The coordinates of the fourth dimension

The fourth dimension is an extremely elusive concept. For the artists, the fourth dimension is not the same as the fourth dimension of the physicists, and both are different as the concept of the mathematicians have, and all of them are different to the fourth dimension of the mystics and spiritualists.

Artists and mystics don't care about the coordinates of reference of the fourth dimension. Physical scientists and mathematicians do care, but from a different perspectives both.

For the physicists, the fourth dimension is mainly the time dimension, although they also embrace the possibility of physical dimension beyond our well-known three-dimensional world.

For the mathematicians, the fourth dimension is just another dimension like the first, the second, the third, etc. Pure mathematicians don't care too much about the physical relevance of their fourth dimension.

The coordinates of Flatland

Flatland: A Romance of Many Dimensions –the short novel that wrote the English schoolmaster Edwin Abbott Abbott in 1884– is a beautiful story of the imaginary people that live in a flat surface, like shadows, but not shadows of anything 3D object like the ones we are used to see everyday. In fact, they don't live and play in any surface as such: their world is planar, and they have nothing around to compare their universe with any other.

Speaking about the second dimension is so ambiguous as speaking about the first one. Dimensions doesn't have any specific order and direction assigned. So its up to us to choose the order. A table top has two dimensions, that's not questionable: length and breath (width). But, in a tabletop, what dimension is length, the first or the second? What dimension is the table's breath (width), the first or the second?

The thee dimensions of the real space

Simply stated, a dimension is a  "space"  where we can measure distance from one point to another. That "space" can be linear if we measure along a line, or that "space" be planar if we need to measure outside a line, but we can stay within a plane. In the case of solid objects, we need three measures to define its spatial properties.

The underlying property among all dimensions is that each dimension is perpendicular to the others, no matter how many dimensions there are. Geometrically, dimensions are represented by axes, such as X-axis, Y-axis, etc. The point of intersection of the axes is called the Origin of coordinates.

Starting from an arbitrary point, O, that we can select as "the origin of coordinates, O", the position of any point P in a 3-dimensional space is easily determined by three real coordinates x, y, and z that we measure from O to P. The x, y, and z measures (values) are from the X-axis, the Y-axis, and the Z-axis.

The distance D that the point P is from the origin O, is given by the equation:

The distance D' between any two points P and P' is computed as:

The next dimension is: the fourth dimension, or the time dimension?

Adding a fourth dimension is easy as adding a third dimension when you have only two. The principles are the same: you add a new axis, a new label or name for that axis, and faithfully assume that that new axis is perpendicular to the previous three. The phrase "faithfully assume" may sound odd, but this comes because there is no way to fully visualize and imagine an axis that is perpendicular to the three axes of our reality. 

For example, in the figure of the small refrigerator above it is very simple for us to visualize its three dimensions, but if we talk of a fourth dimension for this object it is very hard to imagine an extension of this refrigerator that is perpendicular to all of the three dimensions of the pictured fridge.

Any point P in this hyperspace must have 4 measures for the uniqueness of its location. Assume that the coordinates of this point P are P = (x, y, z, f). If we label this fourth dimension with the letter f (can be any letter), then the distance of the point P from the origin O should be given by the equation:

The article Can there be many fourth dimensions? in this Web site is about that with the concept of the fourth dimension it is not clear what is meant by that obscure and esoteric hyperspace.

Adding another coordinate-dimension to a space where every measure is a positive real number (integer or fractional) is a simple exercise in mathematical induction. That means that if we use meters to measure the axes components of the axes X, Y and Z, then the next  F-axis must also be measured in meters. This seems to be simple and straightforward.

However, if we think of the fourth dimension as the time dimension, a problem arises because time is not measured in meters but in seconds. That's the reason why physicists prefer to talk about the space-time 4D hyperspace.

If we are going to talk about time as the fourth dimension, then for the purpose of clarity, let us change the variable f for another more intuitive letter, like t. This time-dimension here, or time-coordinate is problematic because of the mixing of measurement units. We cannot add meters to seconds, or in general, we cannot add distance units to time units. This a problem that is tackled in depth by the theory of relativity, where the concept of time is inevitably embedded in the concept of a four-dimensional universe.

In summary, taking the fourth dimension as the time dimension doesn't give us the same results as when we take the fourth dimension as if it were like any other spatial dimension, where no special theory is needed.