2026-07-25
Geometric origin of the square of the fine structure constant and the parameters of the Bohr atom
This work is part of a more general concept, which views mathematics as the fundamental basis for constructing observable reality. Within this perspective, physical laws are not primary, but rather a manifestation of deeper mathematical structures.
Physics, chemistry, biology, and other natural sciences describe only one possible way in which these structures are realized in the observable Universe. Therefore, the search for new mathematical objects and patterns can lead not only to the expansion of existing physical models, but also to a more general understanding of the principles underlying the construction of reality.
This approach is consistent with the Copenhagen interpretation of quantum mechanics, according to which physical theory describes the observed results of measurements and is not required to directly reflect the "structure of the world." In the proposed concept, mathematical structure is considered primary, while physical objects and laws are one possible form of its manifestation. At the same time, the possibility of the existence of other mathematically consistent structures, which could correspond to different physical laws, is not excluded.
Introduction
In previous work, it was shown that the fine-structure constant can be obtained as a consequence of the geometry of the particle's internal rotation. Within the framework of the proposed model, the internal state is described by the operator \[ J(a)=\j^a, \] and the corresponding projection coefficient takes the form \[ \frac{1}{\gamma} = \alpha_{\mathrm{fs}}. \] Thus, the fine-structure constant appears not only as an experimentally determined constant, but also as a quantity arising directly from the geometric model of the internal motion.
The mathematical basis for the further generalization is the previously introduced split geometry. In it, a single complex rotation is split into two independent rotations in orthogonal idempotent subspaces corresponding to the components \(\ep\) and \(\em\). These rotations are described by the operators \(\j^a\) and \((-\j)^b\), and their product forms the composite split operator \[ J(a,b) = \j^a(-\j)^b = \ep e^{i\pi b} + \em e^{i\pi a}. \] The parameter \(a\) characterizes the internal state of the particle, while the parameter \(b\) describes its external motion. Thus, split geometry allows us to represent the internal and external dynamics as two independent components of a single complete state.
A natural continuation of this construction is the question: what projection coefficient results from the combination of two independent orthogonal motions? In this paper, we consider the complete state \[ J(a,b) = \j^a(-\j)^b. \] At the level of split geometry, it is a composite of two rotations occurring in orthogonal idempotent planes. To determine the relativistic characteristics of this composite state, the corresponding velocities are combined according to the rule of relativistic addition of mutually perpendicular velocities.
It will be shown that such a composition leads to a square Lorentz factor \(\gamma_{\mathrm H}\), whose reciprocal is equal to the square of the fine structure constant: \[ \frac{1}{\gamma_{\mathrm H}} = \alpha_{\mathrm{fs}}^2. \] Thus, the quantity \(\alpha_{\mathrm{fs}}^2\) arises as the relativistic projection coefficient of the composite geometric state formed by internal and external orthogonal motions. From this quantity, the main parameters of the hydrogen atom are then sequentially reconstructed: the velocity of the electron in the first Bohr orbit, the Bohr radius, the angular momentum, and the ground state energy.
Square Lorentz Factor
Fine structure constant \[ \alpha_{\mathrm{fs}}\approx\frac{1}{137} \] is included in almost all the fundamental parameters of the hydrogen atom. In the Bohr model, the electron velocity is proportional to \(\alpha_{\mathrm{fs}}\), the ground state energy is \(\alpha_{\mathrm{fs}}^2\), and the Bohr radius is inversely proportional to \(\alpha_{\mathrm{fs}}\).
In standard theory, these relations follow from the combined use of the Coulomb interaction, angular momentum quantization, and the Schrödinger equation. In this paper, we consider a different approach: the atomic state is constructed as a combination of two independent rotations in an extended basis.
