The Many Dimensions of Dimensions - Part I

Practically everyone knows how to define dimension - it is the minimum number of numbers needed to uniquely define position. For example along a line every point can be described by its distances from the origin, or in a cube every point is given by three numbers labelled x,y,z. It is so simple, and so obvious, and yet wrong. (DISCLAIMER: The statement that it is wrong is partly satirical. There are special examples in which it doesn't work, but the reader should not conclude that this definition is not useful. It is still the definition the is used in almost every field of math and science, and is perfectly acceptable in nearly every situation,)

It is wrong because this definition implies that every compact (or non-infinite) surface or solid object has dimension 0 or dimensions 1. That is because there are known curves (which are too complicated to ever actually be drawn) in which a single number (such as the distance along the curve) can be used to define every point in a solid square, cube, or any other shape.

Although it can never be drawn, there are some nice approximations. (A link to some pictures of these will follow as soon as I find some). For example, take a square. The first approximation - simple draw a line from one corner to the diagonally opposite corner. For the second approximation, draw the same diagonal line, except at the 3/4 distance turn 90 degrees right, go to the edge of the square, turn 90 degrees again, and continue to the next edge, and keep doing this until you return to the original point and then finish the diagonal. For the third approximation redraw the second approximation, except now every time there is a straight line add a square as before. If you were to continue this process forever, you would get closer to the curve that actually goes through every point in the plane, and therefore length along this curve provides a description of every point in the plane, making the plane 1-dimensional!