The paper proposes a new view on the geometry of complex space, based on a Cartesian basis of two independent complex planes, the derivation of which is presented here. This approach allows for a natural construction of a four-dimensional spacetime in which the time coordinate and three spatial coordinates arise as components of a single algebraic structure. On this basis, a velocity vector is introduced for a material point, describing its internal wave motion.

Integrating the velocity vector leads to a helical trajectory of motion and yields the characteristic radius of internal rotation \(r=\lambda/2\pi\). The resulting expression is completely consistent with the previously derived formula for the electron radius, establishing a connection between the new Cartesian basis and the wave model of an elementary particle. Moreover, this construction serves as a starting point for the subsequent derivation of the Lorentz transformations and the origin of the Lorentz factor.

Of particular interest is that the proposed construction combines within a single model both the complex imaginary unit \(i\) and the hyperbolic unit \(\j\), which opens the possibility of jointly using both algebraic structures in describing physical processes.

Complex numbers are traditionally considered as elements of a two-dimensional space with a Cartesian basis \(\{1,i\}\), where \(i^2=-1\). This representation is natural for most problems in analysis and geometry, but it is not the only possible one. This paper proposes an alternative approach based on representing space as the direct sum of two independent complex planes. Each plane has its own real and imaginary axes, while both share a common imaginary unit \(i\). To describe these planes, we introduce two mutually orthogonal basis elements \(\ep\) and \(\em\), dividing the space into two independent complex components. As a result, a natural Cartesian basis arises. \[ \tag{1} \{\ep,i\ep,\em,i\em\}. \] Any element of the new space is represented as the sum of two independent complex numbers. \[ Z=(a+ib)\ep+(c+id)\em, \] where: \(a,b,c,d\in\mathbb R\).

With this construction, hyperbolic unit \[ \tag{2} \j = \ep - \em \] turns out not to be the original object, but a derived element, naturally arising as the difference of two basis elements.

The proposed basis is of interest not only as an independent algebraic construct, but also as a means of describing a more general model of Unit Space. This model postulates that the motion of any material point is always one-dimensional and is described by only two coordinates: the time coordinate and the space coordinate. Consequently, the observed multidimensionality of the world should arise not by increasing the number of independent directions of motion, but by unifying several one-dimensional spaces into a single algebraic structure.

This paper shows that such a structure is naturally formed by two independent complex planes, each corresponding to its own one-dimensional space. Their unification leads to the emergence of a four-dimensional Cartesian basis (1), which is further considered as the mathematical basis for constructing a single four-dimensional space.

Furthermore, the proposed Cartesian basis is directly related to the results obtained in this article, where it was shown that complex and hyperbolic algebras can be considered within a single mathematical construct.

Since the elements \(\ep\) and \(\em\) describe two independent complex planes, they must satisfy the following natural conditions.

The first two equalities mean that repeated projections onto the same complex plane do not change the result. The third expresses the independence of both planes, and the last shows that their sum forms a unit element of space. Consequently, any element of the new space can be uniquely factored with respect to basis (1).

It should be noted that the elements \(\ep\) and \(\em\) are not new mathematical objects. They are well-known idempotents [1-2], widely used in various branches of algebra to decompose space into independent components. In the literature, they are usually denoted by the symbols \(e_+\) and \(e_-\) and are defined as \[\tag{4} \ep\equiv e_+=\frac{1+\j}{2}, \qquad \em\equiv e_-=\frac{1-\j}{2}. \] In this paper, while preserving their mathematical meaning, we use the equivalent notations \(\ep\) and \(\em\), which make the mathematical expressions more compact and clear. The novelty of the proposed approach lies not in the introduction of new algebraic elements, but in their interpretation as a Cartesian basis of two independent complex planes, which allows us to construct a unified four-dimensional coordinate system.

Another important property of the new basis is the existence of a natural norm. Consider an arbitrary element \[\tag{5} J= \ep e^{ig} + \em e^{ih}, \] where \(g\) and \(h\) are independent phase angles of the two complex planes.

Complex conjugation has the form \[\tag{6} J^*= \ep e^{-ig} + \em e^{-ih}. \] Then the modulus is determined by the product \[ |J|^2 = JJ^*. \] Using the idempotent properties (3) \[\tag{7} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \] we obtain \[\tag{8} |J|^2 = \ep + \em = 1. \] Therefore, \[\tag{9} |J|=1. \]

Thus, regardless of the values ​​of the phases \(g\) and \(h\), the element \(J\) always belongs to the unit manifold of the new space. This property is a direct analog of the unit circle in the complex plane, but now it is simultaneously satisfied for two independent complex components. In the future, the preservation of this norm will play a key role in constructing motion vectors and studying their geometric properties. For example, the law of conservation of energy in a closed system will directly follow from it.

