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Saddle Tower
In differential geometry, a saddle tower is a minimal surface family generalizing the singly periodic Scherk's second surface so that it has ''N''-fold (''N'' > 2) symmetry around one axis. These surfaces are the only properly embedded singly periodic minimal surfaces in R3 with genus zero and finitely many Scherk-type ends End, END, Ending, or variation, may refer to: End *In mathematics: **End (category theory) ** End (topology) **End (graph theory) ** End (group theory) (a subcase of the previous) **End (endomorphism) *In sports and games ** End (gridiron footbal ... in the quotient.Joaquın Perez and Martin Traize, The classification of singly periodic minimal surfaces with genus zero and Scherk-type ends, Transactions of the American Mathematical Society, Volume 359, Number 3, March 2007, Pages 965–990 References External links Images of The Saddle Tower Surface Families {{Minimal surfaces Minimal surfaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Saddle Tower Minimal Surfaces
The saddle is a supportive structure for a rider of an animal, fastened to an animal's back by a girth. The most common type is equestrian. However, specialized saddles have been created for oxen, camels and other animals. It is not known precisely when riders first began to use some sort of padding or protection, but a blanket attached by some form of surcingle or girth was probably the first "saddle", followed later by more elaborate padded designs. The solid saddle tree was a later invention, and though early stirrup designs predated the invention of the solid tree, the paired stirrup, which attached to the tree, was the last element of the saddle to reach the basic form that is still used today. Today, modern saddles come in a wide variety of styles, each designed for a specific equestrianism discipline, and require careful fit to both the rider and the horse. Proper saddle care can extend the useful life of a saddle, often for decades. The saddle was a crucial step in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Surface
In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap film, which is a minimal surface whose boundary is the wire frame. However, the term is used for more general surfaces that may self-intersect or do not have constraints. For a given constraint there may also exist several minimal surfaces with different areas (for example, see minimal surface of revolution): the standard definitions only relate to a local optimum, not a global optimum. Definitions Minimal surfaces can be defined in several equivalent ways in R3. The fact that they are equivalent serves to demonstrate how minimal surface theory lies at the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scherk Surface
In mathematics, a Scherk surface (named after Heinrich Scherk) is an example of a minimal surface. Scherk described two complete embedded minimal surfaces in 1834; his first surface is a doubly periodic surface, his second surface is singly periodic. They were the third non-trivial examples of minimal surfaces (the first two were the catenoid and helicoid). The two surfaces are conjugates of each other. Scherk surfaces arise in the study of certain limiting minimal surface problems and in the study of harmonic diffeomorphisms of hyperbolic space. Scherk's first surface Scherk's first surface is asymptotic to two infinite families of parallel planes, orthogonal to each other, that meet near ''z'' = 0 in a checkerboard pattern of bridging arches. It contains an infinite number of straight vertical lines. Construction of a simple Scherk surface Consider the following minimal surface problem on a square in the Euclidean plane: for a natural number ''n'', find a mini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Genus (mathematics)
In mathematics, genus (plural genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a surface. A sphere has genus 0, while a torus has genus 1. Topology Orientable surfaces The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It is equal to the number of handles on it. Alternatively, it can be defined in terms of the Euler characteristic ''χ'', via the relationship ''χ'' = 2 − 2''g'' for closed surfaces, where ''g'' is the genus. For surfaces with ''b'' boundary components, the equation reads ''χ'' = 2 − 2''g'' − ''b''. In layman's terms, it's the number of "holes" an object has ("holes" interpreted in the sense of doughnut holes; a hollow sphere would be considered as having zero holes in this sense). A torus has 1 such h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
End (topology)
In topology, a branch of mathematics, the ends of a topological space are, roughly speaking, the connected components of the "ideal boundary" of the space. That is, each end represents a topologically distinct way to move to infinity within the space. Adding a point at each end yields a compactification of the original space, known as the end compactification. The notion of an end of a topological space was introduced by . Definition Let ''X'' be a topological space, and suppose that :K_1 \subseteq K_2 \subseteq K_3 \subseteq \cdots is an ascending sequence of compact subsets of ''X'' whose interiors cover ''X''. Then ''X'' has one end for every sequence :U_1 \supseteq U_2 \supseteq U_3 \supseteq \cdots, where each ''U''''n'' is a connected component of ''X'' \ ''K''''n''. The number of ends does not depend on the specific sequence of compact sets; there is a natural bijection between the sets of ends associated with any two such sequences. Using this definiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
