What is the minimum number of co-ordinates used to perfectly describe the shape,orientation and position of an n-dimensional object? What is the minimum number of co-ordinates used to perfectly describe the shape,orientation and position of a n-dimensional object? How do I make an approach to this problem? I am confused with the shape and orientation. How many dimensions do we need to measure even common 3-d objects? 
 A: The question is ill posed. A point can exist in an n-dimensional space. You need n coordinates to specify its position, but it has neither shape nor orientation. A piece with some arbitrary fractal surface as a shape would need infinitely many. The next step up from a point would be a sphere. It is a truly n-dimensional object wich can be specified with n+1 coordinates (for example the radius).  The person who posed that question probably had the answer (n+1)*n in mind. This is the number of coordinates necessary to define a simplex -- the smallest polytope that is truly n-dimensional. It does have a completely defined orientation, and because the person who posed the question wanted to prevent something like the sphere as a solution he threw "shape" in there as well, making it quite ill posed.
A: The minimum pieces of information required also depends on where, what dimensional space it is placed, and not only on what dimensional the object is. For example, you place a point on itself, you need nothing. You place it on a line, you need one co-ordinate. You keep it in space, you need three.     
And for determining shape, orientation etc.,consider this: Suppose you want to do it for a sphere whose radius you know. You can know about it's orientation and shape (but which you obviously still know) completely with four pieces of information: The three space co-ordinates of it's centre and it's radius. But this would not have been possible if you did not know that it were a sphere. Then you would probably be getting millions of co-ordinates and still knowing almost nothing. 
One does not worry much about this because most of the time, you can do pretty well physics treating your object as point like. 
