3.1 Proposed 3D hypercomplex number system

We consider a 3D number from the set , in the Cartesian coordinate system (CCS), as (1) such that the set {1, i, j} is basis where i and j are two imaginary numbers, and . This 3D number (1) can also be written using the triplet notation as: . First, we write (1) in one of the standard spherical coordinate system (SCS) as (2) where r ∈ [0, ∞), ϕ ∈ [0, 2π) and θ ∈ [−π/2, π/2]. The function atan2(b, a) computes the four-quadrant inverse tangent arctan(b/a) and utilizes the signs and magnitudes of b and a to determine the correct quadrant for ϕ. Moreover, we can add 2πn for to both angles ϕ and θ without changing the point, i.e., g(r, ϕ, θ) = g(r, ϕ + 2πn, θ + 2πn). Thus, a spherical coordinate triplet (r, ϕ, θ) specifies a single point in a three-dimensional space that has infinitely many equivalent spherical coordinates.

Remark 1. We can fix n = 0 to remove much of the non-uniqueness in the representation of a point in the SCS. In spite of everything, persistent nonuniqueness in representations arises in two specific scenarios within the SCS:

1. Origin (r = 0): In this case, when the radial coordinate r assumes a value of zero, the angular coordinates ϕ and θ become unrestricted, resulting in an infinite representation of the origin as (0, ϕ, θ) in the SCS. This signifies that any non-zero vector (r, ϕ, θ) can be asymptotically approached to a zero vector by allowing its radial component r to approach zero, akin to the behavior observed in polar representations of two-dimensional complex numbers. In other words, variations in the angular coordinates ϕ or θ while situated at the origin do not effectuate any displacement of the vector.
2. Z-axis (θ = ±π/2): In this specific scenario where θ equals π/2 or −π/2, for a fixed r, the azimuth angle ϕ can assume any value, as explicitly revealed by equation (2). Consequently, a vector situated along the Z-axis possesses an infinite array of representations as (r, ϕ, ±π/2). This implies that vectors characterized by diverse values of r, ϕ, and θ can be aligned along the Z-axis by adjusting θ to ±π/2. Once aligned along the Z-axis, alterations in the polar angle ϕ fail to induce any alteration in the orientation of the vector. Thus, in these instances, manipulation of the values of one or more of the other coordinates can be undertaken without inducing displacement of the point in question.

Subsequently, in pursuit of the specific objectives delineated in Remark 2 and aiming to construct new multiplication (16) and division (17) operations with demonstrably sound properties, we propose a meticulously refined definition for (2) as follows: (3) where the π-periodic functions, as shown in Fig 2, are defined for all as (4) (5) where the intervals form a complete, non-overlapping partition of the real line as (6)

We observe that cosπ(θ) = cosπ(θ + nπ) = |cos(θ)| for all and . Moreover, sinπ(θ) = sinπ(θ + π) for θ ≠ −π/2 and sinπ(θ) = −sinπ(θ + π) for θ = −π/2. Therefore, sinπ(θ) = sinπ(θ + nπ) for all and for almost all . Thus, sinπ(θ)/cosπ(θ) = tanπ(θ) = tan(θ), which is differentiable in . The function cosπ(θ) is continuous, while sinπ(θ) is discontinuous at odd multiples of , and both functions are differentiable in . Moreover, within the open interval (−π/2, π/2) of the principal range, both cosπ(θ) and sinπ(θ) are continuous and differentiable. Thus, using these functions, we redefine the SCS representation (2) as (7) where g is g(r, ϕ, θ), ϕ is an azimuth angle, and θ is an elevation angle form XY plane as shown in Fig 3. The significance of the periodicity of the hypercomplex number g, as defined in (7), should be emphasized appropriately. Specifically, it is imperative to recognize that g(r, ϕ, θ) exhibits periodic behavior, as expressed by the relationship g(r, ϕ, θ) = g(r, ϕ + 2mπ, θ + mπ) for all integers . This mathematical characterization underscores the recurrence of the values with respect to variations in the angular coordinates ϕ and θ and elucidates the systematic nature of g within the specified parameter space. Such periodicity holds broader implications for the analytical treatment and interpretation of the hypercomplex number, serving as a fundamental consideration in the study of its properties and behavior. This is to note that the conventional notation of spherical coordinate system is not considered in this work (One can also use the conventional notation of spherical coordinate system where angle θ is measured from the Z-axis. This will also lead to a 3D hypercomplex number system. For more details, see Appendix A.). Thus, from (1), (3) and (7), we can write (8) which represents the following eight cases depending on the three parameters r, ϕ and θ:

