Points, planes etc. embedded in higher-dimensional Euclidean spaces

Question

A common set of definitions for a plane is:

• three non-collinear points
• a line and a point not on that line
• two distinct but intersecting lines
• two parallel lines.

Is it possible to provide a generic set of definitions for higher-dimensional Euclidean spaces? More specifically, given a set of spaces of dimensions $D=\{d_0, d_1, d_2, \dots\}$, what dimension of Euclidean space is required to contain all of these spaces?

Answer 1

$n+1$ points in general positioning, i.e. which are not contained within an $(n-1)$-dimensional space, will define an $n$-dimensional space.

Whenever your spanned space already contains the origin, then that space itself is the searched for Euclidean embedding space. If not then just add that one as further point, and you see that you just need an Euclidean space of one more dimension.

--- rk