How can duality generate a new theorem from an existing geometric theorem?
The sight lines drawn from the image in the reality plane (RP) to the artist's eye intersect the picture plane (PP) to form a projective, or perspective, drawing. The horizontal line drawn parallel to PP corresponds to the horizon. Early perspective experimenters sometimes used translucent paper or glass for the picture plane, which they drew on while looking through a small hole to keep their focus steady.
In projective geometry, the principle of duality allows for the generation of a new theorem from an existing one by systematically interchanging specific geometric terms.12
In the projective plane, this is achieved by swapping the words "point" and "line," as well as "collinear" and "concurrent," throughout a statement. For example, the dual of the theorem "two points determine a line" is "two lines determine a point". In three-dimensional projective geometry, the principle extends to interchanging "point" and "plane," while the line remains its own dual.21
This process functions because projective geometry possesses a lattice structure where points, lines, and planes are ordered by inclusion. By applying these substitutions mechanically to any valid proof, one can derive a new theorem. While some resulting statements may be equivalent to the original or its converse, this principle effectively doubles the number of known theorems at a single stroke.12