In this paper, the problem of the origin of mass is considered primarily from a geometric perspective. We assume that the fundamental characteristic of a particle is the total energy of its internal periodic state, while the rest mass is merely the observed projection of this energy onto a preferred direction in internal space.

The main goal of this paper is to apply this geometric framework first to the electron and then consider its massless limit, which, in terms of its energy properties, may correspond to the photon. In this approach, the electron and photon are not introduced as fundamentally different objects: they are viewed as different orientations of the total energy state, one of which has a nonzero mass projection, while the other does not.

The model is then generalized to an arbitrary particle. Its mass is assumed to be determined simultaneously by the frequency of the internal periodic process and the geometric orientation of the associated energy vector. This allows us to consider the mass spectrum as a possible consequence of different internal frequencies and different projection coefficients.

The geometric construction of the internal motion is based on a previously developed new idempotent basis and a fractional power of the hyperbolic unit. The relationship between the electron's internal frequency, its mass, the internal Lorentz factor, and the fine structure constant was obtained in a previous paper. This article develops this result and proposes interpreting mass not as a primary characteristic of a particle, but as a projection of its total internal energy.

The model is based on two mutually complementary idempotents \[ \tag{1} \ep^2=\ep, \qquad \em^2=\em, \qquad \ep\em=0, \qquad \ep+\em=1. \] Associated with them is the hyperbolic unit \[ \tag{2} \j=\ep-\em, \qquad \j^2=1. \]

In a paper on the geometry of a fractional power of the hyperbolic unit, a representation of the internal periodic state was obtained \[ \tag{3} \boxed{ \j^{-\varpi t} = \ep + \em e^{-i\omega t} }, \qquad \omega=\pi\varpi=2\pi\nu. \]

Therefore, the internal state vector can be written directly as \[ \tag{4} J(t)=\j^{-\varpi t}. \] In expanded form \[ \tag{5} J(t) = \ep + \em\cos\omega t - i\em\sin\omega t. \]

The first component remains constant, while in the plane \[ \tag{6} \left\{ \em,\;i\em \right\} \] periodic internal rotation occurs. Thus, the construction \(\j^{-\varpi t}\) simultaneously contains a fixed direction and a rotational component.

Frequency \(\nu\) characterizes the rate of change of the internal state. In what follows, the energy associated with it will be considered as the modulus of the particle's total energy vector.

Let the total energy corresponding to the internal periodic state of a particle be \[ \tag{7} \boxed{ W=h\nu }. \]

It is fundamentally important that the quantity \(W\) is not directly identified with the rest energy here. It determines the total energy scale of the internal process described by the construction \[ J(t)=\j^{-\varpi t}. \]

We introduce the internal Lorentz factor \[ \tag{8} \gamma_{\mathrm{int}} = \frac{1} {\sqrt{1-\beta_{\mathrm{int}}^2}}, \qquad \beta_{\mathrm{int}} = \frac{v_{\mathrm{int}}}{c}. \]

We write the relationship between the total internal energy and the rest energy as \[ \tag{9} W = \gamma_{\mathrm{int}}mc^2. \] Hence \[ \tag{10} \boxed{ mc^2 = \frac{W} {\gamma_{\mathrm{int}}} } \] and \[ \tag{11} \boxed{ m = \frac{h\nu} {\gamma_{\mathrm{int}}c^2} }. \]

Thus, the mass of a particle is determined by two characteristics: \[ \tag{12} m=m\left(\nu,\gamma_{\mathrm{int}}\right). \] The intrinsic frequency determines the overall energy scale, and the Lorentz factor determines the fraction of this energy that manifests itself as rest energy.

We introduce the angle \(\vartheta\), defined by the relation \[ \tag{13} \cos\vartheta = \frac{1} {\gamma_{\mathrm{int}}}. \] Then \[ \tag{14} \boxed{ mc^2 = W\cos\vartheta }. \]

The total internal energy \(W=h\nu\) is represented by the modulus of the energy vector, and the rest energy \(mc^2\) is its projection onto the chosen direction. Mass becomes not an independent primary quantity, but a geometric characteristic of the internal state.

