Examples of 1 and 2 dimensional.

Question

I understand that the x-axis alone is 1 dimensional. Throw in the y-axis and there are 2 dimensions. Add a z-axis and there are 3 dimensions. Overall I have a good understanding of dimensions. A line is 1 dimensional, a square or rectangle is 2 dimensional, and a cube is 3 dimensional.

My question is what if you throw in parabolas or circles or the absolute value function, etc.? A parabola can only be drawn in 2-space, but lines are also drawn in 2-space unless the slope is 0 or does not exist. A circle is kind of like a parabola, but it is very much like a square, so I am thinking it is 2-dimensional. My conclusion is that the only 1 dimensional object is a straight line, and a point is 0 dimensional, but I am not confident that I am correct. Can you please clear this up for me?

Answer 1

When you talk of a circle you have to make the distinction between (a) circle along with interior usually called a disc and (2) circle with just its circumference. The former is a 2-dimensional object and the latter is a 1-dimensional object. Evev though a point on the circumeference of a circle needs two co-oordinates to specify, if you see careflly only one co-ordinate is free.

By this I mean, take a circle of radius 5 centred at origin, if you know the $x$ co-ordinate to be 2, then y co-ordinate is forced to be $\sqrt{25-4}=\sqrt{21}$.

The dimension is technically defined as the number of independent parameters that are needed to describe the points of a shape. (This is algebro-geometric viewpoint. In topology dimension is defined as what you see when you look only at close enough points. In a circle you will see a small arc which is 1-dimensional. Some insect crawling on a soap bubble will see (when myopic) only a disc, hence a soap bubble is 2-dimensional (a surface).

Answer 2

$x$-axis alone is 1 dimensional. Throw in the $y$-axis and there are 2 dimensions. Add a $z$-axis and there are 3 dimensions.

A rough version of the usual notion of dimension indeed is the following: The number of lines can you draw through a point so that they are all perpendicular to each other (think about the origin and the coordinate axes). The more precise way to say this is to define bases and look at their sizes.

My question is what if you throw in parabolas or circles or the absolute value function, etc.?

When you go beyond flat things (vector spaces), you need to be more careful. I think that differential geometry is the field of mathematics that would, in a way, answer this question. If you consider those shapes as "manifolds", all of them are one dimensional objects.

One alternative notion of dimension is also the Hausdorff dimension which is defined even for some extremely irrelgular objects.