Parameter Space - History

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:

    Those who weep for the happy periods which they encounter in history acknowledge what they want; not the alleviation but the silencing of misery.
    Albert Camus (1913–1960)

    No one is ahead of his time, it is only that the particular variety of creating his time is the one that his contemporaries who are also creating their own time refuse to accept.... For a very long time everybody refuses and then almost without a pause almost everybody accepts. In the history of the refused in the arts and literature the rapidity of the change is always startling.
    Gertrude Stein (1874–1946)

    It’s a very delicate surgical operation—to cut out the heart without killing the patient. The history of our country, however, is a very tough old patient, and we’ll do the best we can.
    Dudley Nichols, U.S. screenwriter. Jean Renoir. Sorel (Philip Merivale)