# 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?

• if the shape is (piecewise) smooth, then countable infinity, I believe. – Yrogirg Aug 20 '12 at 13:36
• Is this question intended as stated? Leaving out the description of the shape would render an answerable question: "What is the minimum number of parameters required to perfectly describe the position and orientation of an arbitrary object in n-dimensional space?" – Johannes Aug 20 '12 at 14:56
• Notice that every object which is indeed describable will be describable on a one dimensional string. You can e.g. tape a video where you explain exactly how the object looks like and where the points are etc. and then take an Edding marker and paint that files data on a long thread, e.g. in Morse code. – Nikolaj-K Aug 20 '12 at 15:22
• If you remove shape, this has an answer--- it's N position variables plus N(N-1)/2 orientation variables. Why are you asking about shape? – Ron Maimon Aug 21 '12 at 3:36

## 2 Answers

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.

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.