Roman Surface
In mathematics, the Roman surface or Steiner surface is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry. This mapping is not an immersion of the projective plane; however, the figure resulting from removing six singular points is one. Its name arises because it was discovered by Jakob Steiner when he was in Rome in 1844. The simplest construction is as the image of a sphere centered at the origin under the map f(x,y,z)=(yz,xz,xy). This gives an implicit formula of : x^2 y^2 + y^2 z^2 + z^2 x^2 - r^2 x y z = 0. \, Also, taking a parametrization of the sphere in terms of longitude () and latitude (), gives parametric equations for the Roman surface as follows: :x=r^ \cos \theta \cos \varphi \sin \varphi :y=r^ \sin \theta \cos \varphi \sin \varphi :z=r^ \cos \theta \sin \theta \cos^ \varphi The origin is a triple point, and each of the -, -, and -planes are tangential to the surface there. The ot ...
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Steiner's Roman Surface
Steiner may refer to: Felix Steiner, German Waffen SS-commander Surname *Steiner (surname) Other uses *Steiner, Michigan, a village in the United States * Steiner, Mississippi * Steiner Studios, film and television production studio in New York City * Steiner's theorem, used to determine the mass moment of inertia around an axis. Also known as parallel axis theorem See also * Poncelet–Steiner theorem *Steiner point (other) * Steiner surface *Steiner system, a type of block design *Steiner tree *Waldorf education, also called Steiner education *The Steiner Brothers The Steiner Brothers are an American professional wrestling tag team consisting of brothers Robert "Rick Steiner" Rechsteiner and Scott "Scott Steiner" Rechsteiner. The brothers wrestled as amateurs at the University of Michigan. The team ma ...

Veronese Surface
In mathematics, the Veronese surface is an algebraic surface in five-dimensional projective space, and is realized by the Veronese embedding, the embedding of the projective plane given by the complete linear system of conics. It is named after Giuseppe Veronese (1854–1917). Its generalization to higher dimension is known as the Veronese variety. The surface admits an embedding in the four-dimensional projective space defined by the projection from a general point in the five-dimensional space. Its general projection to three-dimensional projective space is called a Steiner surface. Definition The Veronese surface is the image of the mapping :\nu:\mathbb^2\to \mathbb^5 given by :\nu: :y:z\mapsto ^2:y^2:z^2:yz:xz:xy/math> where :\cdots/math> denotes homogeneous coordinates. The map \nu is known as the Veronese embedding. Motivation The Veronese surface arises naturally in the study of conics. A conic is a degree 2 plane curve, thus defined by an equation: :Ax^2 + Bxy + Cy^2 ...
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Orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is orientable if such a consistent definition exists. In this case, there are two possible definitions, and a choice between them is an orientation of the space. Real vector spaces, Euclidean spaces, and spheres are orientable. A space is non-orientable if "clockwise" is changed into "counterclockwise" after running through some loops in it, and coming back to the starting point. This means that a geometric shape, such as , that moves continuously along such a loop is changed into its own mirror image . A Möbius strip is an example of a non-orientable space. Various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds oft ...
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Light Bulb
An electric light, lamp, or light bulb is an electrical component that produces light. It is the most common form of artificial lighting. Lamps usually have a base made of ceramic, metal, glass, or plastic, which secures the lamp in the socket of a light fixture, which is often called a "lamp" as well. The electrical connection to the socket may be made with a screw-thread base, two metal pins, two metal caps or a bayonet cap. The three main categories of electric lights are incandescent lamps, which produce light by a filament heated white-hot by electric current, gas-discharge lamps, which produce light by means of an electric arc through a gas, such as fluorescent lamps, and LED lamps, which produce light by a flow of electrons across a band gap in a semiconductor. Before electric lighting became common in the early 20th century, people used candles, gas lights, oil lamps, and fires. Vasily Vladimirovich Petrov developed the first persistent electric arc in 1802, and ...
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Orchid
Orchids are plants that belong to the family Orchidaceae (), a diverse and widespread group of flowering plants with blooms that are often colourful and fragrant. Along with the Asteraceae, they are one of the two largest families of flowering plants. The Orchidaceae have about 28,000 currently accepted species, distributed in about 763 genera. (See ''External links'' below). The determination of which family is larger is still under debate, because verified data on the members of such enormous families are continually in flux. Regardless, the number of orchid species is nearly equal to the number of bony fishes, more than twice the number of bird species, and about four times the number of mammal species. The family encompasses about 6–11% of all species of seed plants. The largest genera are ''Bulbophyllum'' (2,000 species), ''Epidendrum'' (1,500 species), ''Dendrobium'' (1,400 species) and ''Pleurothallis'' (1,000 species). It also includes ''Vanilla'' (the genus of the ...
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Origin (mathematics)
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter ''O'', used as a fixed point of reference for the geometry of the surrounding space. In physical problems, the choice of origin is often arbitrary, meaning any choice of origin will ultimately give the same answer. This allows one to pick an origin point that makes the mathematics as simple as possible, often by taking advantage of some kind of geometric symmetry. Cartesian coordinates In a Cartesian coordinate system, the origin is the point where the axes of the system intersect.. The origin divides each of these axes into two halves, a positive and a negative semiaxis. Points can then be located with reference to the origin by giving their numerical coordinates—that is, the positions of their projections along each axis, either in the positive or negative direction. The coordinates of the origin are always all zero, for example (0,0) in two dimensions and (0,0,0) in three. Ot ...
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