It is assumed that the internal and external motions have the same initial relativistic scale and belong to mutually orthogonal planes of the state. Applying the relativistic addition of the corresponding mutually perpendicular velocities leads tocomposite coefficient
\[ \tag{1} \frac{1}{\gamma_{\mathrm H}} = \alpha_{\mathrm{fs}}^2. \] Here \(\gamma_{\mathrm H}\) is the square Lorentz factor of the composite state described by the operator \[ J(a,b)=\j^a(-\j)^b. \]
It will be shown below that the classical parameters of the first Bohr orbit are consistently reconstructed from this value:
\[ \tag{2} v_1=\alpha_{\mathrm{fs}}c, \] \[ \tag{3} a_0=\frac{\hbar}{m_ec\alpha_{\mathrm{fs}}}, \] \[ \tag{4} m_ev_1a_0=\hbar, \] \[ \tag{5} E_1=-\frac12m_ec^2\alpha_{\mathrm{fs}}^2. \] 1. Two Levels of Motion
The model is based on the state operator
\[ \tag{6} J(t)=\j^a(-\j)^b, \] where parameter \(a\) describes the internal state of the particle, and parameter \(b\) describes its external motion:
\[ \tag{7} a=\varpi t, \qquad \varpi\pi=\omega, \] \[ \tag{8} b=\frac{\arcsin\beta}{\pi}, \qquad \beta=\frac{v}{c}. \] The powers of the hyperbolic unit are expanded into two idempotent components:
\[ \tag{9} \j^a = \ep+\em e^{i\pi a}, \] \[ \tag{10} (-\j)^b = \ep e^{i\pi b}+\em. \] Therefore, the complete operator takes the form
\[ \begin{aligned} J(a,b) &= \left( \ep+\em e^{i\pi a} \right) \left( \ep e^{i\pi b}+\em \right) \\[2mm] &= \ep e^{i\pi b} + \em e^{i\pi a}. \end{aligned} \tag{11} \] Thus, the two components of the total state rotate independently:
\[ \tag{12} J(a,b) = \ep e^{i\phi_{\mathrm{ext}}} + \em e^{i\phi_{\mathrm{int}}}, \] where
\[ \tag{13} \phi_{\mathrm{ext}}=\pi b, \qquad \phi_{\mathrm{int}}=\pi a. \] Geometrically, this state corresponds to two rotations in orthogonal complex planes. For constant moduli of both components, the trajectory of the complete state belongs to a toroidal geometry.
2. Internal Motion of a Free Particle
In the previous work, the Lorentz factor was obtained directly from the geometry of the internal state of the particle. Therefore, in this article, it is not introduced as an independent relativistic hypothesis, but is considered as an already established characteristic of the internal motion operator.
\[ J_a=\j^a. \] For a free particle, a correspondence between the geometric projection coefficient and the fine structure constant was previously obtained:
\[ \tag{14} \frac{1}{\gamma_a} = \alpha_{\mathrm{fs}} = \sqrt{1-\beta_a^2}. \] Here \(\gamma_a\) is the geometrically obtained Lorentz factor of the internal state.
Using the standard relation, we find the dimensionless velocity of the internal motion:
\[ \tag{15} \alpha_{\mathrm{fs}} = \sqrt{1-\beta_a^2}, \] Therefore,
\[ \boxed{ \beta_a = \sqrt{1-\alpha_{\mathrm{fs}}^2} } \tag{16} \] and the corresponding velocity is
\[ \tag{17} v_a = c\sqrt{1-\alpha_{\mathrm{fs}}^2}. \] Thus, the near-light speed of internal motion is not specified separately, but follows from the previously obtained geometric relationship.
Within the model under consideration, the quantity \(\alpha_{\mathrm{fs}}\) is interpreted as the projection coefficient of the internal near-light motion onto the observed massive state of the electron.
It is important to emphasize that the velocity \(v_a\) is not the velocity of the electron along its Bohr orbit. It characterizes the internal motion of a free particle in the extended state space. The observed orbital velocity arises later as a relative projection of the composite internal and external states.
3. Inclusion of External Closed Motion
When an atomic bound state is formed, external motion is added to the internal rotation
\[ \tag{18} \j^a \] external motion is added.
\[ \tag{19} (-\j)^b. \] Assume that the external motion has the same initial relativistic scale:
\[ \tag{20} \beta_a=\beta_b=\sqrt{1-\alpha_{\mathrm{fs}}^2}, \] \[ \tag{21} \gamma_a=\gamma_b=\frac{1}{\alpha_{\mathrm{fs}}}. \] The internal and external motions belong to different planes of state. Therefore, a sequential composition of mutually orthogonal velocities is considered.