1. A vector that represents a point on a unit sphere for r = 1, ϕ = ϕ₀ and θ = θ₀.
2. A circle for r = 1 and ϕ ∈ [0, 2π) at elevation angle θ = θ₀ (e.g., see Fig 1).
3. A semicircle for r = 1 and θ ∈ [−π/2, π/2] at azimuth angle ϕ = ϕ₀ (e.g., see Fig 1).
4. A unit sphere for r = 1, ϕ ∈ [0, 2π) and θ ∈ [−π/2, π/2].
5. A solid unit sphere for 0 ≤ r ≤ 1, ϕ ∈ [0, 2π) and θ ∈ [−π/2, π/2].
6. A conical surface for 0 ≤ r ≤ 1 and ϕ ∈ [0, 2π) at elevation θ = θ₀.
7. A semidisk for 0 ≤ r ≤ 1 and θ ∈ [−π/2, π/2] at azimuth ϕ = ϕ₀.
8. A ray for 0 ≤ r ≤ 1, ϕ = ϕ₀ and θ = θ₀.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. A point P in the considered spherical co-ordinate system, where radius (r), azimuth angle (ϕ rad), and elevation angle (θ rad) measured from XY plane are shown.

https://doi.org/10.1371/journal.pone.0312502.g003

We use the representation in CCS, and in SCS, therefore where this equality is in the sense that both represent the same point. We consider three special cases of (3) as follows:

1. Consider XY plane with θ = 0, as shown in Fig 4(a), which implies (9) and thus for r = 1, 0 ≤ ϕ < 2π represents the unit circle in XY plane.
2. Consider XZ plane with (i) ϕ = 0, as shown in Fig 4(b), right semicircle on the XZ plane for r = 1, −π/2 ≤ θ ≤ π/2, which implies (10) and (ii) ϕ = π, as shown in Fig 4(b), left semicircle for r = 1, −π/2 ≤ θ ≤ π/2, which implies (11)
3. Consider YZ plane with (i) ϕ = π/2, as shown in Fig 4(c), right semicircle on the YZ plane for r = 1, −π/2 ≤ θ ≤ π/2, which implies (12) and (ii) ϕ = 3π/2, as shown in Fig 4(c), left semicircle for r = 1, −π/2 ≤ θ ≤ π/2, which implies (13)

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4.

Euler identities (a) exp(iϕ) for ϕ ∈ [0, 2π) with r = 1 and θ = 0, where a vector that represents a point traces full circle in the direction of arrow as ϕ increases, (b) exp(jθ) for θ ∈ [−π/2, π/2] with r = 1 and ϕ = π for X < 0 (blue semicircle), and ϕ = 0 for X > 0 (red semicircle), where a point moves in the direction of arrow in the right semicircle as θ increases with ϕ = 0, and similarly a point moves in the direction of arrow in left semicircle as θ increases with ϕ = π, (c) exp(jθ) for θ ∈ [−π/2, π/2] with r = 1 and ϕ = 3π/2 (blue semicircle) and ϕ = π/2 (red semicircle). The path traced by exp(jθ) follows a semicircular trajectory, as indicated by the blue and red semicircle in (b) and (c) for the appropriate values of ϕ.

https://doi.org/10.1371/journal.pone.0312502.g004

To obtain the generalized addition and multiplication, we write hypercomplex numbers , using SCS representations (2)–(8), in the triplet notations as (14)

We consider traditional addition operations as (15)

Further, we define a new multiplication operation, named hereby spherical multiplication (SM) as (16)

The proposed SM performs scaling and rotating operations. The additive inverse of g₂ is given by , which leads to the definition of the subtraction operation as . Similarly, the multiplicative inverse of g₂ is defined as , allowing us to define the spherical division (SD) operation as (17)

The complex conjugate of g presented in (1) is defined as (18) such that . The norm of g is defined as and satisfies certain properties, such as positivity, homogeneity, and triangle inequality. Thus, the unified multiplication operation defined in (16) consists of scaling and rotation operations such that ‖g₁g₂‖ = ‖g₁‖‖g₂‖, and ‖g₁ + g₂‖ ≤ ‖g₁‖ + ‖g₂‖. In contrast, a non-distributive elliptic scator algebra that extends complex numbers to a higher number of dimensions fails to satisfy the triangle inequality of the norm [16].

The defined SM (16) reduces to the traditional multiplication when g moves from 3D to 2D by considering c = 0 in (1). The conjugation with respect to i and j can be defined as (19) respectively. The multiplicative inverse of (1) is defined as for every g ≠ 0, which is same as inverse of a quaternion. This can be written in the triplet notation as (20)

Further, we can compute the power of g^ℓ as (21) and (21) becomes (20) when ℓ = −1.

Addition of two complex numbers (e.g., g₂ + g₃) can be written as and thus (22)

We can also compute g₁g₂ + g₁g₃ as (23)

From (22) and (23), it is easy to verify that, in general, g₁(g₂ + g₃) ≠ g₁g₂ + g₁g₃. Moreover, g₁(g₂ + g₃) = g₁g₂ + g₁g₃ if θ₁ = 0, i.e., .

