Hornbein said:
Would you say that a circle is a one dimensional object embedded in a two dimensional space?
A circle is a 1D manifold embedded in a 2D space.
I would say, in terms of the degrees of freedom, for the
equation of a circle, it should capture all of the information one would need to draw it. That would be the center and the radius, so 3 degrees of freedom total, or more if you consider color.
But also the space of circles has 3 dimensions. Any time there is a degree of freedom, I think that degree of freedom can be described as an ability to vary in a dimension.
So I think there is a subtleness about the differences between the terms. For example, if a space is static, like Euclidean space, I wouldn't say that space has degrees of freedom, but it does obviously have dimensions. But if the space can be distorted or varied, or there is a family of spaces based on some parameters, then you could talk about the degrees of freedom.
Likewise, it might be more precise to say a system can vary in N dimensions or is N dimensional rather than to say it
has N dimensions. But I don't think enough people care about that level of specificity in language as long as they can understand each other well enough.
That's my understanding at least.