Orthogonal component of the total energy vector is equal to \[ \tag{15} W_{\perp} = W\sin\vartheta. \]

From the definition of the internal Lorentz factor it follows \[ \tag{16} \sin\vartheta = \sqrt{ 1- \frac{1} {\gamma_{\mathrm{int}}^2} } = \beta_{\mathrm{int}}. \] Therefore \[ \tag{17} \boxed{ W_{\perp} = \beta_{\mathrm{int}}W }. \]

The components form a geometric decomposition \[ \tag{18} W^2 = \left(mc^2\right)^2 + W_{\perp}^2. \] After substituting expression (17) \[ \tag{19} W^2 = \left(mc^2\right)^2 + \beta_{\mathrm{int}}^2W^2. \]

From this we again obtain \[ \tag{20} mc^2 = W \sqrt{ 1- \beta_{\mathrm{int}}^2 } = \frac{W} {\gamma_{\mathrm{int}}}. \] Thus, the energetic and geometric interpretations are in complete agreement.

If the total energy of an internal state is identified with the total relativistic energy of a particle, \[ \tag{21} W=E, \] then \[ \tag{22} E = \gamma mc^2. \]

The relativistic momentum is \[ \tag{23} p = \gamma mv. \] Therefore, \[ \tag{24} pc = \gamma mc^2 \frac{v}{c} = \beta E. \]

Comparing expressions (17) and (24), we obtain \[ \tag{25} W_{\perp}=pc. \] Then the geometric expansion (18) takes the form \[ \tag{26} \boxed{ E^2 = m^2c^4 + p^2c^2 }. \]

This allows us to interpret the rest energy \(mc^2\) as one projection of the total energy, and the dynamic quantity \(pc\) as an orthogonal component.

It should be noted that the spacetime of special relativity has a pseudo-Euclidean geometry. Therefore, the given image should be understood as a visual decomposition of the positive absolute values ​​of the energy components, and not as a literal Euclidean model of Minkowski space.

In previous work the following relation was proposed for the internal motion of the electron: \[ \tag{27} \alpha_{\mathrm{fs}} = \frac{1} {\gamma_{\mathrm{int}}}, \] where \(\alpha_{\mathrm{fs}}\) is the fine-grain constantoh structure.

Hence \[ \tag{28} \gamma_{\mathrm{int}} = \frac{1} {\alpha_{\mathrm{fs}}} \approx 137.036. \] For the electron's energy projection angle, we obtain \[ \tag{29} \cos\vartheta_e = \alpha_{\mathrm{fs}}. \]

The total internal energy of an electron is determined by its internal frequency: \[ \tag{30} W_e=h\nu_e. \] The rest energy is a projection of this quantity: \[ \tag{31} m_ec^2 = W_e\cos\vartheta_e. \]

Taking into account expression (29) \[ \tag{32} \boxed{ m_ec^2 = \alpha_{\mathrm{fs}}W_e = \alpha_{\mathrm{fs}}h\nu_e }. \]

Thus, within the framework of the proposed model, the fine structure constant determines the fraction of the electron's total internal energy that manifests itself as its rest energy.

The orthogonal component is equal to \[ \tag{33} W_{\perp e} = W_e \sqrt{ 1- \alpha_{\mathrm{fs}}^2 }. \] Since \[ \tag{34} \sqrt{ 1- \alpha_{\mathrm{fs}}^2 } \approx 0.99997337, \] the bulk of the total internal energy is attributed to the orthogonal component, while the rest energy is a relatively small geometric projection of it.

