Non-Euclidean Geometry - Planar Algebras

Planar Algebras

In analytic geometry a plane is described with Cartesian coordinates : C = {(x,y) : x, y in R}. The points are sometimes identified with complex numbers z = x + y ε where the square of ε is in {−1, 0, +1}. The Euclidean plane corresponds to the case ε2 = −1 since the modulus of z is given by

and this quantity is the square of the Euclidean distance between z and the origin. For instance, {z : z z* = 1} is the unit circle.

For planar algebra, non-Euclidean geometry arises in the other cases. When, then z is a split-complex number and conventionally j replaces epsilon. Then

and {z : z z* = 1} is the unit hyperbola.

When, then z is a dual number.

This approach to non-Euclidean geometry explains the non-Euclidean angles: the parameters of slope in the dual number plane and hyperbolic angle in the split-complex plane correspond to angle in Euclidean geometry. Indeed, they each arise in polar decomposition of a complex number z.