The classical wave equation in complex form is based on the Euler exponential \(e^{i\omega t}\), which describes the rotation of a point on a planar circle. However, such a representation is limited to two-dimensional space and does not take into account the possibility of more complex oscillatory processes in a multidimensional medium.

To go beyond the planar model, this work introduces a hypercomplex unit \(\jk\) with the property \(\jk^2 = +1\), which makes it possible to transfer the description of wave motion onto a four-dimensional sphere, thereby extending the classical exponential form to a more general, spatially symmetric expression.

The expression currently used for 2D: \[ A(t) = A_0\kern0.5pt e^{i (\omega t - kr)} \]

The proposed expression for 4D: \[ A(t) = A_0\kern1pt \jm^{\varpi t} \] In this work, the following notations are used: \(A_0\) — amplitude, \(\omega\) — angular frequency, \(\varpi\) — doubled frequency, \(t\) — time. In addition, these formulas involve Euler’s number \(\large e\), and the hyperbolic unit \(\large \jk\).

Both formulas represent equations of a circle. In the first one, the number “pi” enters through the exponent, while in the second it is “embedded” inside the hyperbolic unit. Moreover, the second formula implicitly contains the wave vector \(k\) and the length \(r\).

Next, we will examine step by step the method of raising the hyperbolic unit to a power of any real number, obtain the sinus–cosine form of the equation, prove that what we obtain indeed represents a circle in 4D space, and take a closer look at the applications of the resulting formula.

Let \(\jm\) be a hypercomplex unit with the property \(\jm^2=+1\), having a two-point spectrum \(\{+1,-1\}\). Below we derive a formula for \(\jk^{y}\) for a real exponent \(y\) using idempotents. For clarity, the idempotents will be denoted by \(\nu_{\pm}\).

Define the projectors (idempotents) \[\tag{1} \nu_{+} = \frac{1+j}{2}, \qquad \nu_{-} = \frac{1-j}{2} \] They satisfy the standard properties \[\tag{2} \nu_{\pm}^2= \nu_{\pm}, \qquad \nu_{+} \nu_{-} = 0 \\ \nu_{+} + \nu_{-} = 1, \qquad \nu_{+} - \nu_{-} = j. \] Meaning: \(\nu_{+}\) projects onto the eigenvalue \(+1\), and \(\nu_{-}\) onto \(-1\).

Through idempotents, \(\jk\) can be decomposed as \[\tag{3} \jk =(+1) \, \nu_{+} + (-1) \, \nu_{-} \] This immediately follows from the last pair of equalities \( \nu_{+} + \nu_{-}=1 \) and \( \nu_{+} - \nu_{-} = \jk \). By adding and subtracting, we obtain expressions for \(1\) and \(\jk\) through \(\nu_{\pm}\).

For any function \(f\) defined on the spectrum \(\{+1,-1\}\), we have \[ f(\jm) = f(+1) \, \nu_{+} + f(-1) \, \nu_{-} \] Applying \(f\) to \(\jk\) means applying \(f\) to its eigenvalues and then recombining the result using idempotents.

Take \(f(x)=x^{\,y}\) for real \(y\). Then \[\tag{4} \jk^{y} = (+1)^{y} \,\nu_{+} + (-1)^{y} \,\nu_{-} = \nu_{+} + (-1)^{y} \,\nu_{-} \] Here arises the multivaluedness of \((-1)^y\). For the principal branch, it is convenient to set \[ (-1)^{y} = e^{i\pi y}, \] and for an arbitrary branch — \(e^{i(2k+1)\pi y}\), where \(k \in \mathbb{Z}\).

Substitute \(\nu_{\pm} = \dfrac{1\pm j}{2}\) into the expression from the previous step: \[ \begin{aligned} \jk^{y} &= \nu_{+} + e^{i\pi y} \, \nu_{-} = \frac{1+j}{2} + e^{i\pi y} \, \frac{1-j}{2} \\ &= \frac12 (1+e^{i\pi y}) + \frac{\jk}{2} (1-e^{i\pi y}). \end{aligned} \]

For other branches, replace \(e^{i\pi y}\) with \(e^{i(2k+1)\pi y}\).

Here arises the multivaluedness of \((-1)^y\). For the principal branch, it is convenient to set \[ (-1)^y = e^{i\pi y}, \] since such a definition ensures continuous dependence on the parameter \(y\) and naturally reduces to the familiar value \((-1)^1 = -1\). In other words, choosing the exponential form \(e^{i\pi y}\) guarantees that the function \(\jk^{y}\) remains continuous and analytically defined for all real \(y\), without jumps at odd values.

