Definitions - Limits to Thought

Many great mathematicians conceptualised their way to break-through insights by not using Excel ("Excel" is used here, unfairly, as an example of the "status quo" in which it is easy to become entrapped). Kepler, Einstein, Alice in Wonderland and others started with the most general definitions - they always tried to "start at the beginning" and see where it took them.

Kepler, in thinking about the stacking pattern for storing the most cannon balls in a ship hull, started with the problem, "what is a sphere". An answer is the set of all points in space that are less than or equal to a specified distance r from a given point.

Great. So what is "space". And that is where it gets interesting. If you think in a 1D coordinates space, a sphere is a line, of length 2r. If you think in a 2D space, a sphere is the what is enclosed by a circle, of radius r, and if you think in a 3D space co-ordinates a sphere is a volume of radius r. And in 4D?