4. Relativistic Addition of Orthogonal Velocities
BeforeTo move on to mathematical transformations, let's consider the geometric meaning of the proposed model. Unlike the classical approach, where the electron's motion is described by a single velocity, this paper assumes the existence of two mutually perpendicular relativistic motions: internal rotation, associated with the particle's intrinsic structure, and external orbital motion around the nucleus. Their combined effect leads to the appearance of a quadratic Lorentz factor, which is subsequently used to derive the main parameters of the Bohr atom.
Let the first velocity be directed along one coordinate axis:
\[ \tag{22} \beta_x=\beta. \] The second velocity in its own reference frame is directed perpendicular to the first and has the same magnitude \(\beta\). When returning to the original system, its transverse component decreases by a factor of \(\gamma\):
\[ \tag{23} \beta_y=\frac{\beta}{\gamma}. \] Since
\[ \tag{24} \frac{1}{\gamma}=\alpha_{\mathrm{fs}}, \] we obtain
\[ \tag{25} \beta_y=\alpha_{\mathrm{fs}}\beta. \] The square of the magnitude of the resulting velocity is
\[ \begin{aligned} \beta_{\mathrm H}^2 &= \beta_x^2+\beta_y^2 \\[1mm] &= \beta^2+\alpha_{\mathrm{fs}}^2\beta^2 \\[1mm] &= \beta^2 \left( 1+\alpha_{\mathrm{fs}}^2 \right). \end{aligned} \tag{26} \] Substituting
\[ \tag{27} \beta^2=1-\alpha_{\mathrm{fs}}^2, \] we find
\[ \begin{aligned} \beta_{\mathrm H}^2 &= \left( 1-\alpha_{\mathrm{fs}}^2 \right) \left( 1+\alpha_{\mathrm{fs}}^2 \right) \\[1mm] &= 1-\alpha_{\mathrm{fs}}^4. \end{aligned} \tag{28} \] Hence,
\[ \tag{29} 1-\beta_{\mathrm H}^2 = \alpha_{\mathrm{fs}}^4. \] The reciprocal of the resulting Lorentz factor is
\[ \begin{aligned} \frac{1}{\gamma_{\mathrm H}} &= \sqrt{1-\beta_{\mathrm H}^2} \\[1mm] &= \sqrt{\alpha_{\mathrm{fs}}^4} \\[1mm] &= \alpha_{\mathrm{fs}}^2. \end{aligned} \tag{30} \] Thus,
\[ \boxed{ \frac{1}{\gamma_{\mathrm H}} = \alpha_{\mathrm{fs}}^2 } \tag{31} \] and the square of the fine structure constant arises as a result of the composition of two identical, mutually orthogonal relativistic motions.
5. Equivalent Derivation via Lorentz Factors
The same result can be obtained directly by multiplying the Lorentz factors of successive orthogonal transformations:
\[ \tag{32} \gamma_{\mathrm H} = \gamma_a\gamma_b. \] Because
\[ \tag{33} \gamma_a=\gamma_b=\frac{1}{\alpha_{\mathrm{fs}}}, \] That
\[ \tag{34} \gamma_{\mathrm H} = \frac{1}{\alpha_{\mathrm{fs}}^2}. \] From this it follows again
\[ \boxed{ \frac{1}{\gamma_{\mathrm H}} = \frac{1}{\gamma_a} \frac{1}{\gamma_b} = \alpha_{\mathrm{fs}}^2. } \tag{35} \] In operator form, this corresponds to the transition
\[ \tag{36} \j^a \quad\longrightarrow\quad \j^a(-\j)^b. \] One operator creates the coefficient \(\alpha_{\mathrm{fs}}\), and the combined action of two orthogonal operators creates the coefficient \(\alpha_{\mathrm{fs}}^2\).
6. The Complete State and Its Observable Projection
The quantity
\[ \tag{37} \beta_{\mathrm H} = \sqrt{1-\alpha_{\mathrm{fs}}^4} \] is very close to unity. However, it cannot be directly calculated as the velocity of an electron along a circle of radius \(a_0\).
It characterizes the complete motion in the extended state space, containing two orthogonal components:
\[ \tag{38} J_{\mathrm H} = \j^a(-\j)^b. \] The observed motion in ordinary space must be determined by the projection of the complete state onto the external component. Therefore, a distinction is made between:
\[ \tag{39} \text{complete near-wave state} \] and
\[ \tag{40} \text{observed motion of a massive electron}. \] The electron's mass does not disappear. It is determined by the first projection of the internal state:
\[ \tag{41} m_ec^2 = \alpha_{\mathrm{fs}}E_{\mathrm{int}}. \] The additional external rotation changes the overall geometry of the bound state, but the observed projection still has mass \(m_e\).
