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One-dimensional space

From Wikipedia, the free encyclopedia

The number line

A one-dimensional space (1D space) is a mathematical space in which location can be specified with a single coordinate. An example is the number line, each point of which is described by a single real number.[1]

Any straight line or smooth curve is a one-dimensional space, regardless of the dimension of the ambient space in which the line or curve is embedded. Examples include the circle on a plane, or a parametric space curve.

In algebraic geometry there are several structures that are one-dimensional spaces but are usually referred to by more specific terms. Any field is a one-dimensional vector space over itself. The projective line over denoted is a one-dimensional space. In particular, if the field is the complex numbers then the complex projective line is one-dimensional with respect to (but is sometimes called the Riemann sphere, as it is a model of the sphere, two-dimensional with respect to real-number coordinates).

For every eigenvector of a linear transformation T on a vector space V, there is a one-dimensional space AV generated by the eigenvector such that T(A) = A, that is, A is an invariant set under the action of T.[2]

In Lie theory, a one-dimensional subspace of a Lie algebra is mapped to a one-parameter group under the Lie group–Lie algebra correspondence.[3]

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.

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Transcription

How do we know we live in three dimensions? Here's a clue: it's not just that we have to use three coordinates (like x,y,z, or latitude, longitude, altitude) to label every point in space - because we don't! Mathematicians have showed that it's possible to fill up 2d or 3d space using a one-dimensional "space-filling" curve - that means that every point in 3d space can be labelled using just one coordinate: our position along the curve! (it also means that a square and its side contain the same number of points - crazy, right?) So how do we know that we live in three-dimensional space and not on a one-dimensional line curled up so much that it looks three-dimensional? Well, the short answer is that we don't know – but we DO know that it looks 3d. So how do we test that? One way is to look at diffusion of gas - that is, how a gas spreads out over time. We just measure the ratio between volume and radius of the gas cloud: In one dimension, radius and volume are the same! (up to a factor) In 2d, "volume" means area - or radius squared, and in 3d, "volume" is radius cubed, and so on for higher dimensions… and 3d is what we see. So basically, determining how many dimensions we live in is just a bunch of hot air!

Coordinate systems in one-dimensional space

One dimensional coordinate systems include the number line.

See also

References

  1. ^ Гущин, Д. Д. "Пространство как математическое понятие" (in Russian). fmclass.ru. Retrieved 2015-06-06.
  2. ^ Peter Lancaster & Miron Tismenetsky (1985) The Theory of Matrices, second edition, page 147, Academic Press ISBN 0-12-435560-9
  3. ^ P. M. Cohn (1961) Lie Groups, page 70, Cambridge Tracts in Mathematics and Mathematical Physics # 46
This page was last edited on 13 February 2024, at 09:59
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