannot be realized in three-dimensional space without crossing itself.A finite projective plane of order is formally defined as a set of points with the properties that: 1. Any two points determine a line,2. Any two lines determine a point,3. Every point has lines on it, and 4. Every line contains points. (Note that some of these properties are redundant.) A projective plane is therefore a symmetric (, , 1)..

A generalization of an ordinary two-dimensional surface embedded in three-dimensional space to an -dimensional surface embedded in -dimensional space. A hypersurface is therefore the set of solutions to a single equationand so it has codimension one. For instance, the -dimension hypersphere corresponds to the equation .

Let two disks of radius intersect one another perpendicularly and have a diameter in common. If the distance between the centers of the disks is times their radius, then the distance from the center of gravity remains constant and so the object, known as a "two circle roller," rolls smoothly (Nishihara).If the distance of two centers of disk is equal to the radius, then the convex hull produces another figure that rolls smoothly and is known as the oloid (Schatz 1975, p. 122; Nishihara), illustrated above. The oloid is an octic surface (Trott 2004, pp. 1194-1196).For circles of radii , the surface area of the resulting oloid is(the same as that of a sphere with radius ), but no closed form is apparently known for the enclosed volume.

The term "poweroid" has at least two meanings. Sheffer sequences are sometimes called poweroids (Steffensen 1941, Shiu 1982, Di Bucchianico and Loeb 2000). Jackway and Deriche (1996) and Jackway (2002) use the term to mean a function of the formThe case corresponds to the paraboloid and the case is sometimes called a quartoid (Jackway and Deriche 1996).

The surface with parametric equations(1)(2)(3)where is the torsion.The coefficients of the first fundamental formare(4)(5)(6)and those of the second fundamental formare(7)(8)(9)The Gaussian curvature is a somewhat complicated,but the mean curvature is given by(10)

The right conoid surface given by the parametricequations(1)(2)(3)

The surface generated by a twisted curve when rotated about a fixed axis and, at the same time, displaced parallel to so that the velocity of displacement is always proportional to the angular velocity of rotation.

A smooth two-dimensional surface given by embedding the projectiveplane into projective 5-space by the homogeneous parametric equationsThe surface can be projected smoothly into four-space, but all three-dimensional projections have singularities (Coffman). The projections of these surfaces in three dimensions are called Steiner surfaces. The volume of the Veronese surface is .

The envelope of the plane(1)where is the speed of propagation of a wave in the direction (i.e., , , and are the direction cosines) is known as the wave surface of a given medium (Love 1944, p. 299).In an isotropic medium, is independent of , , and and is given by(2)where is the medium density and and are the so-called Lamé constants of the solid. The wave surface is then two-sheeted and both sheets are spheres (Love 1944, p. 299).In the case of anisotropy, the surface in consists of three sheets corresponding to the values of that are roots of(3)where the are functions of , , and in terms of the coefficients of the strain-energy function (Christoffel 1877, Love 1944, p. 299). Green (1839) showed that the wave surface for the most general case of an elastic solid that allows the propagation of purely transverse plane waves consists of a sphere and the two sheets that are the envelope of the plane (1) subject to the condition(4)where..

A surface is -embeddable if it can be placed in -space without self-intersections, but cannot be similarly placed in any for . A surface so embedded is said to be an embedded surface. The Costa minimal surface and gyroid are embeddable in , but the Klein bottle is not (the commonly depicted representation requires the surface to pass through itself).There is particular interest in surfaces which are minimal, complete, and embedded.

A regular surface is called orientable if each tangent space has a complex structure such that is a continuous function.

The corkscrew surface, sometimes also called the twisted sphere (Gray 1997, p. 477), is a surface obtained by extending a sphere along a diameter and then twisting. It can be specified parametrically as(1)(2)(3)The coefficients of the first fundamental formare(4)(5)(6)and those of the second fundamental formare(7)(8)(9)The Gaussian and meancurvatures are(10)(11)

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 two-dimensional submanifold of three-dimensional 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 -dimensional submanifold of an -dimensional manifold, or in general, any codimension-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...

Let be a link in and let there be a disk in the link complement . Then a surface such that intersects exactly in its boundary and its boundary does not bound another disk on is called a compressible surface (Adams 1994, p. 86).

