History
Parameter space contributed to the liberation of geometry from the confines of three-dimensional space. For instance, the parameter space of spheres in three dimensions, has four dimensions -- three for the sphere center and another for the radius. According to Dirk Struik, it was the book Neue Geometrie des Raumes (1849) by Julius Plücker that showed
- ...geometry need not solely be based on points as basic elements. Lines, planes, circles, spheres can all be used as the elements (Raumelemente) on which a geometry can be based. This fertile conception threw new light on both synthetic and algebraic geometry and created new forms of duality. The number of dimensions of a particular form of geometry could now be any positive number, depending on the number of parameters necessary to define the "element".
The requirement for higher dimensions is illustrated by Plücker's line geometry. Struik writes
- geometry of lines in three-space could be considered as a four-dimensional geometry, or, as Klein has stressed, as the geometry of a four-dimensional quadric in a five-dimensional space.
Read more about this topic: Parameter Space
Famous quotes containing the word history:
“... the history of the race, from infancy through its stages of barbarism, heathenism, civilization, and Christianity, is a process of suffering, as the lower principles of humanity are gradually subjected to the higher.”
—Catherine E. Beecher (1800–1878)
“Whenever we read the obscene stories, the voluptuous debaucheries, the cruel and torturous executions, the unrelenting vindictiveness, with which more than half the Bible is filled, it would be more consistent that we called it the word of a demon than the Word of God. It is a history of wickedness that has served to corrupt and brutalize mankind.”
—Thomas Paine (1737–1809)
“To summarize the contentions of this paper then. Firstly, the phrase ‘the meaning of a word’ is a spurious phrase. Secondly and consequently, a re-examination is needed of phrases like the two which I discuss, ‘being a part of the meaning of’ and ‘having the same meaning.’ On these matters, dogmatists require prodding: although history indeed suggests that it may sometimes be better to let sleeping dogmatists lie.”
—J.L. (John Langshaw)