Omnidimensional Convex Polytopes

: The study shows that the volumes and surfaces of n -balls, n -simplices, and n -orthoplices are holomorphic functions of n , which makes those objects omnidimensional, that is well deﬁned in any complex dimension. Applications of these formulas to the omnidimensional polytopes inscribed in and circumscribed about n -balls reveal previously unknown properties of these geometric objects. In particular, for 0 < n < 1, the volumes of the omnidimensional polytopes are larger than those of circumscribing n -balls, and both their volumes and surfaces are smaller than those of inscribed n -balls. The surface of an n -simplex circumscribing a unit diameter n -ball is spirally convergent to zero with real n approaching negative inﬁnity but ﬁrst has a local maximum at n = − 3.5. The surface of an n -orthoplex circumscribing a unit diameter n -ball is spirally divergent with real n approaching negative inﬁnity but ﬁrst has a local minimum at n = − 1.5, where its real and imaginary parts are equal to each other; similarly, is its volume, where the similar local minimum occurs at n = − 3.5. Reﬂection functions for volumes and surfaces of these polytopes inscribed in and circumscribed about n -balls are proposed. Symmetries of products and quotients of the volumes in complex dimensions n and − n and of the surfaces in complex dimensions n and 2 − n are shown to be independent of the metric factor and the gamma function. Speciﬁc symmetries also hold between the volumes and surfaces in dimensions n = − 1/2 and n = 1/2.


Introduction
The notion of a dimension n is intuitively defined as a natural number of coordinates of a point within Euclidean space R n .However, this is not the only possible definition [1,2] of a dimension of a set.For example, analytic continuations from positive dimensions [2][3][4][5] can define negative dimensions [6].Fractional (or fractal), including negative [7], dimensions are consistent with experimental results and enable the examination of transport parameters in multiphase fractal media [8,9].This renders dimension n real, or at least a rational number.Complex [2], including complex fractional [10], dimensions can also be considered.Complex geodesic paths, for example, emerge in the presence of black hole singularities [11] and in entropic motion on curved statistical manifolds [12].Complex wavelengths occur in Maxwell-Boltzmann, and Fermi-Dirac statistics on black hole event horizons [13].Fractional derivatives of complex functions can describe different physical phenomena [14].For example, it was recently shown [15] that the magnetic monopole motion in a 3-simplicial spin ice crystal lattice is limited to a fractal cluster.
In n-dimensional space, n-dimensional objects have (n − 1)-dimensional surfaces which have a dimension of volume in (n − 1)-dimensional space.However, this sequence has a singularity at n = −1.A 0-dimensional point in 0-dimensional space has a vanishing (−1)-surface being a vanishing volume of the (−1)-dimensional void.But the surface of the (−1)-dimensional void is not (−2)-dimensional.It is undefined.This discontinuity hints that thinking about dimension in terms of a point on a number axis, where negative dimensions are analytic continuations from positive ones, may be misleading.Thinking of a dimension as a point on a number semiaxis, similar to a point on a radius with (−n)-dimension corresponding to n-dimension, seems more appropriate.Considering the dimension of a set as the length exponent at which that set can be measured makes the negative dimensions refer to densities as positive ones refer to quantities [5].Thus, the (−2)-dimensional pressure, for example, considered in terms of density (e.g., in units of N/m 2 ) corresponds to a 2-dimensional area (in units of m 2 ) that it acts upon.Following the same logic, gravitational force F = GMm/R 2 acting towards an inside of a 2-dimensional equipotential surface is (−2)-dimensional, whereas centripetal force F = mV 2 /R 1 acting towards an outside of a 1-dimensional curve is (−1)-dimensional.
The notion of distance intuitively defines how far apart two points are.Thus, intuition suggests that it is a non-negative quantity.However, intuition can be misleading (e.g., in aviation, where relying on a sense of orientation [i.e., relying on intuition] can be fatal), and the Euclidean distance admits not only the principal, non-negative, square root but also a negative one.This fact, taken plainly, violates the nonnegativity axiom of the metric.However, diffuse metrics [16], including the Łukaszyk-Karmowski metric [17], are known to violate the identity of the indiscernibles axiom of the metric.This hints that axiomatizing distance as a non-negative quantity may also be misleading.Complex effects (bivalence) extend real effects (classically one value), just as quantum theory extends classical physics.
Prior research [18] presented novel recurrence relations for volumes and surfaces of the three polytopes present in all natural dimensions [19].It was also conjectured in the prior research [18] that they were continuous for n ∈ R within domains of their definitions, whereas the starting points for fractional dimensions can be provided, e.g., using spline interpolation between two (or three, in the case of n-balls) subsequent integer dimensions.The present study shows that these recurrence relations allow removing indefiniteness present in known formulas, thus making them holomorphic functions of a complex dimension n and omnidimensional (i.e., well-defined ∀n ∈ C).
Refs. [13,[20][21][22] outline the framework for the physical explanation of the observed results.The paper is structured as follows.Section 2 summarizes known formulas and recurrence relations for volumes and surfaces of n-balls and regular, convex n-polytopes.Section 3 shows that these formulas and recurrence relations can be extended to all complex dimensions.Section 4 examines the properties of the omnidimensional, regular, convex polytopes inscribed in and circumscribed about n-balls and their reflection functions.Section 5 presents metric-independent relations between volumes and surfaces of these polytopes and n-balls, whereas Section 6 presents particular symmetries between volumes and surfaces for |n| = 1/2.Section 7 concludes and discusses the findings of this study, hinting at their possible applications.

