# Could there be a -1 dimension? [duplicate]

This might be a silly question as it just doesn't make sense to me - but could we have a dimension lower than one? Could we have a 0th dimension? Could we have a -1th dimension?

Are we supporting the 2nd dimension, 1st dimensions existence by being part of the higher multiverse?

Finally, can fields inside our dimension (higgs field) interact with other dimensions? I am sure certain that we think gravity is a force interconnecting this "multiverse" together, if it even exists, but can our own particles and matter interact with dimensions completely different to us? Are we affecting other dimensions in this so called "multiverse"?

Even though this query makes absolutely no sense to me - I'm sure the second part of the question can be answered.

• Dimension numbers are arbitrary, with no physical meaning. – HDE 226868 Jan 4 '16 at 20:09
• Somewhat related : in various formulas relating to dimension numbers (such as the number of faces and sides in measure polytopes and simplexes and such), -1 dimensions is defined as being the empty set. – Slereah Jan 5 '16 at 15:22
• Possible duplicate: physics.stackexchange.com/q/52176/2451 – Qmechanic Jan 5 '16 at 15:24
• "0-dimensional" is not an uncommon phrase. ... a plane is 2-dimensional, a line 1-dimensional, a single point is 0-dimensional. For example, if an atomic lattice is missing one atom, that would be a 0-dimensional defect. – Steeven Jan 5 '16 at 15:56
• Sorry - I asked this question on Astronomy SE so I didn't know there was a duplicate on this site. – Featherball Jan 6 '16 at 7:34

Expanding on my earlier comment:

The numbering of dimensions is arbitrary. There's no significance to the number we choose to call a dimension by. Typical, we don't even refer to the three everyday dimensions (length, width, depth) by number, because there's no point, and they're relative. Dimension 2 (whatever that is) isn't different from Dimension 3.

People often refer to time as "the fourth dimension". I personally strongly dislike this, because

1. It implies that time is similar to spatial dimensions.
2. It makes discussions of spacetimes with more than three spatial dimensions really confusing.

If you're one of these folks, then to you, the numbers have some significance. But there's nothing physical about the number.

In fact, in general relativity, time typically is listed in the metric (i.e. the mathematical entity that describes the curvature of space in a certain way) before the other spatial dimensions - not after them.

I must admit that I'm confused by your third paragraph. We don't know if there are any other dimensions. That means that there's nothing to suggest that the Higgs Field should only propagate in the three we experience. Regarding your remarks afterwards . . . well, in most physicists's opinions, the idea of a multiverse is speculative. Extremely speculative.

Could we have a 0th dimension ? Could we have a -1th dimension ?

In the sense that a dimension number is a label, yes.

However we describe a space as having a given number of dimensions. In this sense a zero dimensional space would be a dimensionless nothing. A -1 dimensional space simply has no meaning in this sense and hence a -1 th dimension has no meaning.

Are we supporting the 2nd dimension, 1st dimensions existence by being part of the higher multiverse?

The dimensions are not related in that way. Again I'd suggest you let go of the view of individual dimensions and thing in terms of a space having a stated number of dimensions.

Finally, can fields inside our dimension (higgs field) interact with other dimensions? I am sure certain that we think gravity is a force interconnecting this "multiverse" together, if it even exists, but can our own particles and matter interact with dimensions completely different to us? Are we affecting other dimensions in this so called "multiverse"?

The dimensions, again, should not be thought of in this way. They are not things that interact with each other. I think you are mistakenly picking up this notion of interactions from the idea that space and time are linked in theoretical space-time models. But this "link" is really a description of the geometric properties of the space we use as our model and which incorporate time as a notional dimension. These are not interactions, but definitions of how that model of space is structured.

Even though this query makes absolutely no sense to me - I'm sure the second part of the question can be answered.

Note that an answer like "it makes no sense" would also be an answer. I think the problem here is that you are treating dimensions as things.

The "right" number of dimensions is a question that starts heated arguments among physicists. If better minds than mine can't agree I'd be loath to express an opinion. I'll go with whatever works for whatever system I need to model, and for an awful lot of purposes Newtonian still works fine.

Mathematically zero-dimensional space is a countable set. I am not aware of any way to make sense of negative dimensions.

Any order of dimensions is arbitrary (it depends on the coordinate presentation of the vector space you happen to use), so considering 1st or 3rd dimension does not really make sense.

For what it is worth, it should probably be mentioned, that there are Dirichlet $p$-branes with dimension $p=-1$ in string theory. However they are just instantons, i.e. $0$-dimensional objects in spacetime, so it is convenient to (somewhat artificially) assign them a spatial dimension $p=-1$.

Colloquially, we say things like, "the dimensions are height, width, and depth," but in math, there are no "dimensions" (plural): There is only "dimension" (singular).

The dimension of a vector space is the maximum size (cardinality) of any linearly independent set of vectors that belong to the space. That is equivalent to the number of components needed to represent a vector from that space.

https://en.wikipedia.org/wiki/Vector_space#Basis_and_dimension

Colloqually, we say "Physical space has three dimensions," but the formal mathematical way to say it is, "The dimension of physical space is three."