Two-dimensional geometries, topologies, trigonometries and physics generated by complex-type numbers
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In the recent monograph , G.L. Naber provides an interesting introduction to the special theory of relativity that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics.
His mathematical model is based on a special indefinite inner product of index one and its associated group of orthogonal transformations (the Lorentz group). Also, in the same monograph, the Hawking, King and McCarthy’s topology  is presented. This topology is physically well motivated and has the remarkable property that its homeomorphism group is essentially just the Lorentz group.
Starting from the remark that the inner product and the topology above can be generated by the so-called hyperbolic complex numbers, in this paper we introduce and study two-dimensional geometries and physics generated in a similar manner, by the more general so-called complex-type numbers, i.e. of the typez=x+qy,q ∉ ℝ, whereq 2=A+q(2B), A,A, B ∉ ℝ fixed.
KeywordsLorentz Group Timelike Vector Dual Complex Newtonian Physic Euclidean Topology
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