Dimension

In mathematics, the word dimension has multiple meanings. More than one meaning applies to polytopes. This article goes through all of them.

Dimension of a vector space
This is perhaps the most common use of the word "dimension". In mathematics, a vector space is a set of objects called vectors, that can be added together or multiplied by a numerical value, called a scalar. The dimension of a vector space is roughly the number of different directions that exist within it.

The dimension of a polytope embedded in a vector space is the dimension of this space. Intuitively, the dimension of a vector space is the amount of different independent directions that exist within it. Most often, polytopes are embedded in some vector space $$\mathbb R^n$$, which is the n-dimensional space whose points have n real coordinates.

More formally: a set of vectors $$\vec{v}_1,\vec{v}_2,\ldots,\vec{v}_n$$ is called linearly independent whenever the only way to add multiples of these vectors to get the zero vector is by making all of the vectors zero. That is,
 * $$a_1\vec{v}_1+a_2\vec{v}_2+\ldots+a_n\vec{v}_n=\vec{0}\text{ implies }a_1,a_2,\ldots,a_n=0.$$

This same set of vectors is said to span the vector space if for any vector $$\vec u$$, there exist scalars $$a_1,a_2,\ldots,a_n$$ such that
 * $$a_1\vec{v}_1+a_2\vec{v}_2+\ldots+a_n\vec{v}_n=\vec{u}.$$

If a set of vectors is linearly independent and spans the vector space, it is called a basis of the vector space. One may prove that all bases of a vector space have the same size, which is then defined as the dimension of the vector space.

For more details, see the Wikipedia articles on linear independence, linear span, and bases.

Rank in polytopes


The rank of a polytope is an intrinsic property that, contrary to previous ones, does not depend on wherever the polytope is embedded. A flag of a polytope is a set of elements that are all incident to one another that cannot be made larger. All flags in a polytope have a common length. This common length minus 2 is defined as the rank of the polytope.

Subtracting 2 is mostly an arbitrary convention that makes it so that rank and other notions of dimension coincide under most circumstances.

Most often, a polytope of rank n will be embedded in an n-dimensional space in any of the previous senses. This isn't always the case: a square tiling has rank 3 but is most naturally embedded in a 2-dimensional space. Conversely, skew polygons like the faces of Petrials have rank 2 but are embedded in a 3-dimensional space.