Inversion in Higher Dimensions
In the spirit of generalization to higher dimensions, inversive geometry is the study of transformations generated by the Euclidean transformations together with inversion in an n-sphere:
where r is the radius of the inversion.
In 2 dimensions, with r = 1, this is circle inversion with respect to the unit circle.
As said, in inversive geometry there is no distinction made between a straight line and a circle (or hyperplane and hypersphere): a line is simply a circle in its particular embedding in a Euclidean geometry (with a point added at infinity) and one can always be transformed into another.
A remarkable fact about higher-dimensional conformal maps is that they arise strictly from inversions in n-spheres or hyperplanes and Euclidean motions: see Liouville's theorem (conformal mappings).
Read more about this topic: Inversive Geometry
Famous quotes containing the words higher and/or dimensions:
“They had their fortunes to make, everything to gain and nothing to lose. They were schooled in and anxious for debates; forcible in argument; reckless and brilliant. For them it was but a short and natural step from swaying juries in courtroom battles over the ownership of land to swaying constituents in contests for office. For the lawyer, oratory was the escalator that could lift a political candidate to higher ground.”
—Federal Writers’ Project Of The Wor, U.S. public relief program (1935-1943)
“Words are finite organs of the infinite mind. They cannot cover the dimensions of what is in truth. They break, chop, and impoverish it.”
—Ralph Waldo Emerson (1803–1882)