Can someone explain intuitively how, for a Galilean universe, $A^4$ is equivalent to $\Bbb{R} \times \Bbb{R}^3$? I am reading Arnold's book on classical mechanics. Obviously, everyone who's studied basic physics feels comfortable with $\Bbb{R} \times \Bbb{R}^3$. This is just a pair $(t,\mathbf{x})$. There are three basic actions one can take. 


*

*Uniform motion with velocity: $g_1(t,\mathbf{x}) = (t, \mathbf{x}  + \mathbf{v}t)$

*Translations: $g_2(t,\mathbf{x}) = (t+s, \mathbf{x}  + \mathbf{s})$

*Rotations: $g_3(t,\mathbf{x}) = (t,G\mathbf{x})$


However, Arnold talks about defining Galilean space using an affine space $A^4$ and nothing is coming to mind that connects the affine space definition of Galilean space to the intuitive one I stated above. 
My Question
Can someone provide an intuitive explanation for how defining Galilean space as an affine space would permit us to define the same kinds of actions as above?  In what way are these two equivalent? 
 A: The Galilean spacetime is indeed the affine space $\mathbb{A}^4$. Affine space can be considered as a 'space with no origin', which makes intuitively sense because why would some point (the origin) be special. For example a trivial Galilean space is $\mathbb{E}\times \mathbb{E}^3$ where $\mathbb{E}$ is Euclidean space.
The $\mathbb{R}\times \mathbb{R}^3$ you have is referred to as Galilean coordinate space. Now define an affine map which preserves the Galilean spacetime structure as 
$$\varphi:\mathbb{A}^4\to\mathbb{R}\times\mathbb{R}^3,\; A_t\mapsto(t(A_t),\mathbf{r}(A_t)),$$
where $A_t$ is a point of simultaneous events in Galilean space. This is called a Galilean chart. With this you can identify the Galilean spacetime with the coordinate space $\mathbb{R}\times \mathbb{R}^3$. Intuitively you attach coordinate system to the affine space $\mathbb{A}^4$ with this map.
So you have this abstract affine space and you attach a coordinate system to it which makes it a coordinate space $\mathbb{R}\times\mathbb{R}^3$. Now all the actions you described can be implemented in the chosen coordinate space.
Edit:
The $g$'s form what is called the Galilean group. This is a mapping $$g:\mathbb{R}\times\mathbb{R}^3\mapsto\mathbb{R}\times\mathbb{R}^3,(t,\mathbf{x})\mapsto(t+s,\mathbf{Gx}+\mathbf{v}t+\mathbf{s}).$$ Also it can be shown that all Galilei charts are of the form $\varphi^´:=g\circ\varphi$ So the $g$'s correspond to change of coordinates in the coordinate space. 