Interestingly, similar to , using the proposed multiplication (16), we can write (24) where from (7) and (14) (25) where {e^iϕ ∣ ϕ ∈ [0, 2π)} represents all points on the unit circle in the XY plane and {e^jθ ∣ θ ∈ [−π/2, π/2]} represents all points on the unit semicircle in XZ plane. In general, and e^iϕ is 2π-periodic, and e^jθ is π-periodic. The 2π modulo operation is applied to ϕ to restrict it to the principal range [0, 2π). Similarly, the π modulo operation is used to adjust θ to lie within the principal range [−π/2, π/2]. Specifically, if θ ∉ [−π/2, π/2], it is normalized to this range by adding or subtracting a multiple of π. Hence, for the elevation angles, we employ the transformation for all as follows: (26) where m is the smallest positive integer such that the result lies within the interval . Therefore, the azimuth angle ϕ is considered 2π-periodic, and the elevation angle θ is considered π-periodic as shown in Fig 5. Therefore, cosπ(θ) = cos(θ) and sinπ(θ) = sin(θ) with . Thus (7) can be written as (27) where [cos(ϕ) + isin(ϕ)] × j sinπ(θ) = e^iϕ × j sinπ(θ) = j sinπ(θ), i² = −1 and j² = −1. Therefore, geometrically, we have defined a multiplication that represents a sphere as the multiplication of a circle in the XY plane with a semicircle in the XZ plane. The resulting sphere can be smoothly traversed without encountering any breaks, reflecting the continuous nature of the circle and semicircle components, as shown in Fig 1.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 5. Plot illustrating the azimuth angle ϕ versus (ϕ mod 2π) in the range −π to π, and the elevation angle θ versus (θ mod π) in the range −π/2 to π/2.

The azimuth angle ϕ is considered 2π-periodic, and the elevation angle θ is considered π-periodic.

https://doi.org/10.1371/journal.pone.0312502.g005

Remark 2. To confine the elevation angle θ within the interval [−π/2, π/2], prevent the occurrence of dual representations, and ensure consistent multiplication outcomes in accordance with (16), even when the sum (θ₁+ θ₂) exceeds the specified range of [−π/2, π/2], and determine the spatial coordinates of a point in three dimensions post-multiplication, an important modification has been implemented in the SCS (2) for elevation angle. Specifically, the conventional trigonometric functions cos(θ) and sin(θ) have been replaced with their respective counterparts cosπ(θ) and sinπ(θ), leading to a new representation of SCS (7). This substitution effectively addresses the specified objectives, offering a systematic approach to confining the elevation angle, avoiding dual representations, ensuring consistent multiplication outcomes, and accurately determining the 3D position of a point following the multiplication process.

Using the defined expressions (1), (3) and (14), we can map the CCS basis to SCS as: (28) where ϕ ∈ [0, 2π). The azimuth angle ϕ in the expression indicates that any subsequent variation in the elevation angle θ results in the displacement of the point from the Z-axis towards the direction specified by ϕ. To derive the distinctive characterization of the complex unit j, subject to the condition j² = −1, a specific angular value, ϕ = π/2, is established in the context of the identity presented in (28). It is imperative to note that, unless explicitly indicated otherwise, this particular angular assignment shall be consistently adhered to throughout the course of this investigation. In addition, we can write (29) such that 1(−1) = −1, i(−i) = 1, j(−j) = 1, and 0(g) = 0. The additive identity 0 and the multiplicative identity 1 are distinct and unique in both CCS and SCS. It is interesting to observe that, . This behaves like a 2D negative unit multiplication factor because for any , , i.e., it changes the angle ϕ by π rad and does not affect θ. Moreover, if ϕ = π/2, and 1 if ϕ = 0. Similarly, for ϕ = 0, and (ij)² = 1 for ϕ = π/2.

Further, if and , then and in general, . In fact, the new imaginary number j has one degree of freedom because it can be written as for any ϕ ∈ [0, 2π) which implies , and thus (30) therefore, it can have infinite number of representations. For examples (i) j² = 1 when ϕ = 0, (ii) j² = −1 when ϕ = π/2, (iii) j² = i when ϕ = π/4, (iv) j² = −i when ϕ = 3π/4. Basically, the multiplication of a complex number, e.g., with specific is another representation of . We observe that these infinite representations of j arise solely from the nonunique representation of a point on the Z-axis in the SCS, as elucidated in Remark 1. These nonuniqueness can be easily avoided by considering θ ∈ (−π/2, π/2) as explained in Remark 3.

