Higher dimension

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Higher dimension as a term in mathematics most commonly refers to any number of spatial dimensions greater than three.

The three standard dimensions are length, width, and breadth (or height). The first higher dimension required is often time, and space-time is the most common example of a four-dimensional space.

In physics and chemistry, the dimensions of a system are referred to as its "degrees of freedom".

1 History
2 Application
3 See also
4 References

[edit] History

The introduction of Cartesian coordinates reduced the three spatial dimensions to three real numbers. The possibility of "geometry of higher dimensions" was thereby opened up: the list of numbers could in principle be longer than three. Applications to geometry awaited the needs of mathematicians.

Historically, the notion of higher dimensions was introduced by Bernhard Riemann, in his 1854 Habilitationsschrift, Über die Hypothesen welche der Geometrie zu Grunde liegen, where he considered a point to be any n numbers $(x_1,\dots,x_n)$ , abstractly, without any geometric picture needed nor implied. He explained the value of this abstraction thus:^[1]

"Solche Untersuchungen, welche, wie hier ausgeführt, von allgemeinen Begriffen ausgehen, können nur dazu dienen, dass diese Arbeit nicht durch die Beschränktheit der Begriffe gehindert und der Fortschritt im Erkennen des Zusammenhangs der Dinge nicht durch überlieferte Vorurteile gehemmt wird."

Loosely translated:

"Abstract studies such as these allow one to observe relationships without being limited by narrow terms, and prevent traditional prejudices from inhibiting one's progress."

The abstract notion of coordinates was preceded by the homogeneous coordinates of August Ferdinand Möbius, of 1827.

[edit] Application

It is commonplace in advanced pure and applied mathematics to study abstract sets and applied models with many dimensions. For instance, the configuration space of a rigid body in Euclidean 3-space is the 6-dimensional group of rigid motions E⁺(3), with 3 dimensions for position (translation) and 3 for orientation (rotation).

Fairly simple constructions yield spaces with arbitrarily high positive integer dimension, and only slightly more sophistication is required to construct spaces of infinite dimension.

In geometric topology, the nature of the difficulties in the subject has turned out to be such that dimensions 3 and 4 are the most resistant (see for example Whitney disc). Therefore in that context higher dimension usually means dimension ≥ 5.