What is the least amount of information needed to describe an n-space solid in m-space? In linear algebra, we are taught firstly that a plane in 3-space can be understood in 6 pieces of information, two 3D vectors. However, after learning about a normal vector, it is understood that a plane only needs 1 vector or 3 pieces of information to represent itself.
So given an N-space solid and in M-space, what is the least amount of information that can define the shape?
 A: First of all, this has nothing to do with information theory. You can't represent all possibilities with a finite number of bits. And you can always map a finite number of real parameters into only one real number.
Second, for a plane in 3D you only need the orientation of the normal vector, so you only need 2 parameters for the orientation. Then you need the position, which can be given by the distance from the plane to the origin (with sign).
Another way of thinking about this is to start with a standard $N$-hypersurface and think about how many parameters you need to transform that into any other possible configuration. In $M$ dimensions, you need $M(M-1)/2$ parameters for rotation. But if you rotate an $N$-hypersurface in the $N$-dimensional space spanned by it, you get the same object. As noted by the user stewbasic in the comment below, the same thing happens for the other $M-N$ dimensions. So we have $N(N-1)/2$ redundant parameters for the $N$-dimensional rotations and $(M-N)(M-N-1)/2$ for the $(M-N)$-dimensional rotations. The total for rotations is then:
$$\frac{1}{2}M(M-1)-\frac{1}{2}N(N-1)-\frac{1}{2}(M-N)(M-N-1)$$
$$= N(M-N)$$
and if you include the position, by the same logic, you need $M$ parameters, where $N$ of them are redundant, so the total is:
$$N(M-N)+M-N$$
$$=(N+1)(M-N)$$