The additive inverse of an element g = a + ib + jc is given by (31) and thus g + g_ai = 0 ⇒ g_ai = −g, e.g., a point P and −P is shown in Fig 6. It is important to note that (−1)g ≠ g_ai because multiplication by −1 changes the angle ϕ by π rad but does not affect θ, while we obtain g_ai from g by mapping ϕ ↦ ϕ + π and θ ↦ −θ as defined in (31), which is equivalent to multiplication by −1 and conjugating with respect to j as defined in (19), i.e., . Therefore ; , and . One can notice that the angle ϕ is wrapped in the multiple of 2π and θ in the multiple of π.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 6. A point in the considered spherical co-ordinate system where radius r = 1, azimuth angle ϕ = π/4 and elevation angle θ = π/6 are shown, and point −P with r = 1, ϕ = 5π/4 and elevation angle θ = 7π/6 = −π/6 are also shown.

https://doi.org/10.1371/journal.pone.0312502.g006

Because the above defined spherical multiplication is not derived from addition, it does not follow the distributive property, and thus, in general, g₁(g₂ + g₃) ≠ g₁g₂ + g₁g₃, likewise addition does not distribute over multiplication, i.e., g₁ + (g₂g₃) ≠ (g₁+ g₂)(g₁ + g₃). This is a desired property of the defined SM because, geometrically, the operation on the left side is different from that on the right side.

Remark 3. As elucidated, when θ assumes values of ±π/2, the azimuthal angle ϕ becomes indeterminate, leading to an infinite set of representations for the complex number j. This inherent indeterminacy underscores the existence of non-uniqueness in the representations. In order to establish complete uniqueness, we impose the constraint θ ∈ (−π/2, π/2), thereby excluding only two endpoints from the interval θ ∈ [−π/2, π/2]. This restriction in (1) ensures that if c ≠ 0, then either a ≠ 0 or b ≠ 0, but both a and b cannot be equal to zero simultaneously. Consequently, a practical assurance of uniqueness in the representation of numbers within the proposed system is attainable such that (32) (33) where from (3) , , , , , functions cosπ(⋅) and sinπ(⋅) are defined in (4) and (5), respectively.

Next, we present two important results as follows:

Result 1. Using the proposed theory with j² = ±1 ⇔ j³ = ±j and j² = ±1 ⇒ j⁴ = 1, for all with mapping θ ↦ θ môd π as defined in (26), we obtain (34) On using j² = −1, j³ = −j and j⁴ = 1, we can easily obtain Euler identity as (35) Interestingly, on using j² = 1, j³ = j and j⁴ = 1, we obtain hyperbolic Euler type identity as (36)

We observe that for all , indicating that e^jθ is π-periodic. This behavior is analogous to e^iϕ = e^i(ϕ+2nπ) = e^{i(ϕ mod 2π)}, which exhibits 2π-periodicity. In both cases, the periodic nature of the exponential function arises from the fundamental periodicity of the complex exponential, with the periodicity determined by the coefficient of the argument π or 2π, respectively. Moreover, from (4) and (5)

Example 1. Now, we consider an interesting example that will be useful in obtaining next result as follows: Let and , where r₁, r₂ > 0. Using new multiplication defined in (16) we obtain . Using (1) to (14), this can be written as g₁g₂ = r₁r₂[cosπ(θ) cos(ϕ) + i cosπ(θ) sin(ϕ) + j sinπ(θ)], and thus, (37)

Moreover, and , thus the behaviour of these two complex numbers is very similar.

Example 2. Consider two complex numbers and . The sum of these two complex numbers is expressed as . Now, let us explore two distinct scenarios:

(i) When r₁ = r₂ = r and ϕ₁ = ϕ₂ = ϕ implying . By simplifying this expression, we obtain: . Here, we have utilized the fact that j(sinπ(θ₁) + sinπ(θ₂)) × e^iϕ = j(sinπ(θ₁) + sinπ(θ₂)).

(ii) In the case where r₁ = r₂ = r and θ₁ = θ₂ = θ, we find that . Simplifying further, we obtain: .

From these two cases, it is evident that: (38) (39) Eqs (38) and (39) illustrate the distinction between the sum of complex numbers in the given cases, where e^iϕ distributes while e^jθ does not distribute.

Example 3. We consider 3D hypercomplex numbers and , where r₁, r₂ > 0. Their respective additive inverses are given by and . Under the multiplication operation defined in (16), we obtain the following results: , , , and . These expressions demonstrate that the multiplication (−g₁)g₂ ≠ g₁(−g₂), and g₁g₂ ≠ (−g₁)(−g₂). Further observations reveal that the multiplication exhibits standard properties: , and . These findings highlight the non-trivial nature of multiplication within this algebraic structure.

Using these observations we present the following result.

Result 2. The distributive property of the defined spherical multiplication over addition (i.e., g₁(g₂+ g₃) = g₁g₂+ g₁g₃) holds if , and . Using (38) we easily obtain .

Result 3. Consider a 3D hypercomplex number g = a + ib + jc, where a, b, c are real numbers. It follows that the exponential of g is given by: . The natural logarithm of e^g is given by: . Therefore, for any 3D hypercomplex number , we obtain (40) The expression (40) reduces to the conventional 2D complex number system if c = 0, leading to θ = 0. This result establishes a fundamental relationship between the exponential, logarithmic, and spherical polar forms of 3D hypercomplex numbers, extending the concepts from complex analysis to a higher-dimensional setting. The spherical polar representation offers a compact and geometric way to visualize and manipulate 3D hypercomplex numbers, analogous to the use of polar coordinates for complex numbers.

