2026-07-24
From Euler's formula to split geometry
How Complex Rotation Reveals the Geometry of the Hyperbolic Unit
The imaginary unit \(i\) is naturally related to rotation in the complex plane. For this unit, as is well known, \(i^2=-1\). But what geometric meaning does the hyperbolic unit \(\j\) have, for which \(\j^2=+1\)? It turns out that it can be viewed not as another variety of imaginary unit, but as an operator rotating only one of the two independent components of space.
The goal of this paper is to show that the hyperbolic unit has a simple and intuitive geometry hidden behind its familiar algebraic notation. It turns out that its powers can be interpreted as independent rotations of two orthogonal complex planes. This representation leads to a new geometric model in which the internal state and external motion are described by a single mathematical object.
1. From the Known to the Unknown
Euler's formula defines a continuous rotation by an angle \(\alpha\):
\[\tag{1} e^{i\alpha}=\cos\alpha+i\sin\alpha. \] When \(\alpha=\pi/2\), an imaginary unit \(i\) appears, and a full rotation returns the object to its original state:
\[\tag{2} e^{i\pi/2}=i, \qquad e^{i2\pi}=1. \] For the hyperbolic unit, a different law holds:
\[\tag{3} \j^2=1, \qquad \j \ne\pm1. \] For integer powers, only alternation of two states occurs:
\[\tag{4} \j ^0=1, \qquad \j ^1=\j , \qquad \j^2=1, \qquad \j ^3=\j , \ldots \] However, a natural question arises: is it possible to fill the gap between \(1\) and \(\j \) and define a continuous power \(\j ^a\)?
2. Two components hidden within the hyperbolic unit
The hyperbolic unit itself naturally generates two idempotents [1-2]:
\[\tag{5} \ep =\frac{1+\j }{2}, \qquad \em =\frac{1-\j }{2}. \] These definitions directly imply their basic properties:
\[\tag{6} \ep ^2=\ep , \qquad \em ^{\,2}=\em , \qquad \ep \em =0, \qquad \ep +\em =1. \] The unit and the hyperbolic unit have almost identical expansions:
\[\tag{7} 1=\ep +\em , \qquad \j =\ep -\em . \] The difference lies only in the sign of the second component. But the sign change can be represented as a simple complex rotation by an angle \(\pi\):
\[\tag{8} -\em = \em e^{i\pi}. \] Therefore, the hyperbolic unit can be written in the form
\[\tag{9} \boxed{ \j = \ep +\em e^{i\pi} }. \] You can read more about this here.
3. The Geometric Meaning of the Hyperbolic Unit
Formula (9) allows us to see the action of \(\j \) geometrically. When going from \(1\) to \(\j \), the \(\ep \) component remains unchanged, and the \(\em \) component rotates by \(180^\circ\):
\[\tag{10} \jmath:\qquad \begin{cases} \mathfrak{e} \longrightarrow \mathfrak{e},\\ \bar{\mathfrak{e}} \longrightarrow \bar{\mathfrak{e}}e^{i\pi} = -\bar{\mathfrak{e}}. \end{cases} \] Thus, the hyperbolic unit acts as a selective rotation operator: it rotates one idempotent component by \(180^\circ\), leaving the other unchanged.
In this reading, the equality \(\j^2=1\) gets a clear explanation. Two consecutive rotations by \(180^\circ\) yield a complete rotation:
\[ \tag{11} \bar{\mathfrak{e}} \overset{\jmath}{\longrightarrow} -\bar{\mathfrak{e}} \overset{\jmath}{\longrightarrow} \bar{\mathfrak{e}}. \] Therefore, the property \(\j^2=1\) can be viewed as the result of a complete complex rotation of one of the two components.
4. Continuous Extension and the Occurrence of the Power \(\j^a\)
If the operator \(\j \) rotates the component \(\em \) by an angle \(\pi\), then its continuous extension must rotate it by an arbitrary angle \(\pi a\). Therefore, it is natural to define
\[\tag{12} \boxed{ \j ^a = \ep +\em e^{i\pi a} }. \] For \(a=0\), \(a=1\), and \(a=2\), this formula returns the initial states:
\[\tag{13} \j ^0 = \ep +\em =1, \] \[\tag{14} \j ^1 = \ep +\em e^{i\pi} = \ep -\em = \j , \] \[\tag{15} \j^2 = \ep +\em e^{i2\pi} =1. \] The notation \(\j ^a\) is also justified by the law of addition of exponents:
\[\tag{16} \begin{aligned} \jmath^{a}\jmath^{c} &= \left( \mathfrak{e} + \bar{\mathfrak{e}}e^{i\pi a} \right) \left( \mathfrak{e} + \bar{\mathfrak{e}}e^{i\pi c} \right) \\ &= \mathfrak{e} + \bar{\mathfrak{e}}e^{i\pi(a+c)} \\ &= \jmath^{a+c}. \end{aligned} \] Therefore, \(\j ^a\) represents the rotation operator of the component \(\em \) by the angle \(180^\circ a\), while the component \(\ep \) remains fixed.
