This work was inspired by a series of discussions with researcher DarQ, who rightly pointed out that the equation of a plane electromagnetic wave (PEW) can be obtained directly in a four-dimensional form — without intermediate transitions through differential operators, rotors, or wave equations. This approach opens the possibility for a more unified geometric description of the wave.

The purpose of this work is to propose an alternative view on the nature of a plane electromagnetic wave, considering it as the motion of a point in a bicomplex velocity space, in which the electric, magnetic, and temporal components are combined into a single four-vector structure. Such representation makes it possible to connect the dynamics of the wave with the internal geometry of the hypercomplex basis and to show that the propagation of the PEW naturally follows from the properties of this space.

This is a generalized form of the complex representation of the wave. Here:

• \(c\) — the speed of light [1];
• \(\jk\) — a hyperbolic unit, the square of which equals plus one;
• \(\v\) — the doubled frequency of the wave: \(\v = 2 f\);
• \(t\) — time.

Based on formula (2), let us make several more assumptions.

• \(\s\) — the time component, the rate of its change (in the direction of the unit of the basis \(1\));
• \(x\) — the “electric” or rotational component (along \(i\));
• \(z\) — the “spatial” or longitudinal component (along \(\jk\));
• \(y\) — the “magnetic” or conjugate component (along \(i\jk\)).

The components \(\s, x, y, z\) are the coordinates of the 4-velocity in the bicomplex space built on the basis \(\{1, i, \jk, i\jk \}\), where \(i^2 = -1,\, \jk^2 = +1\). Here \(\s,z\) form a hyperbolic pair (time + longitudinal direction), and \(x,y\) form a complex pair (electromagnetic polarization).

Then, the expansion of the generalized form of the complex representation of the wave can be written as follows: \[\tag{3} V(t) = \s + i\, x + \jk\, z + i\jk\, y \] Such a vector cannot be regarded as a point in ordinary 4D Cartesian space — only as an expansion over a hypercomplex basis.

The velocity vector \(V = V(t)\) can be decomposed into two lightlike projections \[\tag{4} V = V_{+} + V_{-} \\ V_{+} = V\, \nu_{+}, \quad V_{-} = V\, \nu_{-}, \] using the properties of the idempotents (1, 2) from the first part of this work. Then: \[\tag{5} V_{+} = c\, \nu_{+} \\ V_{-} = c\, e^{i \wt} \nu_{-} \] That is, in lightlike coordinates, the PEW velocity in the \(\nu_{+}\) direction equals the speed of light. Meanwhile, the \(\nu_{-}\) direction, due to the imaginary unit, decomposes into two internal projections \[\tag{6} V_{-} = c\, (\nu_{-}\, \cos \wt + i\nu_{-}\, \sin \wt), \] which provides the rotation of the vector. Thus, one coordinate corresponds to the wave’s translation (drift), while the other two correspond to transverse oscillations.

To obtain such an equation, we must integrate expression (3) \[\tag{7} L(t) = \int V(t)\, dt = \int (\s + i\, x + \jk\, z + i\jk\, y)\, dt, \]

If, geometrically, in the velocity equation we obtained a circle, then after integrating it, we get a spiral propagating along the \(\jm\) axis (the \(Z\) coordinate):

Fig.2. Trajectory of the wave point (bicomplex helical form in subspaces \(i, \jk, i\jk\)

Figure 2 shows a three-dimensional spiral (helix), representing the trajectory of a point describing the propagation of a wave in bicomplex space built on the basis \(\{1, i, \jk, i\jk \}\). It is impossible to display the complete 4D model on a monitor, so it is represented here only in three planes.

This figure demonstrates the structural difference between the classical 2D representation of the PEW: \(e^{i(\wt - kr)}\), and the 4D version presented here: \(\jm^{\v t}\). It contains not only 4D velocities but also the wave’s propagation along one of the coordinates, without separately postulating or manually adding a wave vector. Such a representation appears more natural.

According to classical Maxwell theory, the instantaneous energy density of a plane electromagnetic wave depends on time and varies according to the law \[\tag{9} A(t)^2 \sim \cos^2 \wt \] Thus, energy oscillates over time. However, such a description represents only a temporal modulation of the local field density, not an actual change in the total energy of the system. From a physical point of view, the total energy of the wave is conserved and should be characterized by a stationary value.

In the hyperbolic world, to determine the square (or norm) of a number, it is necessary to multiply it by its conjugate (hyperbolic conjugation): \[\tag{11} V(t) = c\, \jm^{\v t} \\ V^*\!(t) = c\, \jm^{-\v t} \] Then: \[\tag{12} A(t)^2 = V(t)\, V^*\!(t) = c^2 \] Thus, under the accepted norm (a positively defined sum of squares of components), the magnitude of the wave equals the speed of light and does not depend on time, and \(|V(t)|\) is an invariant.

The resulting model shows that a plane electromagnetic wave in bicomplex representation can be described as the motion of a point in a four-dimensional velocity space, where the electric, magnetic, temporal, and spatial components are interconnected by a single hypercomplex basis. This description eliminates the need for the artificial introduction of a wave vector and allows us to consider wave propagation as an inherent property of the geometry of the space itself.

Within this model, the electric and magnetic fields arise as mutually conjugate components of one bicomplex structure, and their dynamics — as rotation and propagation along the hyperbolic axis. It is shown that the Minkowski metric and the lightlike interval naturally follow from this description, and the wave energy has a stationary (invariant) character.

Thus, the presented approach unifies classical electrodynamics and the geometric description of the wave into a single formal system, where the fundamental properties of the PEW appear as a consequence of the hypercomplex symmetry of space rather than as externally imposed equations.