Thus, the proposed 3D hypercomplex number system is a true generalization of the existing 2D complex number system. To obtain the multiplication of two numbers, we can use Result 3 as follows: Let ln(g₁) = ln(r₁) + iϕ₁ + jθ₁ and ln(g₂) = ln(r₂) + iϕ₂ + jθ₂, and thus . Therefore, we conclude that the addition of hypercomplex numbers is naturally defined in the Cartesian coordinates and multiplication is naturally defined in the spherical coordinates through the natural logarithmic addition.

Building upon the constrained periodicity of the hypercomplex number in (7) and the unifying multiplication framework established in (16), we present a novel non-distributive normed division algebra that extends its reach to dimensions previously considered inaccessible. This advancement marks a significant milestone in the field, effectively addressing a longstanding challenge and providing a comprehensive framework for encompassing non-distributive normed division algebras across all dimensionalities. Crucially, the defined spherical multiplication (16) demonstrates seamless backward compatibility with the established complex number multiplication, thereby serving as a generalization of this fundamental operation to the realm of higher-dimensional hypercomplex number systems. To elucidate this compatibility and its broader implications, using the above representations, we present the subsequent results.

Theorem 1. A non-distributive normed division algebra (NDF) is a number system equipped with the fundamental arithmetic operations of addition, subtraction, multiplication, and division. It notably satisfies the norm condition ‖g₁g₂‖ = ‖g₁‖‖g₂‖ and has a dimension of M = 3. Furthermore, its algebraic structure exhibits distributivity for M = 1 and M = 2.

Proof. To establish the theorem, we proceed to demonstrate that the 3D numbers, defined in (1) and (7), constitute a non-distributive field under the operations of addition (15) and multiplication (16). We meticulously verify the following axioms for all elements , as specified in (14):

1. Closure of addition and multiplication: For all , it holds that , and . This is evident from the definitions of addition (15) and multiplication (16).
2. Associativity of addition and multiplication: Both g₁ + (g₂ + g₃) = (g₁ + g₂) + g₃ and yield expressions in and are evident from (15) and (16).
3. Commutativity of addition and multiplication: The commutative natures of g₁ + g₂ = g₂ + g₁ and are apparent from (15) and (16).
4. Additive and multiplicative identities: For every , there exist distinct elements 0 and 1 in , as defined in (28) and (29), such that g + 0 = g and g ⋅ 1 = g.
5. Additive and multiplicative inverses: For every , there exists an element called the additive inverse of g, such that g + g_ai = 0 where g = a + ib + jc and g_ai = −a − ib − jc. For every nonzero element , there exists an element g⁻¹ or 1/g in called the multiplicative inverse of g, such that g ⋅ g⁻¹ = 1. Here, as defined in (20), and is unique.
6. Distributivity of multiplication over addition: Generally, this property does not hold universally in as g₁ ⋅ (g₂ + g₃) ≠ (g₁ ⋅ g₂) + (g₁ ⋅ g₃) or . Specifically, if (implying θ₁ = 0), distributivity is observed in the representation as .

Having rigorously verified all the aforementioned axioms, except for general distributivity, we conclusively establish that the proposed 3D numbers indeed constitute a non-distributive field.

Remark 4. We can observe that the proposed set is an additive Abelian group, and is a multiplicative normed Abelian group (NAG). The set satisfies the fundamental group axioms along with the preservation of the norm, , as follows: (i) closure: , (ii) associativity: (g₁ ⋅ g₂) ⋅ g₃ = g₁ ⋅ (g₂ ⋅ g₃), (iii) identity: 1 ⋅ g = g, (iv) inverses: g ⋅ g⁻¹ = 1, (v) commutativity: g₁ ⋅ g₂ = g₂ ⋅ g₁, and (vi) preservation of the norm: ‖g₁ ⋅ g₂‖ = ‖g₁‖‖g₂‖.

Proposition 1. The following elementary consequences of the field axioms are also being satisfied by the proposed number systems,

1. (−1)[(−1)g] = g
2. (g⁻¹)⁻¹ = g
3. g₁ + g₂ = g₁ + g₃ ⇒ g₂ = g₃
4. g0 = 0
5. [(−1)g₁]g₂ = (−1)(g₁g₂)
6. [(−1)g₁][(−1)g₂] = g₁g₂
7. g₁g₂ = g₁g₃ and g₁ ≠ 0 implies g₂ = g₃
8. g₁g₂ = 0 ⇒ g₁ = 0 or g₂ = 0.

