Points

Serious question:

Does a simple point in space have any dimensions?

For example, the number 0 doesn't have any imaginary dimension to it.

i cant imagine being this autistic desu
i thank the gods every day

points arent real

good numbers man, checked .

a point isnt an entity, its a symbol for a nameless property without a definition
a plane is a nameless entity without a definition and a line is an undefined connection between two points on a plane

That answer sounded kind of memey

A point in our space or a point in a mathematically defined spatial dimension?

Pure a-priori intuition of Space, buddy.

This is a trivial question. By definition, a point has a number of dimensions equal to 0.

It is zero-dimensional, faggot, get over it.
If you want to reason like an idiot then you could call anything any-dimensional.

If a line is two dimensional wouldn't a point be one dimensional?

No- not quite.
Three dimensions is length, width, height.
Two dimensions is length, width.
One dimension is length.
A point is zero-dimensional, because it has none of the above.

think about it in Cartesian coordinates though
a point is a scalar, it has a value on one axis
A line have values in two axes
A plane in three

The Dirac delta function is a point on Re (real numbers) and has an integral across Re (from neg infinity to pos infinity) of a non-zero constant, therefore op is a faggot. Quod erat demonstrandum.

yet there are "real" numbers floating in and around our heads, and our heads are in space

A straight line is ONE-dimensional, not two-dimensional. What is two-dimensional is, for example, a circle, a triangle or a square.

Point = 0D
Line = 1D
Polygon = 2D
Polyhedron = 3D
etc

As I was saying, trivial.

>Line = 1D
>Polygon = 2D

How about a circle? or a torus?

2D and 3D respectively.

A point in N dimensions.

.

What is a point behind a point in a plane?

fucking brainlet, circle is 1-dimensional and torus is 2-dimensional

show me a 1d circle so i can show you why you're wrong.

How would you define a torus of revolution?
Is it self-evident?

But a straight line is defined by two points, you need two pieces of information to define a line

>tfw 0+0=0

dimension of a space is NOT the dimension of the ambient space. it's true that it's natural to picture circle as embedded in 2D plane, that does not mean that the circle itself is 2-dimensional.

>How would you define a torus of revolution?
as a specific embedding of [math]S^1 \times S^1[/math], the cartesian product of two circles, into [math]\mathbb{R^3}[/math].

You can go that route if you want, now tell me what it means for an object to be n-dimensional independent from a vector space.

Little babby brainlet doesn't want to think too outside-the-boxey

that's meaningless dribble then. choose whatever you want.

a topological manifold has dimension n if it's locally homeomorphic to R^n

for a connected manifold it's the dimension of the euclidean space it's locally isomorphic to (this is well-defined by the invariance of domain)
for a CW-complex (simplicial complex) it's the maximum of dimensions of all cells (simplices) contained in the complex (if it exists)
I don't care about other spaces. there's the Hausdorff dimension or something, but I don't know anything about it.

fair enough. just make sure you state that you're talking about parameterization, instead of spanning spaces, since the dimensionality of the space being spanned is usually what is implied when discussing geometric shapes with everyday people.

no. it's not. laymen understand the surface of the earth is 2 dimensional

Yes it is. The earth is flat.

But in all seriousness, the defacto metric is euclidean. In that context, the circle is 2d span of whatever coordinate system you use. Most people define it as a set of points in a plane, which is a great 2d vector space.

the de facto metric in usual manifolds is that induced by a certain typical embedding in R^n, yes. I don't know why you're saying this. it doesn't change that the surface of the earth has dimension 2 and laymen understand that.

the circle isn't a vector space (it's not a span). the fact that it can be embedded in some R^n is irrelevant: any differentiable manifold can be embedded in R^n.