**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.

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.

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 (x_{1},y_{1},z_{1},t_{1}) and (x_{2},y_{2},z_{2},t_{2}) where t_{1} and t_{2} are two different times.

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This is a very interesting discussion. I’m sure that nearly everyone knows what is meant by 3-dimensional. After all, it’s in all of the popular movies now (if you bother paying the ridiculous prices to see those versions!). 2D drawing are probably just as well understood, though I bet that if you were to explain 1 dimensional to a general layperson, it may not be as easily grasped. After all, it is tough to draw something one dimensional because, technically, the line you draw has a thickness to it, and so has a second dimension: width.

I have never paid much attention to a hypercube, though I have heard the term. It is very fascinating, though I am more comfortable to think of the 4th dimension to be time. Wrapping my very 3 dimensional brain around a 4 dimensional hypercube is too much for me! I did like your comment about the boundaries of all the dimensional objects. 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.

Thanks. I copied your last line. Hope you don’t mind.