3.2 Examples of the proposed 3D hypercomplex numbers: Roots computation

To initiate the analysis, we consider the computation of n-th roots for a given polynomial, , using (21). The resulting n-th root is expressed as , where k, ℓ = 0, 1, ⋯, n − 1. This computational approach yields n² roots in , accounting for multiplicity. The systematic enumeration of these roots is accomplished by varying the parameters k and ℓ within the specified ranges k = 0, 1, ⋯, n − 1 and ℓ = 0, 1, ⋯, n − 1. Notably, the azimuth angle ϕ undergoes a modulo operation with respect to 2π, while the elevation angle θ is subject to a modulo operation of π. This mathematical treatment is essential for a comprehensive examination of the solution space, ensuring a thorough coverage of the entire domain within the prescribed intervals. The resulting set of roots manifests diverse configurations within the confines of the 3D hypercomplex numbers in , yielding a total of n² solutions when n is odd and, a total of n² or n² + n solutions when n is even. A presentation of illustrative examples is subsequently provided to elucidate the implications of this computational procedure.

Example 4. For example, let us consider the quadratic equation (QE) x² + 1 = 0. If , then there are no real roots. If (traditionally, ), then there are two roots x = ±i where i² = −1. If , then there are four roots as x² = e^i(π+2πk)e^j(πl) ⇒ x = e^i(π+2πk)/2e^j(πl)/2, for k, l = 0, 1. Therefore, four roots are ; where last two roots represent the same point in Z-axis. The other two roots are in the same Z-axis as which represent the same point in Z-axis. So there are total 6 roots of unity in 3D complex number system. However, it is worth to note that the polynomial, x² + 1 = 0, has infinite number of quaternion roots.

Example 5. Here, we consider x³ − 1 = 0, where and compute its roots as x³ = e^i2πke^jπl ⇒ x = e^i2πk/3e^jπl/3, where k, l = 0, 1, 2. Therefore, roots are ; ; where 2π/3 is same as 2π/3 − π = −π/3 for angle θ. So there are 9 distinct roots of unity in 3D complex number system.

Example 6. Here, we consider x⁴ − 1 = 0, where and compute its roots as x⁴ = e^i2πke^jπl ⇒ x = e^iπk/2e^jπl/4, where k, l = 0, 1, 2, 3. Therefore, roots are ; ;

; where 3π/4 is same as 3π/4 − π = −π/4 for angle θ. Other four roots are . So there are total 20 roots in 3D complex number system.

Example 7. Here, we consider x⁴ + 1 = 0, where and compute its roots as x⁴ = e^{i(π + 2πk)}e^j(πl) ⇒ x = e^{i(π + 2πk)/4}e^j(πl)/4, where k, l = 0, 1, 2, 3. Therefore, roots are ; ; ; . Other four roots are , so there are total 20 roots.

We can conclude the above observation as follows:

Result 4. The number of n- th roots of unity in are (i) n² if n is an odd, and (ii) n² + n if n is an even number.

3.3 Geometrical insights into the generalized hypercomplex number system

We note that algebraically, the additional imaginary axis j considered in behaves similar to i. For example, i² = −1 and j² = −1. Similarly, one can also show that (1 + j)² = 2j and (1 − j)² = −2j. Similar identities are satisfied by i. Moreover, this j axis geometrically plays interesteingly on the hypercomplex numbers. If there is a point in the complex XY plane and if it is multiplied by a unit norm complex number , then that point will rotate counterclockwise by φ, i.e., new point . Similarly, if a 3D point is multiplied by a unit norm point , then it will rotate to new point . Thus in the proposed 3D hypercomplex number system, one can rotate a point in both ϕ and θ directions with desired azimuth and elevation angles.

8.1 Illustrative simulation: Bloch sphere

The Bloch sphere, named after the physicist Felix Bloch, is a fundamental concept in quantum mechanics, particularly in quantum computing and quantum information theory. It provides a geometric representation that visually depicts the quantum states of a two-level quantum system, commonly known as a qubit. The Bloch sphere is a unit sphere where the poles represent the classical states and , and any point on the surface corresponds to a unique quantum state. The azimuthal and polar angles of a point on the Bloch sphere encode the probability amplitudes of the quantum state in the computational basis. This representation is invaluable for illustrating quantum operations, visualizing quantum gates, and understanding the dynamics of quantum systems. The Bloch sphere is a powerful tool for both pedagogical purposes and advanced research in quantum computing and quantum information science.

In the literature, the polar angle θ ∈ [0, π] is measured from the positive Z-axis. However, in our representation θ is the elevation angle measured from the XY plane as shown in Fig 7. The poles of the Bloch sphere typically represent the basis states of the qubit, often denoted as |0〉 and |1〉. A quantum state |ψ〉 can be represented as a superposition of pure states |0〉 and |1〉 as

(51)

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 7. The Bloch sphere representation, where the azimuth angle ϕ ∈ [0, 2π) is measured from the positive X axis and the elevation angle θ ∈ [−π/2, π/2] is measured from the XY plane.