Consider the limit in which the velocity of internal motion approaches the speed of light: \[ \tag{35} \beta_{\mathrm{int}} \rightarrow1. \] Then \[ \tag{36} \gamma_{\mathrm{int}} \rightarrow\infty \] and \[ \tag{37} \cos\vartheta = \frac{1} {\gamma_{\mathrm{int}}} \rightarrow0. \]

Therefore, \[ \tag{38} mc^2 = W\cos\vartheta \rightarrow0. \] In this case, the total energy \[ \tag{39} W=h\nu \] may remain finite.

In the massless limit, the energy vector completely transforms into the orthogonal direction: \[ \tag{40} W_{\perp}\rightarrow W, \qquad mc^2\rightarrow0. \]

This state corresponds to the basic energy properties. Photon: \[ \tag{41} m=0, \qquad E=h\nu, \qquad E=pc. \]

In the proposed geometry, the photon can be viewed as a limiting state in which the total energy vector remains nonzero, but its projection onto the rest energy direction is zero. Therefore, it is not the particle's energy that vanishes, but only its mass projection.

In this sense, the electron and photon can be represented as different geometric states of a single common mechanism. For the electron, the angle \(\vartheta_e\) is slightly different.is derived from the line and there is a small nonzero projection \(m_ec^2\). For a photon, the angle reaches its limiting value \[ \tag{42} \vartheta_\gamma = \frac{\pi}{2}, \] therefore \[ \tag{43} \cos\vartheta_\gamma=0. \]

However, the identification of the \(\j^{-\varpi t}\) construction with the full state of the photon remains a hypothesis for now. For further substantiation, it is necessary to derive the polarization, spin, propagation law, and electromagnetic structure of the photon from this geometry.

For a particle with index \(i\), we write the internal state as \[ \tag{44} J_i(t) = \j^{-\varpi_i t}, \qquad \varpi_i=2\nu_i. \]

The total internal energy of such a particle is \[ \tag{45} W_i=h\nu_i. \] Its mass is determined by the expression \[ \tag{46} \boxed{ m_i = \frac{h\nu_i} {\gamma_i c^2} }. \]

The equivalent geometric form is \[ \tag{47} m_ic^2 = W_i\cos\vartheta_i, \qquad \cos\vartheta_i = \frac{1}{\gamma_i}. \]

Therefore, differences in the particle masses can be due to two factors:

1. differences in the internal frequencies \(\nu_i\), which determine the moduli of the total energy vectors;

2. differences in the angles \(\vartheta_i\), which determine the projections of these vectors onto the direction of rest energy.

The ratio of the masses of two particles is determined by the expression \[ \tag{48} \frac{m_1}{m_2} = \frac{\nu_1}{\nu_2} \frac{\gamma_2}{\gamma_1}. \]

This formula by itself does not yet define the spectrum of elementary particles, since for each particle it is necessary to find an independent condition relating its frequency and geometric projection coefficient. Nevertheless, it defines a general scheme applicable to both massive and massless states.

The algebraic basis for the construction under consideration was laid in the paper on a new idempotent basis. There, it was shown that the complex extension of two complementary idempotents naturally forms a four-dimensional real structure.

The geometry of the fractional power of the hyperbolic unit and the expression \[ \j^{-\varpi t} = \ep + \em e^{-i\omega t} \] were obtained in this work.

The separation of the electron's translational and internal motion was considered in the first part of the geometric model of the electron and in its second part.

The relationship between the internal frequency, electron mass, Lorentz factor, and fine structure constant was proposed in a paper on the origin of mass and the Schrödinger equation.

