﻿ Four-dimensional space

# Four-dimensional space

4D or 4 -D is a common abbreviation for four-dimensional as an indication of a geometric dimension.

4D is an enlargement of the display of objects in the 3D space of our experience reality (length width height, coordinates x, y, z) is an independent auxiliary dimension for unambiguous detection of the position and extension of a body. Using the Cartesian coordinates x, y, z is usually added an axis labeled w.

## The fourth dimension

The introduction of multi-dimensional "hyper- space ", which are our direct experience is not available, results from the areas of higher mathematics or physics. As an illustration of spaces of higher dimension is eg extended the 3D space to add an extra dimension and the body representation projected into the 3D space.

In generalization of the physical theories of Albert Einstein, the "fourth dimension" as by the time occupied adopted and set equal to the 4D space with the term space-time. Binding assumptions of physical theories concerning the homogeneity of space ( which all dimensions in terms of quality and quantity are indistinguishable ) and the Betrachterinvarianz ( according to which the origin of the coordinate system and the choice of the axes have no effect on the presentation of the laws of nature ), however, suggest that the time can not be accepted as a fourth dimension. Forms of space may be distinguished from those of the time by, for example, tries to move along an axis of an observer. Manages the shift axis can not represent time.

## Mathematics and geometrically intuitive derivation

A dimension means a dimension in a direction that can not be represented by the directions of other minor dimensions. As an example of the gradual transition from the point ( 0D) may serve to 3D and 4D:

A point without extension (a circle with radius 0).

In the dimension 1 is a movement in any direction away from the point is possible; one obtains a distance. By way of example, the X -axis of a coordinate system, extension to the left and right.

In dimension 2, we also seek a direction that is not the route, in the simplest case: vertical to the track. This provides us with a coordinate system, with which we can reach any point of a plane. For example, the Y-axis of a coordinate system, extending forwards and backwards.

3 is a dimension for the direction is necessary, which does not lie in the plane ( of dimension 2). We show again (similar to the second hand of a clock ) in all directions of the plane and close all these directions. What remains directions, pointing up or down. In the simplest case, perpendicular to the plane "upwards". This provides us with a coordinate system, with which we can reach any point in space. This is the z-axis of a coordinate system expansion by the top and bottom.

In turn, the direction is required, which is not in the room (from the dimension 3). We show spherically in all directions, we can introduce ourselves and close all these directions. What remains are directions that we with our focused on the detection of three dimensions not mind (more) can imagine. In the simplest case, this is perpendicular to all directions which we can imagine. We expand the space in this direction, we have described a 4-dimensional hyperspace. For this, the W - axis of a coordinate system is introduced with the expansion ana and kata, terms coined by Charles Howard Hinton.

Through such logical considerations can calculate you that a four-dimensional ( hyper-) cube ( tesseract ) has 16 vertices, 32 edges, 24 faces and 8 volumes.

• Hypercube

Each dimension can be thought of as the composition of an infinite number of previous dimension. The straight line, with the dimension 1, then the assembly of an infinite number of points of dimension 0 Applying this idea to the " Fourth Dimension", then this is the composition of infinitely many (three-dimensional ) spaces. The projection of a four-dimensional object is created in three-dimensional space as a "shadow" always in 3D.

## Physically common understanding

According to the above mathematical definition is a four-dimensional coordinate system, a coordinate system with four linearly independent directions. This is to represent our familiar three spatial dimensions and the time dimension.

In Einstein's theory of relativity, space and time are combined into a four-dimensional space-time. The space at any given time is simply a hypersurface ( in special relativity theory is a hyperplane ) in space-time. Thus, the "direction " of the space (and time) in the space-time is not uniquely determined. In fact, the choice of the space hyperplane from the reference system depends. Can be graphically represented in the Minkowski diagrams, details, can also be found under Minkowski space.

However, the space-time - even the non-curved - not Euclidean, since a distance between two points of the flat space-time is not defined by the usual vector norm, but by:

Follows from the negative square, the distances at the time must have imaginary values ​​.

From topological reasons, any fourth spatial dimension is required for the embedding of a curved 3D space.

In a 3D space can be obtained by a different dimension than the time a 4D space formed. This can be achieved by an additional scalar property, as in 1D, or scale as a color scale.

## Cosmological significance

Our learned presentation allows the three-dimensional space. What actual extent of the space surrounding us has in other directions, is the object of investigation of cosmology.

## 4 -dimensional body

• In addition to the tesseract, the four -dimensional hypercube, there is also the Pentachoron, a hyper- pyramid.

## 4D flight guidance

Technology, air traffic - in the air - not only predominantly spatially arranged, but also coordinates over time, including airspeed and sinking in the future. Pilots and computer then no longer act merely as a separation Manager, but as traffic flow managers to denser air traffic and on-time - about thanks to reduced holdings - handle more energy efficient. During a test in February 2012 a broad introduction is not in prospect before 2018.

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