n-dimensional affine space that is not isomorphic with the n-dimensional real space

Question

I'm looking for an example of $n$-dimensional affine space that is isomorphic with $\mathbb{R}^n$ as affine space but not with respect to other properties (for example it has different ordering etc.) Thank you.

EDIT: in practice, I am looking for an affine space that has more or less structure than $\mathbb{R}^n$. Intuitively $\mathbb{R}^n$ has "more structure" than a canonical affine space because, by its field properties, it has a special point (that is the zero with respect to addition). I need an example of affine space different from $\mathbb{R}^n$ but having the same dimension.

Answer 1

If you look on the Wikipedia page http://en.wikipedia.org/wiki/Affine_space you will find an example mentioned in the very first paragraph. Did you look there? Just take any $n$-dimensional subspace of a (larger) vector space and add a fixed vector to it (a shifted subspace). That's an $n$-dimensional affine space. (For example, any line in the plane -- not necessarily containing the origin -- is a 1-dimensional affine space.) If this doesn't contain 0 then it is not isomorphic to ${\mathbf R}^n$ as a vector space since it doesn't have a natural structure as a vector space.

Answer 2

Take the space of possible energy values of $n$ independent particles. This is an affine space of dimension $n$ because you can add a vector in $\mathbb{R}^n$ to any $n$-tuple of energy values, but there is no distinguished origin because there is no distinguished value of energy that we can call zero.