Known Formulas
It is known that the volume of an n-ball (B) is where Γ : C → C is the Euler's gamma function, n ∈ C \ {n = −2k − 2, k ∈ N 0 }, and R denotes the n-ball radius.As the gamma function is meromorphic, volumes of n-balls are complex in complex dimensions.The volume of an n-ball can be expressed [23] in terms of the volume of an (n − 2)-ball of the same radius as a recurrence relation where V(0) B := 1 and V 1 (R) B := 2R.The relation (2) can be extended [18] into negative dimensions as solving Equation (2) for V(n − 2) and assigning a new n as the previous n − 2. A radius recurrence relation [18] (we choose the notation f (n) over f n , as n ∈ C) where n ∈ C, f (0) := 1 and f (1) := 2, allow for expressing the volume n-ball as where " x " is the floor function that yields the greatest integer less than or equal to its argument x.The relation ( 4) can be, analogously as Equation ( 2), extended [18] into negative dimensions as which allows us to define f (−1) = f (0) := 1 to initiate the sequences (4) or (6).The known [23] surface of an n-ball is We note in passing that in this study S(n) denotes the (n − 1)-dimensional surface of an n-dimensional object (that is a property of this object), and not the n-dimensional surface of the (n + 1)-dimensional object, as it is commonly defined in the literature.
The known volume of an n-cube (C) is where n ∈ C and A is the edge length.The known surface of an n-cube is where n ∈ C.
The known [24,25] volume of a regular n-simplex (S) is where n ∈ N. Equation (10) can be written [18] as a recurrence relation with V(0) S := 1, to remove the indefiniteness of the factorial for n < 1. Equation ( 11) can be solved for V(n − 1).Assigning new n as the previous n − 1, yields [18] which also removes the singularity for n = 0 present in the volume (10).An n-simplex has n + 1 (n − 1)-facets [23].Therefore, its surface is The known [23] volume of an n-orthoplex (O) is where n ∈ N 0 .Equation ( 14) can be written [18] as a recurrence relation where n = 0 and V(0) O := 1, to remove the indefiniteness of the factorial for n < 0. Solving Equation (15) for V(n − 1) and assigning new n as the previous n − 1, yields [18] which also removes singularity for n = 0 present in Equation (15).Any n-orthoplex has 2 n facets [23], which are regular (n − 1)-simplices.Therefore, its surface is

