The concept of a configuration may be generalized to higher dimensions, for instance to points and lines or planes in space. In such cases, the restrictions that no two points belong to more than one line may be relaxed, because it is possible for two points to belong to more than one plane.
Notable three-dimensional configurations are the Möbius configuration, consisting of two mutually inscribed tetrahedra, Reye's configuration, consisting of twelve points and twelve planes, with six points per plane and six planes per point, the Gray configuration consisting of a 3×3×3 grid of 27 points and the 27 orthogonal lines through them, and the Schläfli double six, a configuration with 30 points, 12 lines, two lines per point, and five points per line.
A further generalization is obtained in three dimensions by considering incidences of points, lines and planes, or j-spaces (0 ≤ j < 3), where each j-space is incident with Njk k-spaces (j ≠ k). Writing for the number of j-spaces present. a given configuration may be represented by the matrix:
The principle extends generally to n dimensions, where 0 ≤ j < n. Such configurations are related mathematically to regular polytopes.
Read more about this topic: Configuration (geometry)
Other articles related to "dimensions, dimension, higher dimensions, higher":
... See also n-dimensional space#History The possibility of spaces with dimensions higher than three was first studied by mathematicians in the 19th century ... In 1827 Möbius realized that a fourth dimension would allow a three-dimensional form to be rotated onto its mirror-image, and by 1853 Ludwig Schläfli ... Higher dimensions were soon put on firm footing by Bernhard Riemann's 1854 Habilitationsschrift, Über die Hypothesen welche der Geometrie zu Grunde liegen, in which he ...
... Higher dimensions Find the number of non-attacking queens that can be placed in a d-dimensional chess space of size n ... More than n queens can be placed in some higher dimensions (the smallest example is four non-attacking queens in a 3 × 3 × 3 chess space), and it is in fact known that for any k, there are higher dimensions ...
... ("classic" noise) but with a lower computational overhead, especially in larger dimensions ... to address the limitations of his classic noise function, especially in higher dimensions ... Simplex noise scales to higher dimensions (4D, 5D) with much less computational cost, the complexity is for dimensions instead of the of classic noise ...
... This higher dimensional concordance is a relative form of cobordism – it requires two submanifolds to be not just abstractly cobordant, but "cobordant in N" ...
... This can be described in any dimension but is best done in three where parallels can be drawn with more familiar objects, before being applied to higher dimensions ... In two dimensions the geometric interpretation is trivial, as the space is two dimensional so has only one plane, and all bivectors are associated with it differing only by a scale factor ... In three dimensions there are three properties of a bivector that can be interpreted geometrically The arrangement of the plane in space, precisely the attitude of the plane (or ...
Famous quotes containing the words dimensions and/or higher:
“It seems to me that we do not know nearly enough about ourselves; that we do not often enough wonder if our lives, or some events and times in our lives, may not be analogues or metaphors or echoes of evolvements and happenings going on in other people?or animals?even forests or oceans or rocks?in this world of ours or, even, in worlds or dimensions elsewhere.”
—Doris Lessing (b. 1919)
“Art, it seems to me, should simplify. That, indeed, is very nearly the whole of the higher artistic process; finding what conventions of form and what detail one can do without and yet preserve the spirit of the wholeso that all that one has suppressed and cut away is there to the readers consciousness as much as if it were in type on the page.”
—Willa Cather (18731947)