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]] There are three conventional spatial , and as such is used to explain Space-time in Einstein's theories of Special Relativity and General Relativity . When a reference is used to four-dimensional co-ordinates, it is likely that what is referred to is the three spatial dimensions plus a time-line. If four (or more) spatial dimensions are referred to, this should be stated at the outset, to avoid confusion with the more common notion that time is the Einsteinian fourth dimension. If time is the "fourth dimension", an additional Spatial dimension would be referred to as the Fifth Dimension . The implications of another ''spatial'' dimension are now discussed. This would be Orthogonal to the other three spatial dimensions. The cardinal directions in the three known dimensions may be referred to as up/down (altitude), north/south (latitude), and east/west (longitude). When speaking of the fourth spatial dimension, an additional pair of terms is needed. Attested terms include Ana / Kata (sometimes called Spissitude or spassitude), vinn/vout (used by Rudy Rucker ), and upsilon/delta. CONCEPTS The fourth spatial dimension and orthogonality A right angle is defined as one quarter of a revolution and "orthogonal" (from the Greek) refers to co-ordinates or functions that are at right angles to each other. Cartesian geometry arbitrarily chooses orthogonal directions through space, which means that they add height. The fourth dimension is therefore the direction in space that is at right angles to these three observable directions. Vectors The fourth spatial dimension can be thought of in terms of Vectors , analogous to arrows, fixed from some single place in space which we call the ''origin'', that point to other places. These are called geometric vectors. A '' Point '' is a zero-dimensional object. It has no extension in space, and no properties. If one were to think of this point as a geometric vector, like an arrow, it would have no length. This vector is called the Zero Vector . A '' Line '' is a one-dimensional object. If we pick some nonzero vector in some direction, this vector has some definite length. That vector has a head at some point in space and a tail at the origin. If we think of stretching that vector so it is twice as long, three times as long, and so on and even stretching it backwards so it takes all possible lengths it can (even ''zero'' length, to get the zero vector), we get a single line with one dimension of length. All the vectors that describe points on this line are said to be ''parallel'' to each other. Even though any line we can draw must have some small thickness (so that we can see it), this theoretical line does not. A '' Plane '' is a two-dimensional object. It has a finite length and breadth but no thickness — somewhat like a sheet of paper (but paper too has some thickness). Thinking of a plane in terms of vectors can be a little more challenging. If we think of taking one vector and moving it so that its tail is touching the head of the first and forming a vector with its tail at the origin and the head at the head of the repositioned second vector, we have a reasonable way of talking about adding vectors. If we have two vectors that are not parallel, we can talk about all the points we can reach by stretching either of the vectors (or not stretching them), and, adding these vectors together, these points form a plane. We say that the two vectors ''span'' the plane. '' Space '', as we perceive it, is three-dimensional. We can think of putting a line together with a "stack" of planes. These planes are "stuck together" like a sandwich, with the line passing through them like a skewer. To get to some point in space, we can imagine traveling up the line and then moving across the plane to the point. We then have three vectors to think about, one to travel some distance up the line and two to get to some point in space. The fourth spatial dimension, then, can be described by "sticking together" several three-dimensional spaces in a row. To get to some point in the four-dimensional space, one travels along the three-dimensional spaces, and also across the fourth dimension. The total number of vectors involved is four. Mathematically, the 4-dimensional spatial equivalent of conventional 3-dimensional geometry is the Euclidean 4-space , a 4-dimensional Normed Vector Space with the Euclidean norm. The "length" of a vector : expressed in the standard basis is given by |
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