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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?

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if the shape is (piecewise) smooth, then countable infinity, I believe. – Yrogirg Aug 20 '12 at 13:36
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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
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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. – Nick Kidman Aug 20 '12 at 15:22
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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

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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.

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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.

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