Holomorphizing Known Formulas
The known formulas presented in the preceding section can be extended to all complex dimensions.We note that volumes and surfaces of n-cubes are already defined ∀n ∈ C and thus are holomorphic.We also note that a square root is bivalued in the complex domain and by choosing complex analysis, we enter into bivalence by its very nature Therefore, volumes and surfaces of n-balls and regular, convex n-polytopes are also bivalued in complex dimensions n.We call the branch with positive values in positive real dimensions the principal branch.
Theorem 1.The volume of an n-ball is a holomorphic, bivalued function of a complex dimension n.
However, now we can use any of the backward recurrence relations (3) or ( 5) with ( 6) to determine the values of the n-ball volume outside this domain: ∀n On the other hand so the poles of the meromorphic gamma function Γ(n/2 + 1) present in Equation ( 22), now defined in the sense of a limit of a function, vanish, which completes the proof.
Corollary 1.The surface of an n-ball is a holomorphic, bivalued function of a complex dimension n.
Proof.If the volume of an n-ball is a holomorphic, bivalued function by Theorem 1, then, using Equation ( 7), the surface of an n-ball is also a holomorphic function.
Using Equations ( 22) and ( 7) the surface of an n-ball is given by Theorem 2. The volume of a regular n-simplex is a holomorphic, bivalued function of a complex dimension n.
Proof.Expressing the factorial in the volume of a regular n-simplex (10) by the gamma function extends its domain of applicability to complex dimensions On the other hand, comparing the volume (25) with the recurrence relation ( 12) yields which, after setting ñ := n + 1, yields (28) and recovers the volume (25) .Thus, we have proved that the recurrence relation ( 12) corresponds to the generalized n-simplex volume (25) within this domain.
However, now we can use the recurrence relation (12) to determine the values of the n-simplex volume outside this domain: ∀n + 1 ∈ C we can find V(n) S .For example, even though n √ n/ √ n 3 is undefined for n = 0, we can determine that V(0) S = ±1 using the recurrence relation (12) with V(1) S = ±A obtained from the volume (25).
On the other hand, so the poles of the meromorphic gamma function Γ(n + 1) present in the volume (25), now defined in the sense of a limit of a function, vanish, which completes the proof.
Corollary 2. The surface of a regular n-simplex is a holomorphic, bivalued function of a complex dimension n.
Proof.If the volume of a regular n-simplex is a holomorphic, bivalued function by Theorem 2, then its surface, using Equation ( 13), is also a holomorphic, bivalued function.Using ( 13) and ( 25), the surface of a regular n-simplex is given by (cf.Appendix A) 3 is undefined for n = 1, we can determine that S(1) S = ±2 directly from Equation ( 13) knowing that V(0) S = ±1.
Theorem 3. The volume of an n-orthoplex is a bivalued, holomorphic function of a complex dimension n.
Proof.Expressing the factorial in the volume of an n-orthoplex (14) with the gamma function extends the domain of applicability of Equation ( 14) to complex dimensions On the other hand, comparing Equation (31) with the recurrence relation ( 16) whereas setting ñ = n + 1 in Equation (33) yields which recovers n-orthoplex volume ( 14), as nΓ(n Thus, we have proved that the recurrence relation ( 16) corresponds to the general n-orthoplex volume (31) within this domain.
However, now we can use the recurrence relation (16) to determine the values of the n-orthoplex volume outside this domain: ∀n + 1 ∈ C we can find V(n) O .On the other hand so the poles of the meromorphic gamma function Γ(n + 1) present in (31), now defined in the sense of a limit of a function, vanish, which completes the proof.
Corollary 3. The surface of an n-orthoplex is a holomorphic, bivalued function of a complex dimension n.
Proof.If the volume of a regular n-simplex is a holomorphic, bivalued function by Theorem 2, then, using Equation ( 17), the surface of an n-orthoplex is a holomorphic, bivalued function.
Using Equations ( 17) and (31), the surface of an n-orthoplex is given by For n ∈ R \ Z : n < 0, n-orthoplex bivalued surface (36) is imaginary.Thus, the volumes and surfaces of n-balls, n-simplices, and n-orthoplices, shown in Figures 1 and 2 for n ∈ R, are holomorphic functions of n, which makes those objects omnidimensional.They can be explicitly expressed in terms of the complex dimension (cf.Appendix B).

Omnidimensional Polytopes Inscribed in and Circumscribed about n-Balls
First, we will introduce the definition of a reflection function.

Definition 1.
For each function F : C → C of a complex argument n given by the formula where j = 1, 2, . . ., l ∈ N, a j , b j , c j , d j ∈ R : a j ≥ 0, b j > 0, and g(n) is some nonzero function of n, we define its reflection function Proof.For n ∈ R, the functions (37) and (38) can be expressed as e (c j +d j n) ln(a j +b j n)/2 g(n), where θ n = atan2(0, n), and θ −n = atan2(−0, −n), as lim y→0 − atan2(y, x) = −π and lim y→0 + atan2(y, x) = π for x < 0. Without loss of generality, we consider positive branches of F(n) (40) and F(n) (41) We note here that ∀n ∈ R, such that n = 0 so that and that ∀n ∈ C and for the principal value of θ Therefore, Each regular, omnidimensional polytope discussed in Section 3 can be inscribed in and circumscribed about an n-ball.This section presents their volumes, surfaces, and reflection functions (38).Particular values of their volumes and surfaces are listed in Tables 1 and 2. Unless stated otherwise, n ∈ R and k ∈ N. Unit diameter n-balls are assumed.

