One-dimensional space
In physics and mathematics, a sequence of n numbers can be understood as 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.[1]
One-dimensional geometry
Lines
The real number line and the real projective line are one-dimensional, though the latter is homeomorphic to a circle.
In algebraic geometry there are several structures which are technically one-dimensional spaces but referred to in other terms. For a field k, it is a one-dimensional vector space over itself. Similarly, the projective line over k is a one-dimensional space. In particular, if k = ℂ, the complex number plane, then the complex projective line P1(ℂ) is one-dimensional with respect to ℂ, even if it is also known as the Riemann sphere.
More generally, a ring is a length-one module over itself. Similarly, the projective line over a ring is a one-dimensional space over the ring. In case the ring is an algebra over a field, these spaces are one-dimensional with respect to the algebra, even if the algebra is of higher dimensionality.
Polytopes
The only regular polytope in one dimension is the line segment, with the Schläfli symbol { }.
Hypersphere
The hypersphere in 1 dimension is a pair of points,[2] sometimes called a 0-sphere as its surface is zero-dimensional. Its length is
where is the radius.
Coordinate systems in one-dimensional space
The most popular coordinate systems are the number line and the angle.
References
- ↑ Гущин, Д. Д. "Пространство как математическое понятие" (in Russian). fmclass.ru. Retrieved 2015-06-06.
- ↑ Gibilisco, Stan (1983). Understanding Einstein's Theories of Relativity: Man's New Perspective on the Cosmos. TAB Books. p. 89.
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