Duality

The concept of duality can be understood geometrically in an n-dimensional projective setting by considering both the subspace that an object occupies and the complementary subspace that the object concurrently does not occupy. The dimensionalities of these two components always sum to n, and they represent the space and antispace associated with the object. The example shown in Figure 1 demonstrates the duality between homogeneous points and planes in a four-dimensional projective space. The quadruplet of coordinates $$(p_x, p_y, p_z, p_w)$$ can be interpreted as a vector pointing from the origin toward a specific location on the projection subspace $$w = 1$$. This vector corresponds to the one-dimensional space of the point that it represents. The dual of a point materializes when we consider all of the directions of space that are orthogonal to the single direction $$(p_x, p_y, p_z, p_w)$$. As illustrated by the figure, these directions span an $$(n - 1)$$-dimensional subspace that intersects the projection subspace at a plane when $$n = 4$$. In this way, the coordinates $$(p_x, p_y, p_z, p_w)$$ can be interpreted as both a point and a plane, and they are duals of each other.

When we express the coordinates $$(p_x, p_y, p_z, p_w)$$ on the vector basis as $$p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$, it explicitly states that we are working with a single spatial dimension representing a point, and the ambiguity in the coordinates is removed. Similarly, if we express the coordinates on the antivector basis as $$p_x \mathbf e_{423} + p_y \mathbf e_{431} + p_z \mathbf e_{412} + p_w \mathbf e_{321}$$, then we are working with the three orthogonal spatial dimensions representing a plane. In each case, the subscripts of the basis elements tell us which basis vectors are present in the representation, and this defines the space of the object. The subscripts also tell us which basis vectors are absent in the representation, and this defines the antispace of the object. Acknowledging the existence of both the space and the antispace of any object and assigning equal meaningfulness to them allows us to explore the nature of duality to its fullest. A vector $$p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ is never only a point, but both a point and a plane simultaneously, where the point exists in space, and the plane exists in antispace. Likewise, an antivector $$p_x \mathbf e_{423} + p_y \mathbf e_{431} + p_z \mathbf e_{412} + p_w \mathbf e_{321}$$ is never only a plane, but both a plane and a point simultaneously, where the plane exists in space, and the point exists in antispace. If we study only the spatial facet of these objects, then we are missing half of a bigger picture.