The positive Z axis corresponds to the state |0〉, the negative Z axis corresponds to the state |1〉, and any point on the Bloch sphere can be uniquely represented by a pair of angles (ϕ, θ).

https://doi.org/10.1371/journal.pone.0312502.g007

Thus, the quantum state |ψ〉 is uniquely mapped to a point in the Bloch sphere that can be represented in SCS as (52)

Therefore, using (52), we can write quantum states in the Bloch sphere as (53) (54) (55) where set {|0〉, |1〉}, {|+〉, |−〉}, and {|i〉, |−i〉} represent the Z, X, and Y bases, respectively. Thus, for example, using the proposed multiplication (16) and division (17), we obtain and define valid multiplication of quantum states and their inverses as (56) (57) (58)

From (51) and (52), we observe that the designated pair of quantum states, identified by antipodal locations on the Bloch sphere, forms a basis. This basis can be expressed as (59) where the defining characteristic of these states lies in their distinct angular coordinates ϕ and θ. Notably, the well-established X, Y, and Z bases manifest as specific cases within this framework. Moreover, the global phase of quantum states lacks physical significance and observable consequences. Mathematically, if |ψ〉 represents a valid quantum state, then any state of the form e^iϕ|ψ〉 is considered physically equivalent to |ψ〉. Consequently, states such as , , and are deemed equivalent.

In quantum mechanics, any unit norm vector in the complex vector space represents a valid quantum state for a qubit, a fundamental principle in quantum theory. A unit vector in its associated Hilbert space describes the state of a quantum system, and for a qubit, the general quantum state form is |ψ〉 = α|0〉 + β|1〉. Here, α and β are complex numbers, while |0〉 and |1〉 are basis states. The condition ‖α‖² + ‖β‖² = 1 ensures normalization, making any unit norm vector in represent a valid quantum state for a qubit. The principle of superposition permits the combination of quantum states, resulting in a valid quantum state as long as the resulting vector is normalized. This normalization guarantees that the probabilities of all possible outcomes sum to 1. Mathematically, given two quantum states |ψ₁〉 and |ψ₂〉, their superposition and normalization remain valid for all such that . This consistency ensures that the probabilities associated with different measurement outcomes align with the principles of quantum mechanics.

We postulate that analogous to the principles of superposition and normalization, the proposed vector multiplication of two quantum states yields a valid quantum state. In mathematical terms, considering two given quantum states |ψ₁〉 and |ψ₂〉, their vector multiplication |ψ₃〉 = |ψ₁〉 × |ψ₂〉 is deemed a valid quantum state. In contrast to vector addition scenarios, normalization is unnecessary, as vector multiplication preserves the norm. The physical interpretation of multiplication can be considered as (i) Evolution: the operation |ψ₁〉 × |ψ₂〉 can represent the transition of a state |ψ₁〉 under the influence or operator |ψ₂〉, resulting in a new state |ψ₃〉, akin to time evolution in quantum mechanics. (ii) Interaction: If |ψ₁〉 and |ψ₂〉 represent states of two interacting quantum systems, |ψ₁〉×|ψ₂〉 represents the state resulting from their interaction, capturing the effects of entanglement or mutual influence. Traditional quantum mechanics relies on linear unitary evolution. Our nonlinear approach may offer new ways for state evolution, potentially uncovering novel insights or behaviors where linear approximations are insufficient.

There are a few physical interpretations that align with the nonlinear nature of the influence. We can imagine quantum states represented by vectors as magnetic moments or electric charges, with the influence vector representing an external magnetic or electric field, causing nonlinear changes in orientation similar to magnetic moments aligning with an external field. Quantum states could also represent the spin states of particles, where the influencing vector represents the spin of another particle, modeling interactions akin to spin-spin coupling with nonlinear dynamics. In quantum computation, the influencing vector can transform quantum states as qubits, simulating quantum gate operations with nonlinear characteristics that extend beyond standard unitary gates. Additionally, quantum states evolving in nonlinear potentials could be influenced by vectors representing aspects of the potential or interactions within the system, reflecting the nonlinear evolution relevant to fields like nonlinear optics or chaotic quantum systems. Finally, quantum states in open systems interacting with an environment, where the influencing vector represents environmental effects, can model the complex, nonlinear evolution of states under decoherence or dissipation using multiplication.

Moreover, since the spherical multiplication forms an Abelian group, preserves the norm, and provides an invertible nonlinear map, it has unique properties for representing certain types of invertible nonlinear quantum gates in quantum systems. This method represents an alternative avenue for representing a quantum state as the multiplication of given states, thereby serving as a potential complement to the prevailing superposition paradigm, which is scalar multiplication and vector addition. It is noteworthy to mention that a non-distributive scator algebra has previously been employed to model the evolution and collapse of quantum wave functions [17].

