A movement requires 1 dimension, a rotation requires 2 dimensions, a what requires three dimensions? Movement
A zero-dimensional object cannot move. A one-dimensional object can move in one dimension (the x-axis). A two-dimensional object can move in two dimensions (the x-axis and y-axis). A three dimensional object can move in three dimensions (the x-, y-, and z-axis).
In other words: There are n different ways for an n-dimensional object to move.
Rotation
A zero- and a one-dimensional object cannot rotate. A two-dimensional object can rotate in one way (in the xy-plane). A three dimensional object can rotate in three ways (in the xy-, xz-, and yz-plane). And a four dimensional object can rotate in 6 ways (xy-, xz-, yz-, xw-, yw-, and zw-plane).
In other words: There are n(n-1)/2 different ways for an n-dimensional object to rotate.
Now for the question, what do you call the next thing?
I do know that the next thing has some properties if you follow the same pattern. But I have no idea what it is and I was hoping you guys could help me out.
Here are some of the properties that I have extrapolated:
The next thing
A zero-, one- and two-dimensional object cannot do it. A three dimensional object can do it in one way (along xyz). A four dimensional object can do it in four ways (along xyz, xyw, xzw and yzw. And if you want to extrapolate further... an n-dimensional object can do it in n(n-1)(n-2)/6 different ways.
So yeah, what do you call this thing?
 A: One way of interpreting this question is as follows:
The Lie group $\mathbb R^n$ acts effectively and transitively on the smooth manifold $\mathbb R^n$; each of these is $n$-dimensional. The special orthogonal group $SO(n+1)$ acts effectively and transitively on $S^n$; the former has dimension $n(n+1)/2$, and $S^n$ has dimension $n$.
Can we improve this? Are there, for instance, $n(n-1)(n-2)/6$-dimensional groups that act on $n$-dimensional manifolds?
If the group $G$ is compact, the answer is a firm no. It is a result of Montgomery and Zippin that a compact topological group acting effectively and transitively on a manifold of dimension $n$ must have dimension at most $n(n+1)/2$. If you want $G$ to be a Lie group acting smoothly, this is easier: first prove it if $G$ acts by isometries on a Riemannian manifold by induction; then if you have a transitive effective action on $M$, average the metric over $G$ to get a $G$-invariant metric on $M$, hence an action by isometries. That $G$ is compact is important here!
I don't know what bounds one has in the case that $G$ is non-compact - I'd like to hear from somebody if they do. 
