How can I generalize PGA to higher dimensional Euclidean space?

Projective GA
1

According to ‘A Guided Tour to the Plane-Based Geometric Algebra PGA’, PGA is applicable to R_{d, 0, 1}. But I am confused about how I can apply PGA to scenario where d > 3.

For example, in 2D PGA:

  • Lines are vectors
  • Points are bivectors

In 3D PGA:

  • Planes are vectors
  • Lines are bivectors
  • Points are trivectors

How this should look like for 4D PGA? Should I treat volumes as vectors? If so, what should be the geometric interpretation of it?

2

How about interpreting the vector as the orientation of the volume in 4D? That is: a signed size + direction of the volume (3D subspace) orthogonal to (all) three independent directions in the 3D volume.

3

In other words, the 1-vector in 4D PGA is the vector that stays fixed for all rotations of the volume.

4

Thanks! I think that makes sense and align with the intuition in 3D and 2D as well.