7. Observed Orbital Velocity
For the internal state, the projection coefficient is
\[ \tag{42} \alpha_{\mathrm{int}} = \frac{1}{\gamma_a} = \alpha_{\mathrm{fs}}. \] For the complete atomic state:
\[ \tag{43} \alpha_{\mathrm H} = \frac{1}{\gamma_{\mathrm H}} = \alpha_{\mathrm{fs}}^2. \] The relative coefficient of the external projection is determined by the ratio of the total coefficient to the internal coefficient:
\[ \begin{aligned} \beta_{\mathrm{orb}} &= \frac{\alpha_{\mathrm H}} {\alpha_{\mathrm{int}}} \\[1mm] &= \frac{\alpha_{\mathrm{fs}}^2} {\alpha_{\mathrm{fs}}} \\[1mm] &= \alpha_{\mathrm{fs}}. \end{aligned} \tag{44} \] Therefore, the observed velocity of the external motion is
\[ \boxed{ v_{\mathrm{orb}} = \alpha_{\mathrm{fs}}c. } \tag{45} \] This coincides with the velocity of an electron in the first orbit in the Bohr model.
This resolves the apparent contradiction. The total state is almost light-like and almost wave-like:
\[ \tag{46} \beta_{\mathrm H}\approx1, \] but its observable external projection is nonrelativistic:
\[ \tag{47} \beta_{\mathrm{orb}} = \alpha_{\mathrm{fs}} \ll1. \] 8. The Classical Radius of the Electron
Previously, a spatial scale was obtained for the internal motion
\[ \tag{48} r_e = \alpha_{\mathrm{fs}} \frac{\hbar}{m_ec}. \] This expression coincides with the classical radius of the electron:
\[ \tag{49} r_e = \frac{e^2}{4\pi\varepsilon_0m_ec^2}. \] Indeed, from the definition of the fine structure constant
\[ \tag{50} \alpha_{\mathrm{fs}} = \frac{e^2} {4\pi\varepsilon_0\hbar c} \] it follows
\[ \begin{aligned} \alpha_{\mathrm{fs}} \frac{\hbar}{m_ec} &= \frac{e^2} {4\pi\varepsilon_0\hbar c} \frac{\hbar}{m_ec} \\[1mm] &= \frac{e^2} {4\pi\varepsilon_0m_ec^2}. \end{aligned} \tag{51} \] 9. Obtaining the Bohr radius
When moving from a single internal rotation to a combined internal and external state, the coefficient changes from
\[ \tag{52} \alpha_{\mathrm{fs}} \] to
\[ \tag{53} \alpha_{\mathrm{fs}}^2. \] The corresponding external spatial scale is defined by the expression
\[ \tag{54} R_{\mathrm H} = \frac{r_e}{\alpha_{\mathrm{fs}}^2}. \] Substituting formula (48), we obtain
\[ \begin{aligned} R_{\mathrm H} &= \frac{ \alpha_{\mathrm{fs}} \hbar }{ m_ec\alpha_{\mathrm{fs}}^2 } \\[1mm] &= \frac{\hbar} {m_ec\alpha_{\mathrm{fs}}}. \end{aligned} \tag{55} \] But
\[ \tag{56} a_0 = \frac{\hbar} {m_ec\alpha_{\mathrm{fs}}} \] is the Bohr radius. Hence,
\[ \boxed{ R_{\mathrm H}=a_0. } \tag{57} \] The relationship between the three characteristic scales takes the form
\[ \tag{58} r_e = \alpha_{\mathrm{fs}} \bar\lambda_C, \] \[ \tag{59} a_0 = \frac{\bar\lambda_C} {\alpha_{\mathrm{fs}}}, \] where
\[ \tag{60} \bar\lambda_C = \frac{\hbar}{m_ec} \] is the reduced Compton wavelength of the electron.
