And if you take quaternions further, you end up with something you can generalize to any number of dimensions, which also unifies algebra and geometry in very n

And if you take quaternions further, you end up with something you can generalize to any number of dimensions, which also unifies algebra and geometry in very nice ways: Clifford algebra. https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/ https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-v... > Here we see the hierarchy of Clifford Algebras each with its own characteristic pattern of components, that appear in numbers corresponding to Pascal’s triangle. Two-dimensional clifs, Cl2 (the even sub-algebra) are mathematically isomorphic to complex numbers. Three-dimensional clifs, Cl3 are isomorphic to Pauli matrices used to model subatomic particles. Cl4 is isomorphic to Dirac matrices that model relativistic subatomic particles. All of that assumes every element of the metric signature has the same sign; that is, the dot product of any basis vector with itself is always positive for all of them or negative for all of them. If you pick one dimension to have a sign opposite the others, that dimension is your time dimension and you're now in Minkowski space. The consequences of doing that is literally the entirety of Special Relativity. If you construct a Clifford algebra in Minkowski space, that's Spacetime algebra: https://arxiv.org/abs/1411.5002 https://arxiv.org/abs/1411.5002 > We present a comprehensive introduction to spacetime algebra that emphasizes its practicality and power as a tool for the study of electromagnetism. We carefully develop this natural (Clifford) algebra of the Minkowski spacetime geometry, with a particular focus on its intrinsic (and often overlooked) complex structure. Notably, the scalar imaginary that appears throughout the electromagnetic theory properly corresponds to the unit 4-volume of spacetime itself, and thus has physical meaning. The electric and magnetic fields are combined into a single complex and frame-independent bivector field, which generalizes the Riemann-Silberstein complex vector that has recently resurfaced in studies of the single photon wavefunction. The complex structure of spacetime also underpins the emergence of electromagnetic waves, circular polarizations, the normal variables for canonical quantization, the distinction between electric and magnetic charge, complex spinor representations of Lorentz transformations, and the dual (electric-magnetic field exchange) symmetry that produces helicity conservation in vacuum fields. This latter symmetry manifests as an arbitrary global phase of the complex field, motivating the use of a complex vector potential, along with an associated transverse and gauge-invariant bivector potential, as well as complex (bivector and scalar) Hertz potentials. Our detailed treatment aims to encourage the use of spacetime algebra as a readily available and mature extension to existing vector calculus and tensor methods that can greatly simplify the analysis of fundamentally relativistic objects like the electromagnetic field. More prosaically, it allows you to write all of the Maxwell equations in a single equation. http://www.av8n.com/physics/clifford-intro.htm http://www.av8n.com/physics/clifford-intro.htm