This paper combines these results into a single energy scheme: \[ J(t) = \j^{-\varpi t}, \qquad W=h\nu, \qquad mc^2 = \frac{W}{\gamma} = W\cos\vartheta. \]

Until now, it has been assumed that the internal state of a particle is characterized by a single fundamental frequency \(\omega\). However, the geometric construction does not require that rotation occur only in the fundamental mode. We can assume the existence of a family of internal states whose angular frequencies are multiples of some fundamental frequency: \[ \tag{49} \omega_n = n\omega_0, \qquad n=1,2,3,\ldots \]

Since \[ \omega_0 = \pi\varpi_0 = 2\pi\nu_0, \] for each mode we obtain \[ \tag{50} \varpi_n = n\varpi_0, \qquad \nu_n = n\nu_0. \]

Then the internal state of the particle is written as \[ \tag{51} J_n(t) = \j^{-n\varpi_0t}. \] Using the geometric representation of the fractional power of the hyperbolic unit, we obtain \[ \tag{52} \boxed{ J_n(t) = \j^{-n\varpi_0t} = \ep + \em e^{-in\omega_0t} }. \]

The number \(n\) determines the number of complete internal revolutions completed during the period of the fundamental mode. For n=1, we obtain the ground state: J_1(t) = j^{-ππt}. For n>1, the rotational component traverses the same geometric circle with a higher angular frequency.

The total internal energy of the nth mode is W_n=h. Since nu_n = n, we obtain W_n = nW_0. We obtain 55. boxed{ W_n = nh. W_n = nW_0.\]

Thus, the fundamental frequency \(\nu_0\) defines the minimum energy scale of internal motion, and the coefficient \(n\) determines the number of its harmonic mode.

The mass of a particle in state \(n\) is determined by the projection of the total internal energy: \[ \tag{56} m_nc^2 = \frac{W_n} {\gamma_n}. \] Therefore, \[ \tag{57} \boxed{ m_n = \frac{nh\nu_0} {\gamma_n c^2} }. \]

The equivalent geometric form is \[ \tag{58} m_nc^2 = W_n\cos\vartheta_n, \qquad \cos\vartheta_n = \frac{1} {\gamma_n}. \]

From expression (57), two possible mechanisms for the formation of the mass spectrum follow. If the internal Lorentz factor is the same for all modes, \[ \tag{59} \gamma_n = \gamma_0, \] then \[ \tag{60} m_n = nm_0. \] In this case, the masses form a linear harmonic spectrum.

More generally, each mode may have its own geometric orientation of the energy vector: \[ \tag{61} \gamma_n \neq \gamma_0. \] Then \[ \tag{62} \frac{m_n}{m_1} = n \frac{\gamma_1} {\gamma_n}. \]

Therefore, the observed mass spectrum can be determined simultaneously by the number of the internal mode and its projection coefficient. Even if the frequencies form a simple harmonic series, the particle masses do not necessarily have to be integer multiples of the fundamental mass.

For an arbitrary particle with index \(i\), we can write \[ \tag{63} J_i(t) = \j^{-n_i\varpi_0t}, \] \[ \tag{64} W_i = n_ih\nu_0, \] \[ \tag{65} \boxed{ m_i = \frac{n_ih\nu_0} {\gamma_i c^2} }. \]

In this scheme, different particles can be viewed as different harmonic and geometric states of a single fundamental internal process. The number \(n_i\) determines the energy mode, and the quantity \(\gamma_i\) determines the fraction of the total energy that manifests itself as the rest mass.

The massless limit holds for any harmonic mode. If \[ \tag{66} \gamma_n \rightarrow \infty, \] then \[ \tag{67} m_n \rightarrow0, \] although the total energy remains finite: \[ \tag{68} W_n = nh\nu_0. \] Therefore, a photon can also correspond to one of the harmonic modes whose energy vector has no projection onto the direction of the rest mass.

At this stage, the integer nature of \(n\) is an additional hypothesis. It must be justified by the condition of internal trajectory closure, boundary conditions, or the requirement of uniqueness of the state. If such a condition is found, the discreteness of internal frequencies and energies will become a consequence of the model's geometry, rather than a separate quantum postulate.

In the proposed approach, mass is not the primary quantity of matter. The primary quantity is the internal periodic state of a particle, which has a frequency and a corresponding total energy.