That's why
\[ \boxed{ a_0 = \frac{r_e}{\alpha_{\mathrm{fs}}^2}. } \tag{61} \] 10. Angular Momentum
Using the obtained values
\[ \tag{62} v_{\mathrm{orb}} = \alpha_{\mathrm{fs}}c \] and
\[ \tag{63} a_0 = \frac{\hbar} {m_ec\alpha_{\mathrm{fs}}}. \] Classical angular momentum is
\[ \tag{64} L=m_ev_{\mathrm{orb}}a_0. \] After substitution we get
\[ \begin{aligned} L &= m_e \left( \alpha_{\mathrm{fs}}c \right) \frac{\hbar} {m_ec\alpha_{\mathrm{fs}}} \\[1mm] &= \hbar. \end{aligned} \tag{65} \] Hence,
\[ \boxed{ m_ev_{\mathrm{orb}}a_0=\hbar. } \tag{66} \] Thus, the Bohr condition for the first orbit arises as a consequence of the combined determination of velocity and radius, rather than being introduced by a separate postulate.
11. Ground State Energy
The resulting square Lorentz factor allows us to reconstruct not only the observed electron velocity, Bohr radius, and angular momentum, but also the energy scale of the hydrogen atom.
Using the previously obtained relation
\[ \frac{1}{\gamma_{\mathrm H}} = \alpha_{\mathrm{fs}}^2, \] as well as the standard expressions for kinetic and Coulomb potential energies, we obtain the ground state energy:
\[ \boxed{ E_1 = -\frac12m_ec^2\alpha_{\mathrm{fs}}^2. } \tag{67} \] This corresponds to the ground state energy of the hydrogen atom in the non-relativistic approximation. Thus, the square Lorentz factor determines not only the spatial parameters of the atom but also the energy scale of the bound state.
12. The Unified Role of the Value \(\alpha_{\mathrm{fs}}^2\)
The resulting value simultaneously characterizes several different aspects of the atomic state.
First, it is the inverse Lorentz factor of the total geometric motion:
\[ \tag{68} \frac{1}{\gamma_{\mathrm H}} = \alpha_{\mathrm{fs}}^2. \] Second, it is equal to the square of the observed orbital velocity:
\[ \tag{69} \frac{v_{\mathrm{orb}}^2}{c^2} = \alpha_{\mathrm{fs}}^2. \] Thirdly, it defines the energy scale of the atom:
\[ \tag{70} \frac{2|E_1|}{m_ec^2} = \alpha_{\mathrm{fs}}^2. \] Therefore, we can write a single relationship:
\[ \boxed{ \frac{1}{\gamma_{\mathrm H}} = \frac{v_{\mathrm{orb}}^2}{c^2} = \frac{2|E_1|}{m_ec^2} = \alpha_{\mathrm{fs}}^2 } \tag{71} \] In this model, this coincidence is not accidental: all three quantities come from a single geometric composition of two orthogonal motions.
13. Geometric Transition Chain
The complete inference logic can be represented in a compact form:
\[ \tag{72} \j^a \quad\Longrightarrow\quad \frac{1}{\gamma_a} = \alpha_{\mathrm{fs}}, \] \[ \tag{73} \j^a(-\j)^b \quad\Longrightarrow\quad \frac{1}{\gamma_{\mathrm H}} = \alpha_{\mathrm{fs}}^2, \] \[ \tag{74} \frac{ \alpha_{\mathrm{fs}}^2 }{ \alpha_{\mathrm{fs}} } = \alpha_{\mathrm{fs}} \quad\Longrightarrow\quad v_{\mathrm{orb}} = \alpha_{\mathrm{fs}}c, \] \[ \tag{75} \frac{ r_e }{ \alpha_{\mathrm{fs}}^2 } = a_0, \] \[ \tag{76} m_ev_{\mathrm{orb}}a_0 = \hbar, \] \[ \tag{77} E_1 = -\frac12m_ec^2\alpha_{\mathrm{fs}}^2. \] Thus, the parameters of the first Bohr orbit are related to a single geometric mechanism.
14. Physical Interpretation
The resulting picture differs from both the purely corpuscular and the purely wave descriptions.
In ordinary space, the electron manifests itself as a massive particle with parameters
\[ \tag{78} m=m_e, \qquad v=\alpha_{\mathrm{fs}}c, \qquad R=a_0. \] However, its complete state in the extended basis contains two almost light-like orthogonal motions:
\[ \tag{79} J_{\mathrm H} = \j^a(-\j)^b. \] Therefore, the complete state is almost wave-like:
\[ \tag{80} \beta_{\mathrm H} = \sqrt{1-\alpha_{\mathrm{fs}}^4} \approx1. \] The observed corpuscular characteristics arise from the projection of this state onto ordinary space.
