@bellinterlab. I’m not exactly clear what you mean when you say:
" in PGA, Euclidean objects reduce in dimension as the grade of the blades representing them increases"
It does indeed appear at first sight that in the dual construction (i. e., the 1-vectors represent planes) that increasing the grade decreases the dimension of the associated geometry. However, it’s my belief that a closer look at what we mean by “dimension” shows that the dimension always increases with grade.
In PGA whatever primitive is represented by 1-vectors, has dimension 0, and the higher grades are built by wedging these 0-dimensional objects together . This is familiar to us when the 1-vectors are points, the wedge is the join operation, and wedging two 1-vectors gives a 1D line, and wedging 3 gives a 2D plane. This plane as 2-vector is a composite object and can be thought of as being made up of all the points that lie in the plane.
When planes are 1-vectors, then the higher grades are also built up by wedging 1-vectors, but now the wedge is meet and planes are considered indivisible, i. e., 0-dimensional. The wedge of 2 planes is their intersection line (1D), and the wedge of 3 planes is their intersection point (2D). This point as 2-vector is a composite object and can be considered as made up of all planes that pass through it.
I’ll stop here and refer you to a previous post here that includes more details on this question.
Please post again if anything is still puzzling.