This contribution explores some analogies between special relativity and geometrical tools developed by the ancient Greeks. The kinematics of one-dimensional elastic collisions is solved with simple ruler-and-compass constructions on conic sections. Then, a thought-provoking relation involving Lorentz transformations, Archimedes’ law of the lever and Einstein’s formula for the relativistic mass is put forward. The familiarity with classical geometry is useful in developing intuitions on deep concepts of modern physics and can be profitable for high school or basic undergraduate teaching. Moreover, it is fascinating to establish a bridge connecting beautiful ideas separated by two millennia.

This study proposes an explanation for the physical power flow in planar circuits by analogy to theoretical two-dimensional circuits using a new mathematical model based on Geometric Algebra (GA) and 2D Maxwell’s equations. In contrast with traditional 3D physics in the observable real world, the magnetic field can be defined as a bivector instead of an axial vector allowing to obtain the Poynting Vector directly in a 2D flat world, where physical variables of planar circuits can be obtained. This approach is presented here for the first time to the best of the author’s knowledge. Previous investigations have focused on simplifications and symmetries of real 3D circuits studied mainly in the phasor and frequency domain. In this work, the electromagnetic power flow phenomenon is analyzed on a completely 2D time-domain basis and derived directly from the undisputed Maxwell equations, formulated in two dimensions. Several cases of special interest in AC multi-phase circuits are presented using the proposed technique, bringing a new simplified approach to the measurement of power flow exchange between the source and the load. It suggests a new way to understand energy propagation from a purely physical point of view.

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Using geometric algebra and calculus to express the laws of electromagnetism we are able to present magnitudes and relations in a gradual way, escalating the number of dimensions. In the one-dimensional case, charge and current densities, the electric field E and the scalar and vector potentials get a geometric interpretation in spacetime diagrams. The geometric vector derivative applied to these magnitudes yields simple expressions leading to concepts like divergence, displacement current, continuity and gauge or retarded time, with a clear geometric meaning. As the geometric vector derivative is invertible, we introduce simple Green’s functions and, with this, it is possible to obtain retarded Liénard–Wiechert potentials propagating naturally at the speed of light. In two dimensions, these magnitudes become more complex, and a magnetic field B appears as a pseudoscalar which was absent in the one-dimensional world. The laws of induction reflect the relations between E and B, and it is possible to arrive to the concepts of capacitor, electric circuit and Poynting vector, explaining the flow of energy. The solutions to the wave equations in this two-dimensional scenario uncover now the propagation of physical effects at the speed of light. This anticipates the same results in the real three-dimensional world, but endowed in this case with a nature which is totally absent in one or three dimensions. Electromagnetic waves propagating exclusively at the speed of light in vacuum can thus be viewed as a consequence of living in a world with an odd number of spatial dimensions. Finally, in the real three-dimensional world the same set of simple multivector differential expressions encode the fundamental laws and concepts of electromagnetism.

With the goal of developing didactic tools, we consider the geometrization of the addition of velocities in special relativity by using Minkowski diagrams in momentum space. For the case of collinear velocities, we describe two ruler-and-compass constructions that provide simple graphical solutions working with the mass-shell hyperbola in a 1+1-dimensional energy-momentum plane. In the spirit of dimensional scaffolding, we use those results to build a generalization in 1+2 dimensions for the case of non-collinear velocities, providing in particular a graphical illustration of how the velocity transverse to a boost changes while the momentum remains fixed. We supplement the discussion with a number of interactive applets that implement the diagrammatic constructions and constitute a visual tool that should be useful for students to improve their understanding of the subtleties of special relativity.