A Closer Look at the Meaning of Dimensions

Understanding Dimensions

All of us have a notion of dimensions.  We measure the number of kilometers we jog, give appropriate price to a piece of land, and are recommended to drink at least 8 glasses of water a day. In talking about these things, we talk about dimensions.  In determining the number of kilometers we jog, we measure length. The price of land is based on its area and other factors. In counting the number of glasses of liquid we drink, we are talking about volume.  Length, area, and volume are examples of measurements in 1, 2, and 3 dimensions.

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In mathematics, a dimension (of a space or object) is the least number of coordinates needed to specify a point within it. For example, on the number line, which is 1 dimension, we only need one number to determine a point. The number 5 corresponds to the point that is 5 units to the right hand side of 0.

In addition, we are also familiar with the Cartesian Coordinate system. In the Cartesian plane, we need two coordinates to determine a point. The ordered pair (4,3) means a point that is four units away from the y-axis and 3 units away from the x-axis.

Lastly, we also talk of of the triples (x, y, z) in the Cartesian coordinate space, and most of you have an idea what do we mean by a point with coordinates (4, 1, -2) in the coordinate space.  From the discussion above, a line has 1 dimension, a plane has 2 dimensions, and a space has 3 dimensions. Needless to say, a point has no dimensions — no length, no width, and no height.

Representing the Dimensions

Aside from the facts mentioned above, notice that the dimensions can be represented by geometrically points, lines, squares, and cubes. It should also be noted that the representation of objects in the next dimension can be constructed by connecting the objects in the lower dimensions. Connecting two points (0 dimension) produce a line (1 dimension). Connecting two lines (1 dimension) produce a square (2 dimensions). Connecting two squares produce a cube (3 dimensions). This only means two cubes (3 dimensions) must be connected to produce a representation of 4 dimensions. Yes, that representation is called the hypercube.

Dimensions

Dimension Levels

In addition, A 2D square is bounded by a line, a 3D cube is bounded by squares, and a 4D hypercube is bounded by cubes. So, you could say that n-dimensional objects are bounded by (n-1)-dimensional objects. (Taken from Shaun‘s comment below).

Application of 0 and 4 Dimensions

We have mentioned the practical examples of the three dimensions above: length, area, and volume.  But what about 0 and 4 dimensions?

A point has 0 dimensions, so we can say that it can be a location, a particular point somewhere. A point can be a location on a map, a city . Of course, in reality, cities are large therefore cannot be considered an object with 0 dimension.

For four dimensions, the most practical is to add time to the three dimensional space. For example, if a fly is trapped inside a transparent cube, then, we can determine its location using three coordinates. But, the fly is also moving, it means that at two different times, the locations of the fly are different.

In four dimensional space, we can represent the two coordinates by (x1,y1,z1,t1) and (x2,y2,z2,t2) where  t1 and t2 are two different times.

Image via Wikipedia

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