This separation allows us to simultaneously preserve:
- the near-light dynamics of the complete state;
- the non-zero observed mass of the electron;
- the non-relativistic velocity of motion in the atom;
- the Bohr radius;
- the quantum of angular momentum;
- the energy of the ground state of hydrogen.
The electron in the atom in this interpretation is not a classical point literally orbiting at near-light speed. The near-light motion of the multicomponent complete state is near-light, while its spatial projection reproduces the parameters of a massive particle.
15. Initial Assumptions and Limits of Derivation
It is necessary to separate the mathematical consequences of the model from its initial hypotheses.
The following assumptions are made in this article:
- The internal state of the particle is described by the operator \[ \j^a; \]
- The external closed motion is described by the operator \[ (-\j)^b; \]
- Both motions have the same initial parameter \[ \beta=\sqrt{1-\alpha_{\mathrm{fs}}^2}; \]
- The motions belong to mutually orthogonal state planes;
- Standard relativistic transformations of orthogonal velocities are applied to their composition;
- The observed orbital velocity is determined by the relative projection of the total and internal states: \[ \beta_{\mathrm{orb}} = \frac{\alpha_{\mathrm H}} {\alpha_{\mathrm{int}}}. \]
After making these assumptions, the formulas for \(\alpha_{\mathrm{fs}}^2\), \(v_{\mathrm{orb}}\), \(a_0\), \(L\), and \(E_1\) follow sequentially.
At the same time, the identification of the internal motion parameter with the fine structure constant and the projection rule require further physical justification. They are elements of the proposed model, not consequences of special relativity alone.
Conclusions
This paper considers an atomic state formed by two independent orthogonal motions:
\[ \tag{81} J_{\mathrm H} = \j^a(-\j)^b. \] Each motion is characterized by the inverse Lorentz factor.
\[ \tag{82} \frac{1}{\gamma} = \alpha_{\mathrm{fs}}. \] Successive relativistic composition of two orthogonal motions leads to the product:
\[ \tag{83} \frac{1}{\gamma_{\mathrm H}} = \frac{1}{\gamma_a} \frac{1}{\gamma_b} = \alpha_{\mathrm{fs}}^2. \] The ratio of the total projection coefficient to the internal one gives the observed velocity of the electron:
\[ \tag{84} v_{\mathrm{orb}} = \alpha_{\mathrm{fs}}c. \] From the intrinsic scale
\[ \tag{85} r_e = \alpha_{\mathrm{fs}} \frac{\hbar}{m_ec} \] and the total coefficient \(\alpha_{\mathrm{fs}}^2\), we obtain the Bohr radius:
\[ \tag{86} a_0 = \frac{r_e}{\alpha_{\mathrm{fs}}^2} = \frac{\hbar} {m_ec\alpha_{\mathrm{fs}}}. \] Then the angular momentum quantum is automatically reconstructed.
\[ \tag{87} m_ev_{\mathrm{orb}}a_0 = \hbar \] and the ground state energy:
\[ \tag{88} E_1 = -\frac12m_ec^2\alpha_{\mathrm{fs}}^2. \] Thus, the square of the fine structure constant is interpreted as the geometric result of the unification of internal and external orthogonal motions. The total state remains almost wave-like, while its observed projection reproduces the classical parameters of a massive electron in a hydrogen atom.
The main result can be represented in one chain:
\[ \boxed{ \j^a(-\j)^b \Longrightarrow \alpha_{\mathrm{fs}}^2 \Longrightarrow \left\{ \begin{aligned} v_{\mathrm{orb}}&=\alpha_{\mathrm{fs}}c,\\ a_0&=\dfrac{\hbar}{m_ec\alpha_{\mathrm{fs}}},\\ L&=\hbar,\\ E_1&=-\dfrac12m_ec^2\alpha_{\mathrm{fs}}^2. \end{aligned} \right. } \tag{89} \] The proposed construction suggests that the Bohr atom is only the simplest manifestation of a more general geometric structure. The same approach can be extended to excited states, multielectron systems, particle interactions, and other quantum phenomena, where the geometry of orthogonal motions can play the role of a unifying mathematical principle.