Flatness
From Free net encyclopedia
The intuitive idea of flatness is important in several fields.
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Flatness in mathematics
The flatness of a surface is the degree to which it approximates a mathematical plane. The term is generalized for higher-dimensional manifolds to describe the degree to which they approximate the Euclidean space of the same dimensionality. See curvature.
Flatness in homological algebra and algebraic geometry means, of an object <math>A</math> in an abelian category, that <math>- \otimes A</math> is an exact functor. See flat module or, for more generality, flat morphism.
Flatness in cosmology
Template:Mergeto In cosmology, the concept of "curvature of space" is considered. A space without curvature is called a "flat space" or Euclidean space.
A question often asked is "is the Universe flat"? According to Albert Einstein's theory of relativity, it probably is curved and warped due to gravity.
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Flatness in mechanical engineering
Template:Mergeto Joseph Whitworth popularized the first practical method of making accurate flat surfaces during the 1830s, using engineer's blue and scraping techniques on three trial surfaces. By testing all three pairs against each other, it is ensured that the surfaces become flat. Using two surfaces would result in a concave surface and a convex surface. Eventually a point is reached when many points of contact are visible within each square inch, at which time the three surfaces are uniformly flat to a very close tolerance.[1]
Up until his introduction of the scraping technique, the same three plate method was employed using polishing techniques, giving less accurate results. This led to an explosion of development of precision instruments using these flat surface generation techniques as a basis for further construction of precise shapes.
References
- Wayne R. Moore, Foundations of Mechanical Accuracy, Moore Special Tool Company, Bridgeport, CT (1970)
- Joseph Whitworth, Plane Metalic Surfaces, Longman, Brown, and Co., London (1858)
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Flatness in liquids
Template:Mergeto A carbonated beverage becomes flat when it loses enough of its carbon dioxide that there is no more "fizz" left, although this refers to the intrinsic properties of the substance, rather than the geometric properties of the liquid.
On planet earth, the flatness of a liquid is a function of the curvature of the earth, and from trigonometry, can be found to deviate from true flatness by approximately 19.6 nanometers over an area of 1 square meter. This is using the earths mean radius at sea level, however a liquid will be slightly flatter at the poles..