Doubly nilpotent numbers in the 2D plane

Dual numbers, split-quaternions, split-octonions, and other number systems with nilpotent spaces have received sporadic yet persistent interest, beginning from their roots in the 19th century, to more recent attention in connection with supersymmetry in physics. In this paper, a number system in the 2D plane is investigated, where the squares of its basis elements p and q each map into the coordinate origin. Modeled similarly to an original concept by C. Musès, this new system will be termed “PQ space” and presented as a generalization of nilpotence and zero. Compared to the complex numbers, its multiplicative group and underlying vector space are equipped with as little as needed modifications to achieve the desired properties. The locus of real powers of basis elements pα and qα resembles a four-leaved clover, where the coordinate origin at (0, 0) will not only represent the additive identity element, but also a map of “directed zeroes” from the multiplicative group. Algebraic and geometric properties of PQ space are discussed, and its naturalness advertised by comparison with other systems. The relation to Musès' “p and q numbers” is shown and its differences defended. Next to possible applications and extensions, a new butterfly-shaped fractal is generated from a recursion algorithm of Mandelbrot type.