Dimension of an object?

Question

One simple way to define dimension is "the number of numbers required to describe an object."

If we consider the set of circles, we can describe each of them by one number -- radius, or circumference, or area. So we say these are one dimensional. Rectangles can be described by two numbers, so they are two dimensional. A 10x10 red-green-blue picture has 300 dimensions.

It's easy to determine, for example, that two numbers are a maximum required to describe a rectangle. But how do we know it's a minimum as well? For example we could map real numbers to ordered pairs of real numbers by extracting every other digit of the input. E.g.: 23.7114812589 maps to (3.14159, 2.71728)

Why are mappings like this "illegal" ways to reduce dimension? Is there a formal way to define dimension so that rectangles are definitely two dimensional objects?

Edit/Clarification: Based on answers/comments below and some additional reading, I think it is more correct to refer to the "space" of all circles (outside of a coordinate system) as being one dimensional. The space has one dimension, not the circles themselves (well actually, circles may be considered one dimensional too - but the first one is the context I meant). Likewise, I'm referring to the space of all un-oriented rectangles above, not the rectangle itself.

Answer 1

There are formal ways to define dimension, indeed, lots of them, depending on the context. As already noted in a comment, your description of why a circle is one-dimensional is not really correct though. The one-dimensionality of a circle is not a function of the fact that a circle itself can be described by a single number (such as radius), but that to describe the points on the circle takes a single number --- e.g. the usual parameterization $(\cos\theta,\sin\theta)$ of a circle requires just one number, the angle $\theta$. A sphere has two-dimensions because it takes two parameters to describe a point on the sphere (e.g. latitude and longitude). Of course, the sphere itself is again determined by one number, its radius (but that doesn't make it one-dimensional).

To make this more precise, one has to use ideas from topology, as mentioned in Fredrik Meyer's answer. A key point is that the map we use to parameterize the circle/sphere/whatever should be continuous (in fact even more; at least in a neighbourhood of each point it should also admit a continuous inverse); the kind of bijection you indicate between points on the $(x,y)$-plane and points on the line will not be continuous).

There are other ways to define dimension that don't use parameters, but use different topological properties.

Of these, perhaps the notion of covering dimension is the most basic. To get the idea of it, think first about a ruler: we can divide it up into inch markings, and at any point there will be at most two different inch-long intervals meeting each other. Now think about a brick wall: if the brick layer did a bad job (just stacking bricks on top of one another in columns) then there will be points where four different bricks are in contact, but even if they lay the bricks properly, there will be points where three different bricks are in contact (this is what you usually see in a brick wall, when a single brick in one layer will sit on top of two bricks in the layer below); you can't avoid having points where three bricks meet. (Here I am ignoring the mortar in between the bricks.)

Now think about making a solid pyramid out of stone blocks: even if you arrange the blocks as stably as possible, you will have points where four stone blocks meet.

So: paving a line (one-dimensional) forces some points to have to paving stones meeting; paving a plane (two-dimensional) forces some points to have three paving stones meeting; paving a solid (three-dimensional) forces some points to have four paving stones meeting; you can probably see the pattern!

This is a more topological, less analytical, definition of dimension, which is important in theoretical investigations of dimension. (For example, it underlies the Cech definition of cohomology in topology.)

There is also the notion of Hausdorff dimension, which captures the idea of dimension in terms of the amount of volume that a figure occupies. It can give non-integral values of dimension, and so is used for discussing the dimension of fractals.

Answer 2

It's true that you can encode two real numbers into one, but as you mention it is "cheating", as the resulting number does not describe anything measurable in your figure.

I'm not even sure there is a clean number of dimension as you are looking for, because then what would be the dimension of something with a real paramater and an integer paramater ? or a real parameter and a parameter from $\{1,2,3\}$ ? I think you need additional structure to define a clean notion of dimension, like vectorial space for instance.

Answer 3

At least in differential geometry and topology, it is common to define dimension as (loosely speaking) the number of coordinates needed to give a point.

So for example, you can use the angle from the x-axis as coordinates on the unit circle, so it is one-dimensional. To give a point on a sphere you need both latitude and longitude, so a sphere is 2-dimensional. Space is 3-dimensional since you need three coordinates.

We don't want to allow "cheating maps", because we want our maps to be continious (in some topology).

Answer 4

There are more qualified answers but I think I can add somethings.

First of all, if $V$ is a vector space, and we are in $ZFC$ it has bases and all bases have same amount of elements, let's say $n$. Then, we define dimension of this vector space $V$ as $n$.

Secondly, let's approach with measurement. If we have a line, double it and we have $2$ line. If we have a rectangle, double it, and we have $4$ rectangle. You can visualize this with the picture bellow. So, we can say, an object is $n$ dimensional if, let's say we will have $m$ of it when we double it, $log_2m = n$. We can think about this fractal. If we do this fractal and double it, we will have $3$ copy of it. So, it's dimension is $log_23 (= 1.585...)$. Also, instead of doubling one can multiply the measures of object by $k$ and dimension will be $log_km$ with same definition of $m$.