Mass characterizes only that part of the total energy state that is projected onto a specific direction in rest space. Therefore, the same total internal energy at different angles can correspond to different masses.

Changing the frequency changes the length of the energy vector. Changing the internal Lorentz factor changes its orientation. Mass depends on both factors simultaneously.

In the limiting case of internal motion at the speed of light, the mass projection vanishes, but the energy is conserved. This allows the photon to be included in the general geometric scheme as a massless limit of a periodic internal state.

Construction \[ J(t) = \j^{-\varpi t} = \ep + \em e^{-i\omega t} \] defines the internal periodic state of the particle in a complex-extended idempotent basis. The frequency of this state determines the total internal energy \[ W=h\nu. \]

The rest mass is interpreted as a geometric projection of the total internal energy: \[ mc^2 = \frac{W}{\gamma} = W\cos\vartheta. \] Thus, mass is not the initial energy scale of a particle, but a derived characteristic of the orientation of its total internal state.

The orthogonal component of the total energy vector is equal to \[ W_{\perp} = \beta W. \] When \(W\) is identified with the total relativistic energy, this quantity coincides with \(pc\), which leads to the relation \[ E^2 = m^2c^4 + p^2c^2. \] Thus, the geometric expansion is consistent with the standard relationship between energy, momentum, and mass.

For the electron, the previously proposed relationship \[ \frac{1} {\gamma_{\mathrm{int}}} = \alpha_{\mathrm{fs}} \] gives \[ m_ec^2 = \alpha_{\mathrm{fs}}h\nu_e. \] Within the model, the fine-structure constant determines the projection coefficient of the electron's total internal energy onto ndirection of the rest energy.

For \(\gamma\rightarrow\infty\) and \(v_{\mathrm{int}}\rightarrow c\), the mass projection vanishes, while the energy \(W=h\nu\) remains finite. This limit corresponds to the fundamental energetic properties of the photon and allows us to consider massive and massless particles as different geometric states of a single internal mechanism.

An additional hypothesis has been put forward about the existence of harmonic modes of internal rotation: \[ \omega_n = n\omega_0, \qquad n=1,2,3,\ldots \] They correspond to the family of states \[ J_n(t) = \j^{-n\varpi_0t} = \ep + \em e^{-in\omega_0t}. \]

The total internal energy of the (n)th mode is \[ W_n = nh\nu_0, \] and its mass projection is determined by the expression \[ \boxed{ m_n = \frac{nh\nu_0} {\gamma_n c^2} }. \] Therefore, the observed mass can depend simultaneously on the fundamental internal frequency, the harmonic mode number, and the geometric projection coefficient.

If the internal Lorentz factor is the same for all modes, then the masses form a linear spectrum \[ m_n=nm_1. \] In a more general case, \[ \gamma_n\neq\gamma_1, \] and the mass ratio takes the form \[ \frac{m_n}{m_1} = n \frac{\gamma_1} {\gamma_n}. \] Therefore, even with an integer-valued internal frequency spectrum, the particle masses do not necessarily have to be integer multiples of the fundamental mode mass.

The integer-valued nature of \(n\) can be related to the closure condition of the internal state: \[ J_n(t+T_0)=J_n(t). \] For \[ T_0 = \frac{2\pi}{\omega_0} \] this condition requires \[ e^{-i2\pi n}=1, \] which is satisfied for integers \(n\). In this case, the discreteness of internal frequencies and energies acquires a geometric basis.

Thus, within the framework of the proposed scheme, an arbitrary particle is characterized by an internal mode \(n\), a fundamental frequency \(\nu_0\), and a projection coefficient \(1/\gamma_n\). Electrons, photons, and other particles can be considered as different harmonic and geometric states of a single periodic process.

The proposed model does not yet determine specific values ​​of \(n\) and \(\gamma_n\) for known particles. Further development requires obtaining independent geometric or dynamic conditions linking internal modes with mass, spin, charge, and other quantum characteristics.