Dimensions

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Basic Description

A dimension can be thought of as a direction of motion. The first dimension has one direction, say "forward and backward". The second dimension has two: "forward and backward", "up and down". The third dimension has three directions: "forward and backward", "up and down", "right and left".


Tug-of-war is a one dimensional game; you can only go forward and backward. A game like Mario is two dimensional, which is quite apparent when you can't avoid obstacles by stepping out of the screen. Real life is 3D, you can go "forward and backward", "up and down", "right and left".


Dimensionality is a property of both objects and spaces, however the dimension of an object can be different than that of the space it is in.


Additionally, there is the zeroth dimension and fractal dimensions. The zeroth dimension has no directions of motion. The zeroth dimension is a point without a "forward and backward", "up and down", or "right and left". Fractal dimensions are discussed in depth here.

Dimensions of Objects

Image 1.
Image 1.


Every object has an intrinsic dimensionality. Take the line; it has only one direction along it, meaning it is 1 dimensional. The circle is merely a line segment connected front to back. If you are on the circle, you can only go in one direction: around it. So the circle is 1 dimensional as well. On the disk (best thought of as a filled in circle), there are two directions akin to "up and down", "right and left", making the disk two dimensional. The image to the right displays these shapes.

Whether the line is drawn on a 2 dimensional sheet of paper, or made in 3 dimensions as, say, a string, the line is always 1 dimensional. Similarly, the disk, drawn on paper, or made in 3 dimensions, like a flattened quarter, is always 2 dimensional. Althought a string and a flattened quarter are 3 dimensional because they do still have length, width, and height, they can be used as analogies for an ideal 1 dimensional line and 2 dimensional disk within 3 dimensional space.


Dimensions of Spaces

Just as objects have dimensions, so do the spaces that objects are in. The dimensions of a space can, as discussed above, be thought of as a direction. We live in 3 dimensional space, meaning we have three directions in which we can move. Mario lives in 2 dimensional space; he can only "move up and down", "left and right". Humans are 3 dimensional objects; It takes three basic directions to describe our bodies:"upward and downward", "leftward and rightward", "forward and backward". Mario is 2 dimensional, and can be described with "upward and downward", "leftward and rightward". Thus, we have the same dimensions as the space where we live, as does Mario. Nevertheless, this is not always the case for objects. Ideally, a line drawn on a piece of paper is a 1 dimensional object that is embedded in the 2 dimensional space of the page. Thus, an object can exist in a space with a higher number of dimensions than that of the itself.


Four Dimensions

1D

Each dimension is represented mathematically by a coordinate variable. Any point specified by one variable, such as x=8, is located in 1 dimension, and can be plotted on the 1 dimensional number-line, also known as the x-axis, at the point (8), as shown below.

Image 2.

2D

If the point is defined by two coordinates:

x=8

y=7


then it can no longer be placed as a point solely on the number-line. It now has two coordinates, One stating that it is located at 8 in the x direction, and the other placing it at 7 in the y direction. Thus, we plot the point as (8,7) on the 2 dimensional xy-plane, which may be familiar from math class. We merely added another direction to the 1 dimensional number-line to arrive at the 2 dimensional plane.


Image 3.


3D

If a point has three coordinates:

x=8

y=7

z=6

Then to capture all of this information on our graph, a third dimension, the z direction, must be added, say out of the paper the graph is drawn on. Our new point, (8,7,6), is at 8 in the x direction, 7 in the y direction, and 6 in the z direction, as illustrated in The graph below.

Image 4.

4D

Before directly discussing four dimensions, let us introduce an analogue in 2 and 3 dimensions. Notice that, in Image 5, (8,7,6) and (8,7,0) have the same x and y coordinates. If we look at them from directly along the z-axis, the two points overlap, appearing as though they were a single point at (8,7) on a 2 dimensional xy-plane (Image 6). As such, they are at the same position in a two dimensions, but have different positions in a third. It is this third coordinate, their z coordinate, which distinguishes them as different points.

Image 5. The points (8,7,6) and (8,7,0) plotted in thre dimensions.
Image 5. The points (8,7,6) and (8,7,0) plotted in thre dimensions.
Image 6. A different view of the same points in Image 5. The red point, at (8,7,0), is seen directly behind the blue point, located at (8,7,6).
Image 6. A different view of the same points in Image 5. The red point, at (8,7,0), is seen directly behind the blue point, located at (8,7,6).

Just as in the previous examples, a 4th dimension can be thought of as merely adding anther coordinate variable describing the location of the point. A point (8,7,6,5) is at 8 in the x direction, 7 in the y direction, and 6 in the z direction, and 5 in a fourth direction, which we will call the w direction. The two points (8,7,6,5) and (8,7,6,20) would appear in the same place as each other on a 3D graph in much the way as the points (8,7,6) and (8,7,0) appeared to be in the same place as each other on the 2 dimensional xy-plane. For (8,7,6,5) and (8,7,6,20), we must simply remember that they have an other direction in which they are not at the same place, even though in the three directions we know of, they are.

One way to do this would be to give the points different coordinates in time. For instance, an object could be at the point (8,7,6) at 5 seconds after you start measuring, thus (8,7,6,5), and then another one is there at 20 seconds, thus (8,7,6,20).

n Dimensions

Infinitely many dimensions exist, and when talking about and arbitrary dimension, the term "n-dimensional" is used. The use of multiple dimensions has many concrete applications as well.


In Physics, multiple dimensions can describe the motion of n particles in 3 dimensions. Rather than considering n particles each with 3 dimensions, one can consider 1 system with 3n dimensions, comprised of all the particles.


  • Particle 1 is at (x1 , y1 , z1)
  • Particle 2 is at (x2 , y2 , z2)
  • Particle 3 is at (x3 , y3 , z3)

Or:

  • System 1 is at (x1 , y1 , z1 , x2 , y2 , z2 , x3 , y3 , z3)


Similarly, in Economics, an economy is described interns of the number of products it produces. Each product has a quantity, so the economy as a whole has one value for each product it produces, leading to a multidimensional description of the state of the economy. The following, hypothetical economy has five attributes, or dimensions.


  • There are v units of product 1
  • There are w units of product 2
  • There are x units of product 3
  • There are y units of product 4
  • There are z units of product 5

Thus:

  • The economy is in the state (v, w, x, y, z)


References

Image: http://wikieducator.org/MathGloss/C/Cartesian_plane

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