Gaussinen

Gaussinen is a fictional concept in speculative mathematics that generalizes the classical Gaussian integers. It is not a recognized object in real-world mathematics, but it is used in world-building and theoretical discussions to explore extensions of two-dimensional integer lattices and quadratic forms. The term implies a family of structures that mimic the arithmetic of complex numbers while introducing additional parameters.

In the Gaussinen framework, elements are represented as ordered pairs (a, b), where a and b come from a chosen base ring R. A fixed central element k in R determines the multiplication rule: (a, b) · (c, d) = (ac + k bd, ad + bc). Conjugation is defined by (a, b)∗ = (a, −b), and the Gaussinen norm is N(a, b) = a^2 − k b^2. The norm is designed to be multiplicative, N(xy) = N(x)N(y), mirroring the role of a norm in Gaussian integers.

Variants arise by selecting different base rings and different values of k. If R = Z and k = −1, the construction recovers the ordinary Gaussian integers, while other choices yield alternative two-dimensional integer-like rings with distinct factorization properties. Depending on k, Gaussinen rings may be principal ideal domains, UFDs, or fail to have unique factorization, which makes them suitable for exploring Diophantine questions in fictional settings.

Applications in fiction often use Gaussinen structures to illustrate lattice geometry, factorization challenges, or cryptographic analogies within a mathematically flavored universe. In non-fiction contexts, Gaussinen serves as a didactic device to discuss how changing the fundamental quadratic form affects arithmetic.

See also: Gaussian integers, quadratic forms, imaginary quadratic fields, norm.