The introduced basis admits a natural geometric interpretation. We will consider the idempotent \(ep\) as the direction of time, and the three remaining basis components as orthogonal spatial coordinates. Then the following correspondence is established between the elements of the basis and the space-time coordinates: \[\tag{10} \begin{aligned} \ep &\longleftrightarrow ct,\\ i\ep &\longleftrightarrow x,\\ \em &\longleftrightarrow y,\\ i\em &\longleftrightarrow z. \end{aligned} \] Therefore, the basis \[ \{\ep ,\;i\ep ,\;\em,\;i\em\} \] acquires the meaning of a four-dimensional space \(\{ct,x,y,z\}\), in which the time coordinate is not a separate quantity, but one of the components of a single algebraic structure.

Based on this representation, we introduce the velocity vector \[\tag{11} V(t)=c\left(\ep +\em e^{i\omega t}\right), \] where \(c\) is the speed of light, \(\omega\) is the angular frequency of the internal motion, and \(t\) is time.

Using Euler's formula, \[ e^{i\omega t}=\cos(\omega t)+i\sin(\omega t), \] we obtain \[\tag{12} V(t) = c \left(\ep + \cos(\omega t)\,\em + \sin(\omega t)\,i\em\right) \] Therefore, in coordinate representation, this vector has the form \[\tag{13} V(t)= \begin{pmatrix} c\\ 0\\ c\cos(\omega t)\\ c\sin(\omega t) \end{pmatrix}, \] which corresponds to motion in space-time \[\tag{14} (ct,x,y,z). \] This is shown in the following figure.

Thus, in the model under consideration, a point mass moves continuously along the time coordinate at the speed of light, simultaneously executing an internal circular motion in the \((y,z)\) plane. The \(x\) coordinate remains equal to zero and can arise when transitioning to another reference frame. The article "Origin of the Lorentz Factor" shows that it is precisely this geometric interpretation that leads to the natural emergence of the Lorentz transformations and the corresponding Lorentz factor.

Since the quantity \(V(t)\) defines the motion vector, the corresponding displacement vector can be obtained by integrating over time: \[\tag{15} L(t)=\int V(t)\,dt. \] Substituting the expression for \(V(t)\), we have \[\tag{16} L(t) = c \int \left(\ep +\em e^{i\omega t}\right)dt. \] Hence \[\tag{17} L(t) = ct\,\ep + {c\over i\omega}\em e^{i\omega t} + C. \]

Since \(1/i=-i\), then choosing a zero constant of integration, we obtain \[\tag{18} L(t) = ct\,\ep - {c\over \omega}e^{i\omega t}\, i\em. \] Expanding the complex exponential, we find \[\tag{19} L(t) = ct\,\ep + {c\over \omega}\sin(\omega t)\, \em - {c\over \omega}\cos(\omega t)\, i \em. \]

Therefore, in space-time coordinates \((ct,x,y,z)\), this displacement vector has the form \[\tag{20} L(t)= \begin{pmatrix} ct\\ 0\\ {c\over \omega}\sin(\omega t)\\ -{c\over \omega}\cos(\omega t) \end{pmatrix}. \] This means that the displacement has a uniform time component \(ct\), while the spatial component describes a circle of radius \(c/\omega\) in the plane \((y,z)\). Let's look at this radius in more detail.

The resulting expression for the radius of the spatial part of the motion has an important physical interpretation. Since for the wave process \[\tag{21} \lambda={2\pi c\over \omega}, \] then the quantity \(c/\omega\) can be written as \[\tag{22} {c\over \omega}={\lambda\over 2\pi}. \] Therefore, the radius of the internal circular motion is \[\tag{23} r={\lambda\over 2\pi}. \]

Thus, the new Cartesian interpretation of the basis leads not only to a helical trajectory in spacetime, but also to the natural emergence of a characteristic radius. \[\tag{25} r_e=r={\lambda\over 2\pi}. \] Then the displacement vector can be rewritten as follows: \[\tag{26} L(t)= \begin{pmatrix} ct\\ 0\\ r\sin(\omega t)\\ -r\cos(\omega t) \end{pmatrix}. \]

This result is of particular interest, since the radius of the electron arises directly from the geometry of motion in the new Cartesian basis. This eliminates the need to introduce additional assumptions about the particle's internal structure: the radius is determined by integrating the velocity vector and the wave relationship between wavelength and frequency.

Thus, the new Cartesian interpretation of the basis connects the geometry of the internal motion of a particle with the previously obtained wave model of the electron radius and serves as a natural transition to the further derivation of the Lorentz transformations.

In standard physics, rotation in the \((ct,x)\) plane is not simply a rotation, but a Lorentz transformation (a boost). It is hyperbolic, not trigonometric. However, in the context of this work, where \(ct\) is considered a fully-fledged spatial coordinate in the new basis, such a Euclidean rotation becomes an acceptable geometric transformation. This fundamentally distinguishes this approach from classical relativity.