5. The second operator: \((-\j)^b\)
A completely symmetric construction arises for \(-\j \):
\[\tag{17} -\j = -\ep +\em = \ep e^{i\pi}+\em . \] Now the \(\ep \) component rotates, while the \(\em \) component remains stationary. Therefore,
\[\tag{18} \boxed{ (-\j)^b = \ep e^{i\pi b}+\em }. \] Similar relations hold for this operator:
\[\tag{19} (-\j)^0=1, \qquad (-\j)^1=-\j , \qquad (-\j)^2=1, \] \[\tag{20} (-\j)^b(-\j)^d = (-\j)^{b+d}. \] So, \(\j \) rotates the \(\em \) component by \(180^\circ\), while \(-\j \) rotates the \(\ep \) component by the same angle. Their fractional powers define the corresponding partial rotations.
6. Two independent rotations
Since the idempotent components are orthogonal,
\[\tag{21} \ep \em =0, \] Two selective rotations can be combined into a single product:
\[\tag{22} \begin{aligned} \jmath^{a}(-\jmath)^{b} &= \left( \mathfrak{e} + \bar{\mathfrak{e}}e^{i\pi a} \right) \left( \mathfrak{e}e^{i\pi b} + \bar{\mathfrak{e}} \right) \\ &= \mathfrak{e}^{2}e^{i\pi b} + \bar{\mathfrak{e}}^{2}e^{i\pi a} \\ &= \mathfrak{e}e^{i\pi b} + \bar{\mathfrak{e}}e^{i\pi a}. \end{aligned} \] Thereby we obtain the main relation:
\[\tag{23} \boxed{ \j ^a(-\j)^b = \ep e^{i\pi b} + \em e^{i\pi a} }. \] One mathematical object contains two independent phase motions. Parameter \(a\) controls the rotation of the \(\em\) component, and parameter \(b\) controls the rotation of the \(\ep\) component.
7. Relationship with Exponential Form
For an idempotent factorization, the following rule holds
\[\tag{24} f\!\left( \ep x+\em y \right) = \ep f(x)+\em f(y). \] Applying it to the exponent, we get
\[\tag{25} e^{\,i\left( \ep \pi b+\em \pi a \right)} = \ep e^{i\pi b} + \em e^{i\pi a}. \] Hence,
\[\tag{26} \boxed{ e^{\,i\left( \ep \pi b+\em \pi a \right)} = \j ^a(-\j)^b }. \] Conclusion
The hyperbolic unit \(\jmath\) contains two natural idempotent directions. Its decomposition \(\jmath=\mathfrak{e}-\bar{\mathfrak{e}}\) shows that the transition from \(1\) to \(\jmath\) is equivalent to rotating only one of the two independent components of the space by \(180^\circ\).
Continuously continuing this operation leads to the operator \(\jmath^a\), which rotates the component \(\bar{\mathfrak{e}}\) by an angle of \(180^\circ a\). Similarly, the operator \((-\jmath)^b\) rotates the component \(\mathfrak{e}\) by an angle of \(180^\circ b\).
Their product combines two independent selective rotations:
\[\tag{27} \jmath^{a}(-\jmath)^{b} = \mathfrak{e}e^{i\pi b} + \bar{\mathfrak{e}}e^{i\pi a}. \]
The following formula shows that the classical Euler formula is a special case of the proposed split geometry. When the phases of both subspaces coincide (\(a=b\)), the two independent rotations merge into one, and the operator \(\jmath^{a}(-\jmath)^{b}\) becomes the usual complex exponential \(e^{i\pi a}\). This is where split geometry gets its name: it splits the Euler rotation into two independent complex subspaces, each with its own phase.
\[\tag{28} \jmath^{a}(-\jmath)^{a} = e^{i\pi a}, \quad \text{if}\,\, a=b. \] Thus, the hyperbolic unit acquires a natural geometric meaning: it acts not as an analog of the imaginary unit, but as an operator of selective rotation of two independent idempotent components of space.
This property explains the name of the proposed split geometry. If the classical Euler formula describes rotation in one complex plane, then the operator \(\jmath^{a}(-\jmath)^{b}\) naturally splits it into two independent complex subspaces corresponding to the idempotents \(\mathfrak{e}\) and \(\bar{\mathfrak{e}}\). Each of these subspaces has its own phase and rotation, forming a single geometric structure.
In the next section, it will be shown that such a splitting allows us to naturally isolate the real and imaginary components of the operator and introduce the split-cosine and split-sine functions, which are directlya generalization of classical trigonometric functions to two independent complex subspaces.
Materials used
- Wikipedia. idempotent.
- Wikipedia. Idempotent (ring theory).