One-dimensional spaceIn physics and mathematics, a sequence of n numbers can specify a location in n-dimensional space. When n = 1, the set of all such locations is called a one-dimensional space. An example of a one-dimensional space is the number line, where the position of each point on it can be described by a single number. In algebraic geometry there are several structures that are technically one-dimensional spaces but referred to in other terms. A field k is a one-dimensional vector space over itself.
Skew linesIn three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel. A simple example of a pair of skew lines is the pair of lines through opposite edges of a regular tetrahedron. Two lines that both lie in the same plane must either cross each other or be parallel, so skew lines can exist only in three or more dimensions. Two lines are skew if and only if they are not coplanar. If four points are chosen at random uniformly within a unit cube, they will almost surely define a pair of skew lines.
Whitney embedding theoremIn mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney: The strong Whitney embedding theorem states that any smooth real m-dimensional manifold (required also to be Hausdorff and second-countable) can be smoothly embedded in the real 2m-space, \R^{2m}, if m > 0. This is the best linear bound on the smallest-dimensional Euclidean space that all m-dimensional manifolds embed in, as the real projective spaces of dimension m cannot be embedded into real (2m − 1)-space if m is a power of two (as can be seen from a characteristic class argument, also due to Whitney).
Lebesgue covering dimensionIn mathematics, the Lebesgue covering dimension or topological dimension of a topological space is one of several different ways of defining the dimension of the space in a topologically invariant way. For ordinary Euclidean spaces, the Lebesgue covering dimension is just the ordinary Euclidean dimension: zero for points, one for lines, two for planes, and so on. However, not all topological spaces have this kind of "obvious" dimension, and so a precise definition is needed in such cases.
Extra dimensionsIn physics, extra dimensions are proposed additional space or time dimensions beyond the (3 + 1) typical of observed spacetime, such as the first attempts based on the Kaluza–Klein theory. Among theories proposing extra dimensions are: Large extra dimension, mostly motivated by the ADD model, by Nima Arkani-Hamed, Savas Dimopoulos, and Gia Dvali in 1998, in an attempt to solve the hierarchy problem. This theory requires that the fields of the Standard Model are confined to a four-dimensional membrane, while gravity propagates in several additional spatial dimensions that are large compared to the Planck scale.
Brane cosmologyBrane cosmology refers to several theories in particle physics and cosmology related to string theory, superstring theory and M-theory. Brane The central idea is that the visible, three-dimensional universe is restricted to a brane inside a higher-dimensional space, called the "bulk" (also known as "hyperspace"). If the additional dimensions are compact, then the observed universe contains the extra dimension, and then no reference to the bulk is appropriate.
Zero-dimensional spaceIn mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space. A graphical illustration of a nildimensional space is a point. Specifically: A topological space is zero-dimensional with respect to the Lebesgue covering dimension if every open cover of the space has a refinement which is a cover by disjoint open sets.
Schlegel diagramIn geometry, a Schlegel diagram is a projection of a polytope from into through a point just outside one of its facets. The resulting entity is a polytopal subdivision of the facet in that, together with the original facet, is combinatorially equivalent to the original polytope. The diagram is named for Victor Schlegel, who in 1886 introduced this tool for studying combinatorial and topological properties of polytopes. In dimension 3, a Schlegel diagram is a projection of a polyhedron into a plane figure; in dimension 4, it is a projection of a 4-polytope to 3-space.
