blade

A blade is a term often used to describe a basis entity in the space defined by a geometric algebra. Since a geometric algebra is a multi-graded space, the basis entities also have multiple grades. To distinguish the various graded entities, the blades are often prefixed by their grade. For example a grade-$k$ basis entity would be called a $k$-blade.

The number of linearly independent $k$-blades in a particular geometric algebra is dependent on the number of dimensions of the manifold on which the algebra is defined. For an $n$-dimensional manifold, the number of $k$-blades is given by the binomial coefficient.

$$N_k=\left(\begin{array}{c}\hfill n\hfill \\ \hfill k\hfill \end{array}\right)$$

The total number of basis blades of all grades in a geometric algebra defined on an $n$-manifold is then:

$$N=\sum _{k=0}^n{N}_k=2^n$$