A surface generated by the parametricequations(1)(2)(3)The above image uses and .The coefficients of the first fundamental formare(4)(5)(6)the coefficients of the second fundamentalform are(7)(8)(9)the area element is(10)and the Gaussian and meancurvatures are given by(11)(12)

A nonuniform rational B-spline surface of degree is defined bywhere and are the B-spline basis functions, are control points, and the weight of is the last ordinate of the homogeneous point .NURBS surfaces are implemented in the WolframLanguage as BSplineSurface[array].

A surface such as the Möbius strip or Klein bottle (Gray 1997, pp. 322-323) on which there exists a closed path such that the directrix is reversed when moved around this path. The real projective plane is also a nonorientable surface, as are the Boy surface, cross-cap, and Roman surface, all of which are homeomorphic to the real projective plane (Pinkall 1986).There is a general method for constructing nonorientable surfaces which proceeds as follows (Banchoff 1984, Pinkall 1986). Choose three homogeneous polynomials of positive even degree and consider the map(1)Then restricting , , and to the surface of a sphere by writing(2)(3)(4)and restricting to and to defines a map of the real projective plane to .In three dimensions, there is no unbounded nonorientable surface which does not intersect itself (Kuiper 1961, Pinkall 1986)...

A capsule is a term coined here for a stadium of revolution, i.e., a cylinder with two hemispherical caps on either end.The capsule is implemented in the Wolfram Language as CapsuleShape[x1, y1, z1, x2, y2, z2, r].A capsule with cap radius and cylinder height has volumeand surface area

A surface parameterized in variables and is called smooth if the tangent vectors in the and directions satisfywhere is a cross product.

A surface given by the parametric equations(1)(2)(3)The coefficients of the coefficients of the firstfundamental form are(4)(5)(6)and the second fundamental form coefficientsare(7)(8)(9)giving area element(10)and Gaussian and meancurvatures(11)(12)

A projection of the Veronese surface into three dimensions (which must contain singularities) is called a Steiner surface. A classification of Steiner surfaces allowing complex parameters and projective transformations was accomplished in the 19th century. The surfaces obtained by restricting to real parameters and transformations were classified into 10 types by Coffman et al. (1996). Examples of Steiner surfaces include the Roman surface (sometimes know as "the" Steiner surface; Coffman type 1) and cross-cap (type 3).The Steiner surface of type 2 is given by the implicit equationand can be transformed into the Roman surface or cross-cap by a complex projective change of coordinates (but not by a real transformation). It has two pinch points and three double lines and, unlike the Roman surface or cross-cap, is not compact in any affine neighborhood.The Steiner surface of type 4 has the implicit equationand two of the three..

The Klein bottle is a closed nonorientable surface of Euler characteristic 0 (Dodson and Parker 1997, p. 125) that has no inside or outside, originally described by Felix Klein (Hilbert and Cohn-Vossen 1999, p. 308). It can be constructed by gluing both pairs of opposite edges of a rectangle together giving one pair a half-twist, but can be physically realized only in four dimensions, since it must pass through itself without the presence of a hole. Its topology is equivalent to a pair of cross-caps with coinciding boundaries (Francis and Weeks 1999). It can be represented by connecting the side of a square in the orientations illustrated in the right figure above (Gardner 1984, pp. 15-17; Gray 1997, pp. 323-324).It can be cut in half along its length to make two Möbius strips (Dodson and Parker 1997, p. 88), but can also be cut into a single Möbius strip (Gardner 1984, pp. 14 and 17).The above picture..

Algebraic geometry is the study of geometries that come from algebra, in particular, from rings. In classical algebraic geometry, the algebra is the ring of polynomials, and the geometry is the set of zeros of polynomials, called an algebraic variety. For instance, the unit circle is the set of zeros of and is an algebraic variety, as are all of the conic sections.In the twentieth century, it was discovered that the basic ideas of classical algebraic geometry can be applied to any commutative ring with a unit, such as the integers. The geometry of such a ring is determined by its algebraic structure, in particular its prime ideals. Grothendieck defined schemes as the basic geometric objects, which have the same relationship to the geometry of a ring as a manifold to a coordinate chart. The language of category theory evolved at around the same time, largely in response to the needs of the increasing abstraction in algebraic geometry.As a consequence,..

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