Alan Watts transcending duality: https://youtu.be/F2I0vWoFNq0
Principle of DualityEdit
A projective plane C may be defined axiomatically as an incidence structure, in terms of a set P of points, a set L of lines, and an incidence relation I that determines which points lie on which lines. These sets can be used to define a plane dual structure.
Interchange the role of “points” and “lines” in
C = (P, L, I)
to obtain the dual structure
C∗ = (L, P, I∗),
where I∗ is the inverse relation of I. C∗ is also a projective plane, called the dual plane of C.
If C and C∗ are isomorphic, then C is called self-dual. The projective planes PG(2, K) for any field (or, more generally, for every division ring(skewfield) isomorphic to its dual) K are self-dual. In particular, Desarguesian planes of finite order are always self-dual. However, there are non-Desarguesian planes which are not self-dual, such as the Hall planes and some that are, such as the Hughes planes.
In a projective plane a statement involving points, lines and incidence between them that is obtained from another such statement by interchanging the words “point” and “line” and making whatever grammatical adjustments that are necessary, is called the plane dual statement of the first. The plane dual statement of “Two points are on a unique line” is “Two lines meet at a unique point”. Forming the plane dual of a statement is known as dualizing the statement.
If a statement is true in a projective plane C, then the plane dual of that statement must be true in the dual plane C∗. This follows since dualizing each statement in the proof “in C” gives a corresponding statement of the proof “in C∗”.
The Principle of Plane Duality says that dualizing any theorem in a self-dual projective plane C produces another theorem valid in C.
The above concepts can be generalized to talk about space duality, where the terms “points” and “planes” are interchanged (and lines remain lines). This leads to the Principle of Space Duality.
These principles provide a good reason for preferring to use a “symmetric” term for the incidence relation. Thus instead of saying “a point lies on a line” one should say “a point is incident with a line” since dualizing the latter only involves interchanging point and line (“a line is incident with a point”).
The validity of the Principle of Plane Duality follows from the axiomatic definition of a projective plane. The three axioms of this definition can be written so that they are self-dual statements implying that the dual of a projective plane is also a projective plane. The dual of a true statement in a projective plane is therefore a true statement in the dual projective plane and the implication is that for self-dual planes, the dual of a true statement in that plane is also a true statement in that plane.
As the real projective plane, PG(2, R), is self-dual there are a number of pairs of well known results that are duals of each other. Some of these are:
Desargues’ theorem ⇔ Converse of Desargues’ theorem
Pascal’s theorem ⇔ Brianchon’s theorem
Menelaus’ theorem ⇔ Ceva’s theorem
|douglasleethompson Blogger, Theoretical Philosopher, Alchemist, Righteous Dude: Douglas Lee Thompson