Abstraction: dimension

The easiest approach to visualising multi dimensional space is to look at the models we have of one, two and three dimensional space, and then throw away the "real world" and extend the "mathematical world" to more dimensions. It is not in the slightest bit clear at the outset that it will be "obvious" or "easy" to spot the right way to extend the "mathematical world" to more dimensions, but generally speaking it is usually fairly clear what to do.

Perhaps we are most familiar mathematically with the mathematical model of "two dimensional space". This is the familiar x,y coordinates system that goes back to Descartes. So we start with this.

You might say that a point on the plane has x and y coordinates, which are both real numbers. Except that this isn't quite the fully abstract mathematical version of what's going on. What you are probably thinking of is a map (e.g.~on paper) and measuring the coordinates of the point on the map. A better mathematical description is that the plane is the set of points. So what is a point? The usual definition is to say a point (in mathematical two dimensional space) is a pair of real numbers $(x,y)$. Thus the mathematical plane really is the set of pairs of real numbers, ${ℝ}^2=\left\{(x,y):x,y\in ℝ\right\}$.

In the same way, mathematical one dimensional space is ${ℝ}^1$ or $ℝ$, and mathematical three dimensional space is ${ℝ}^3$ or the set of triples from $ℝ$.

Now it is "obvious" that mathematical $n$ dimensional space should be ${ℝ}^n$.

This is not enough though, because we need to do things to points in mathematical $n$ dimensional space. ("It's not what things are, but what they do.") For example, we might want to calculate the distance of a point from the origin, or to add a $n$ dimensional vector quantity to another $n$ dimensional vector quantity. Most of the basic processes have "obvious" analogues.

It is entertaining and educational to imagine sequences of similar objects in ever-increasing numbers of dimensions. For example, the triangle is a 2-dimensional figure with 3 vertices and 3 sides and one face. The tetrahedron is a 3-dimensional figure with 4 vertices and 4 sides (all triangles) and one solid "3-dimensional face" (a tetrahedron). There are in fact six edges on a tetrahedron. Based on a combination of guesswork and looking for the nicest possible patterns, one might consider a table of "$n$ dimensional triangles" ($n$-triangles for short).

I've included a 0-triangle (a point), and a 1-triangle (a line segment). You should be able to see the pattern. And then you can "visualise" higher dimensions by imagining how the higher dimensional figures arise.

You can do the same thing for squares. The pattern is a 2 dimensional square has four points, four lines and one square. A 0 dimensional square is a point, a one dimensional square is a line, and a 3 dimensional square is a cube. Then what?

Fractional dimensional objects are "trendy". Very trendy, because they correspond to "fractals". I suspect if it wasn't for the fashion for fractals, Gowers would not have written this section. But as it is, it fits his pattern very well.

First, to see why fractional dimension might be regarded as silly, note that $n$-triangles or $n$-squares don't really make sense if $n$ is not a whole number. (Or I don't think they do.) What is a $5/2$-triangle, i.e.~a triangle in $5/2$ dimensions?

However the abstraction process is about writing down properties of an object as rules or axioms saying what the objects "do". We don't have to give a long long list of lots of detailed properties. We can abstract more, which means to say we remove some of the key properties we had previously expected "dimension" to have and focus on only a few of the other properties. Under these circumstances, fractional dimension makes sense!

The key property that is preserved when we consider fractional dimensions is the idea that area goes as the square of length, volume goes as the cube of length, and so on. Therefore something has fractional dimension $q$ if the "amount of stuff it has" goes as the $q$th power of length when the object is scaled.

Let's see how this works. The volume of a cube is a three dimensional quantity. What this means is that we choose to measure the "amount of stuff" of the cube as its volume. If the cube as linear dimension (i.e. length of a side) one, then its volume is also $1$. Multiplying the linear dimension by two gives a cube of volume $8$. Since $8=2^3$ the concept of "volume of the cube" is three dimensional. Now suppose we change our idea of "amount of stuff" and say it is the surface area than matters. A unit cube has surface area $6$. Doubling linear dimensions makes the surface area go to $6\times 4=24$ (six faces of area four). Since $24/6=4=2^2$ "surface area of a cube" is a two dimensional concept.

Fractals "squeeze" some extra stuff into every unit of length. For example the Koch snowflake squeezes four units into every three units of length. The easiest way of calculating its fractal dimension is by seeing that, scaling a Koch snowflake by tripling 1-dimensional length multiplies its overall "length" (the amount of "stuff" in the Koch snowflake) by $4$. If $d$ is the dimension, the expected increase in amount of "stuff" is a factor of $3^d$, so

taking logs (to any base)

giving

You can imagine a Koch-type star, made from the surface of a regular tetrahedron, where each face is marked into four triangles and the middle triangle replaced by another regular tetrahedron of half the linear dimensions, and then the same for the three exposed faces of this, and so on.

If we measure "amount of stuff" as being the surface area, then doubling linear dimension increases the amount of stuff by six. For this $2$-fold increase of linear dimension we get an increase by a factor of

where $d$ is the dimension. Solving, we get

Another example, that of the Hilbert curve, is shown on a separate web page.

As Gowers says, all this depends on abstracting a certain property of "dimension" and ignoring the others. This property that is chosen may not be appropriate for all situations, and other abstract ideas of dimension are sometimes used, and these may even give different answers.

title: Explain the notion of dimension and give examples.

Conclusions: $n$ dimensions make sense in the mathematics alone. There may or may not be "models" in the real world. This doesn't even matter!