8.2 Visual representation: Point cloud image 1

A point cloud image is composed of a collection of N points within a three-dimensional (3D) coordinate system, where the geometric coordinates are designated as x, y, and z, and are stored in an N × 3 matrix as which is equivalent to set , and corresponding squared image is . The considered example, depicted in Fig 8, encompasses the following elements: (i) the top-left image in the top row, with dimensions 5184 × 3, along with its corresponding squared image on the right; and (ii) the bottom-left image in the bottom row, measuring 20402 × 3, which represents two spheres with radii r = 1 (outer sphere) and r = 1/2 (inner sphere). The squared image that depicts this arrangement is presented on the right. It is noteworthy that, in this representation, the radius of the outer sphere remains constant, while the radius of the inner sphere is modified to 1/4.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 8.

A point cloud image consist of a set of points in 3D coordinate system with geometric coordinates stored in N × 3 matrix (i) Top-row left image (5184×3) and its squared image on right. (ii) Bottom-row left image (20402 × 3) represents two spheres of radius r = 1 (outer sphere) and r = 1/2 (inner sphere) and its squared image on right where outer sphere radius remain same, and inner sphere radius changed to 1/4.

https://doi.org/10.1371/journal.pone.0312502.g008

8.3 Visual representation: Point cloud image 2

In this case, we delve into a set of point cloud images, specifically focusing on the top-row left image with dimensions 5184 × 3, i.e., with N = 5184. Subsequent images in the top row are generated by systematically varying the phase angle θ within the range [0, π], with intervals set at π/24, using the proposed multiplication method, i.e., with θ_ℓ = ℓ × π/24. This phased transformation results in a sequence of point cloud images, each corresponding to a distinct value of θ within the specified range. Intriguingly, the right-most image of the the bottom row is identical to the left-most image of the top row, as visually represented in Fig 9. This recurrence signifies a specific condition or point in the phased evolution of the point cloud, demonstrating the cyclical nature of the proposed multiplication process. The detailed exploration of this progression allows for a comprehensive understanding of the impact of varying the phase angle on the resultant point cloud images.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 9. A set of point cloud images: From the leftmost image (5184 × 3) of the top-row other images are obtained by clanging the phase angle θ ∈ [0, π] in step of π/24.

The right-most image of the bottom-row is the same as leftmost image of the top-row.

https://doi.org/10.1371/journal.pone.0312502.g009

8.4 Visual representation: Point cloud image 3

A collection of point cloud images representing a solid cube (132651 × 3) is depicted in Fig 10. Two distinct variations are explored:

1. (a) Commencing with the top-leftmost image in the top row, 24 additional images are generated by modulating the phase angle ϕ within the interval ϕ ∈ [0, 2π], with increments set at 2π/24, i.e., with ϕ_ℓ = ℓ × 2π/24. In particular, the rightmost image in the bottom row mirrors the leftmost image in the top row. The variation in ϕ signifies the rotation around the Z-axis that does not alter the shape of a three-dimensional object.
2. (b) Alternatively, the top left image in the top row undergoes modification by altering the phase angle θ within the range θ ∈ [0, π], with increments of π/24, producing 24 distinct images. In this case as well, the bottom-rightmost image in the bottom row corresponds to the top-leftmost image in the top row.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 10.

A set of point cloud images of solid cube (132651 × 3): (a) From the leftmost image in the top row, other images are obtained by varying the phase angle ϕ within the range ϕ ∈ [0, 2π], with intervals set at 2π/24. The rightmost image of the bottom row is identical to the leftmost image of the top row. (b) The leftmost image of the top-row is varied by changing the phase angle θ within the range θ ∈ [0, π] in increments of π/24 to produce other images. The right-most image of the bottom-row is identical to the left-most image of the top-row.

https://doi.org/10.1371/journal.pone.0312502.g010

8.5 Dynamic earth phenomena simulation

Our investigation yields a noteworthy observation, elucidating a correlation between the spherical coordinates system delineated by (3), as depicted in Fig 11, and dynamic Earth phenomena such as its rotation around the Z-axis and the flux dynamics of the magnetic field. Within this framework, a continual increase in the azimuth angle ϕ over a 2π period faithfully emulates the rotational motion of Earth along its axis, distinguishing between the north and south poles. Currently, the uninterrupted evolution of the elevation angle θ over a π period manifests the flow pattern of magnetic flux along longitudes. It is noteworthy to mention that in this representation the origin of the flux is at the south pole (θ = −π/2) and its termination is at the north pole (θ = π/2). This discerning analogy establishes a meaningful connection between the mathematical representation of spherical coordinates and the physical phenomena associated with the rotation of Earth and the flow of magnetic flux, thereby contributing to a nuanced understanding of their interrelation.

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 11. Spherical coordinates system representation by (3) where continuous increase in azimuth angle ϕ with period 2π depicts spin of Earth around Z-axis (north and south pole), and continuous increase in elevation angle θ with period π depicts flow of magnetic flux along longitudes with source at south pole (θ = −π/2) and sink at north pole (θ = π/2).

https://doi.org/10.1371/journal.pone.0312502.g011