Expanding the exponential into sine and cosine, we can write \[\tag{6} e^{i\pi y} = \cos(\pi y) + i \sin(\pi y), \] substituting which into (5) gives: \[\tag{7} \jk^{y} = \frac12 \left( 1 + \cos\pi y + i \sin\pi y \right) + \frac{j}{2} \left( 1 - \cos\pi y - i \sin\pi y \right) \]

Everything matches the base rules \(\jk^0=1\), \(\jk^1=j\), \(\jk^2=+1\).

Thus, expression (7) is a four-dimensional generalization of Euler’s formula, and \(\jk\) plays the role of a “second orthogonal direction” in hypercomplex space.

This notation reflects the “two-point” nature of the spectrum of \(\jk\): part corresponding to eigenvalue \(+1\) remains unchanged, while the part corresponding to \(-1\) is multiplied by the phase factor \((-1)^y\).

It should also be noted that without the imaginary unit such a decomposition (9) is impossible.

Let us show that for all real y the radius vector \(\jk^y\) in \(ℝ^4\) equals 1.

Sum the squares of the coordinates: \[\tag{10} a^2 + b^2 + c^2 + d^2 = \frac14 \left[ (1 + \cos \pi y)^2 + (\sin \pi y)^2 + (1 - \cos \pi y)^2 + (\sin \pi y)^2 \right] = 1 \] Thus, the norm of \(\jk^y\) equals one for any \(y\), which corresponds to a circle of unit radius.

All points \(\jk^y\) lie on the 4D unit sphere \(S^3\). Since linear dependencies \((a + c = 1, b + d = 0)\) hold, the set \(\jk^y\) forms a circle — the intersection of the sphere \(S^3\) with a two-dimensional plane.

In Figure 1, the motion of a point along the circle representing the change in y is shown. It is impossible to display the full 4D model on a monitor, so it is presented here in the first three planes: \(a, b, c\).

A certain similarity can be noticed between formula (9) and the classical Euler formula (6). At the same time, expression (9) extends the familiar two-dimensional exponential into four-dimensional space, whereas Euler’s formula describes rotation only in a plane. Nevertheless, such a two-dimensional form has long been successfully used to describe wave processes.

We propose to consider a generalization in which the exponential \(e^{i\omega t}\) is replaced by the hypercomplex power \(\jk^{\varpi t}\). Such a transition naturally transfers the wave equation from the planar model to a four-dimensional, more symmetric form. The corresponding transformation is shown in the following formula:

The reverse wave is usually represented with a minus sign in the exponent. In our case, one can proceed in the same way. In the expansion into cosines and sines, only the sign of the odd functions changes: \[\tag{12} \jk^{-\varpi t} = \frac12 \left[ (1 + \cos\wt) - i \sin\wt + \jk (1 - \cos\wt) + i \jk \sin\wt \right] \]

From a geometric point of view, the rotation defined by the exponential \(e^{i\omega t}\) describes the motion of a point along a circle in the two-dimensional plane of complex numbers. In turn, the expression \(j^{\varpi t}\) defines a similar rotation but in four-dimensional space, where the trajectory of the point is a circle “embedded” in the unit sphere \(S^3\). Such an extension gives the wave equation a more general geometric form and reflects the symmetry of multidimensional oscillatory processes.

This work shows that the hypercomplex unit \(\jk\), possessing the property \(\jk^2 = +1\), can be raised to an arbitrary real power and thereby describe rotation in four-dimensional space. The resulting expression \(\jk^y\) forms a closed trajectory on the unit four-dimensional sphere, which geometrically corresponds to a circle — analogous to that described in the two-dimensional case by the complex exponential \(e^{i\omega t}\). Such a generalization makes it possible to naturally extend the familiar planar wave model into a higher-dimensional space.

The proposed replacement of the Euler exponential with the hypercomplex power \(\jk^{\varpi t}\) allows describing wave processes not only as oscillations in a plane, but also as rotations in a four-dimensional medium, where each component reflects the interrelated parameters of amplitude, phase, and an additional hypercomplex coordinate. This opens the way to a more general and symmetric representation of wave equations and can serve as a basis for constructing an extended algebra of space-time oscillations.