# Thread: Can math describe zero-dimensional space?

1. Crunchy Cat: can you think of a way to represent an absence of dimension so the representation uses no dimensions?
Since this representation will be in zero dimensions, what do you think it should look like? I can't picture it myself.

You say that "n" represents a dimension with no locations. What kind of dimension is it? Does it have a definition of distance, or not?
I think that without reference to locations (i.e. points in n), you can't even define distance, so your representation has no metric (but maybe that's ok). I can't see that it's all that useful without locations, however.
If n is some kind of space, what does it contain? Does it have any further structure apart from being dimensional, and in what sense?

You see, you need to be a bit more definitive than "call it n". As mentioned, a graph with one vertex can be called "G"; since it's a graph it inherits all the structure of a graph--it's simple, connected, has a path from the vertex to itself of distance 0, (distance or path length is defined) etc. The location of the vertex in a 1-vertex graph is arbitrary, it's "in G"; with more than one vertex you should identify them with labels like a,b since there can be a path from a to b or from b to a, a or b can have loops on them, etc.
You could make a 1-vertex graph more complex by having multiple loops from the single vertex to itself, or even an infinite number of such loops--but what does that achieve, or even say about the structure?

2. Describing a space that has n dimensions isn't that hard. We understand that real numbers describe physical things like length, width, depth and so on. So we use three numbers with labels like x,y,z in that case. We use two labels for 2-dimensional spaces, otherwise known as surfaces, in three dimensions (i.e. n + 1).
So those two cases are R3 and R2, respectively, and with R1 there's one label. So what about R0, what do you label objects with in a 0-dimensional "real" space?

R0 contains only itself; from the set of real numbers it contains no real number, it's the empty set 'in' the reals. There is exactly one element in it, but it isn't a real number, it's the 0-ball. It looks something like a graph with one vertex and a possibly infinite number of cycles of length 1 from the vertex to itself.

3. So sets of numbers and mathematics; since the last paragraph in my last post hasn't been corrected yet, I'll try being a bit more pedantic.

Notice, you have to have a space with at least one more dimension to describe elements with a given dimension, lines and 'distances' don't make sense in less than one, so we have to be in R1 at least, and here, any section of the real line is bounded by two real numbers, points like x1 and x2 which have one less dimension; so we say the interior or line segment is bounded by two objects with zero dimension, or two 0-spheres.
Notice also that this means we have a graph with two vertices and one edge, and just as there are an infinite number of points in a unit interval of the reals, there are an infinite number of 2-cycles in this graph, the same as the order.

But where points like x1 and x2 are in R1 (because they're the surface of a 1-ball), the boundary of objects in R0 (the set of real numbers with no dimension) needs one less dimension, which is -1.
Looking at the graph with one vertex,, the concept of a negative dimension might be (speculation) tied to the difference in path length between the simple (no loops), and the at least one loop version.
But a boundary with a negative dimension is fairly abstract, I would say. Anyway, there's a relation between objects with an interior and their boundaries, which is that if the former has n dimensions, the latter needs n-1, so the n-ball has an (n-1)-sphere boundary. The 0-ball has therefore a "boundary" with -1 dimensions.

4. A correction: in graph theory a cycle is defined as a walk through a graph such that no edge is traversed more than once, the start vertex is the same as the end vertex, and at least three vertices are in the walk.
So a graph with one vertex has no cycles, but the vertex is connected to itself so there's a walk of length 0 which is a circuit.
Likewise in a connected graph with only two vertices there is a circuit of length 2.

But in mathematics the idea of dimension is contextual; the context of a 0-dimensional space is tricky, because you have to ask what kind of context is there with no dimensions to "express" it in? Does zero dimensional mean nonzero contextual? Maybe not.

5. Originally Posted by arfa brane
Crunchy Cat: can you think of a way to represent an absence of dimension so the representation uses no dimensions?
Since this representation will be in zero dimensions, what do you think it should look like? I can't picture it myself.
Nope, I cannot. But that was my point in the response to the OP. The best that can be done is to wrap the zero dimension in an aritificial container such as R0, but that's that of course is not a direct representation of an actual absence of dimension.

6. Originally Posted by Crunchy Cat
The best that can be done is to wrap the zero dimension in an aritificial container such as R0, but that's that of course is not a direct representation of an actual absence of dimension.
R0 isn't artificial, it's a mathematical consequence of set theory.

A real number has 'dimension' in two ways; first of all, every real number has a value, and since 0 (or 0.0) is a real number, the value is just an abstract 'distance' from 0.
Obviously 0 is distance 0 from itself. The distance between two real values is the absolute value of their difference, such that for a,b ∈ R, |a - b| is (the real value of) the distance. So all real numbers have a magnitude = distance from 0.
This has an easily understood geometric interpretation: if S = { x ∈ R : |x - a| < δ } then S = { x ∈ R : a - δ < x < a + δ } is the set of points on the real number line between a - δ and a + δ not including either point.

We have:
(a,b) = { x ∈ R : a < x < b},
[a,b] = { x ∈ R : a ≤ x ≤ b}, and
B(a,δ) = { x ∈ R : a - δ < x < a + δ } = S.

The first set is the open interval with endpoints (boundary) a and b. The second set is the closed interval (interior + boundary) with endpoints a and b, and the third is the open ball centred on a with radius δ.
If δ 'shrinks' to 0, a is still a, but only if it's embedded in a space such that a has a value (a zeroth dimension, say).
But if we define B(a,δ) as the set of points whose distance from a is less than δ, then if δ = 0, all the points in B must be a distance less than 0 from a.