Fractal-type sets in the four-dimensional space using bicomplex and hyperbolic numbers (Q2300488)

This paper provides a way to construct fractal sets from the point of view of bicomplex numbers \(\mathbb{BC}\). Bicomplex numbers forms a \(4\)-dimensional real algebra that generalizes complex numbers. Differently from quaternions, bicomplex numbers are commutative but contain zero divisors. Because of zero divisors, many classical concepts need to be redefined. Many definitions rely on the so-called subset of hyperbolic numbers, the bicomplex counterpart of real numbers. Starting from these main definitions it is possible to reconstruct: -- a partial ordering; -- a notion of angle; -- a notion of slope, straight line and segment. It is also possible to review many classical transformations in this setting. In particular, many classical cases are discussed in the paper. At the end some classical fractal sets are generalized in the context of bicomplex numbers and it is explained how these new constructions give a new viewpoint on the classical constructions.