Regular n-Simplices Inscribed in n-Balls
The diameter D BCS of an n-ball circumscribing a regular n-simplex (BCS) is known [25] to be where A is the edge length.Hence, the edge length A SIB of a regular n-simplex inscribed (SIB) in an n-ball (B) with a diameter D is so that the regular n-simplex volume (25) becomes For n / ∈ Z : n < −1, the inscribed n-simplex volume (49) is imaginary and divergent with n approaching negative infinity.It is complex for −1 < n < 0, with the real part equaling the imaginary part for n = −1/2.It is zero for n = −k, and for 0 < n < 1, it is larger than the volume of the circumscribing n-ball.The volume (49) does not have a reflection function (38), as d in the definition (39) vanishes.
Similarly, the surface (30) of a regular inscribed n-simplex with edge length A given by Equation (48) is as shown in Figure 4.For n / ∈ Z : n < −1 the inscribed n-simplex surface (50) is imaginary and divergent with n approaching negative infinity.It is complex for −1 < n < 0, with the real part equaling the imaginary part for n = −1/2.It is zero for n = −k, k ∈ N 0 .The surface (50) does not have a reflection function (38), as d in the definition (39) vanishes.
Volumes (49) and surfaces (50) of n-simplices inscribed in n-balls are shown in Figures 3 and 4.

Regular n-Simplices Circumscribed about n-Balls
The diameter D BIS of an n-ball inscribed in a regular n-simplex (BIS) is known [25] to be where A is the edge length.Hence, the edge length A SCB of a regular n-simplex circumscribed (SCB) about an n-ball (B) with a diameter D is so that its volume (25) becomes For n < 0, the volume ( 53) is complex with a nonzero imaginary part, whereas both branches are left-handed spirals converging to zero with n approaching negative infinity.For 0 < n < 1, the volume ( 53) is smaller than the volume of an inscribed n-ball.It is zero for n = −k, and real for n = −(2k + 1)/2, where it amounts to ≈ ≈ ±{−0.3549,−0.3359, 0.2996, 0.2626, −0.2283, . ..} The reflection function (38) of the volume (53) is and the volumes (56) and ( 53) are equal for n = (2k + 1)/2, k ∈ N 0 .Similarly, the surface (30) of a regular circumscribed n-simplex with edge length A SCB (52) is The surface (57) is zero for n = −k, k ∈ N 0 , and for 0 < n < 1, it is smaller than the surface of the inscribed n-ball.The surface (57) is complex for n < 0, with both branches being left-handed spirals converging towards zero with n approaching negative infinity.However, it is initially divergent to achieve a modulus maximum of about 2.9757 at n ≈ −3.4997 (numerically calculated; numerical calculations were performed in this study using Euler gamma function of complex argument defined for |n| < 172) and a real maximum of about 2.976 at n = −3.5.For n = −(2k + 1)/2 the imaginary part of the surface (57) vanishes and it amounts to ≈ ≈ ∓{2.1, 2.809, 2.976, 2.854, 2.588, . ..}. (58) The real part of the surface (57) is equal to the imaginary part in modulus for n = −1/2 The reflection function (38) of the surface (57 and it equals the volume (53) for n = (2k + 1)/2, k ∈ N 0 .Volumes (53) and surfaces (57) of n-simplices circumscribed about n-balls are shown in Figures 5 and 6.

n-Orthoplices Inscribed in n-Balls
The diameter D BCO of an n-ball circumscribing an n-orthoplex (BCO) is known [26] to be where A is the edge length.Hence, the edge length A OIB of an n-orthoplex inscribed in an n-ball (OIB) with a diameter D is so that its volume (31) becomes The inscribed n-orthoplex volume (63) is real, vanishes for n = −k, and for 0 < n < 1 it is larger than the volume of the circumscribing n-ball.It does not have a reflection function, as Equation (63) does not satisfy the definition (37).
Similarly, the surface (36) of the inscribed n-orthoplex with the edge length A given by Equation (62) becomes For n / ∈ Z : n < 0, the inscribed n-orthoplex surface (64) is imaginary and oscillatory divergent with n approaching negative infinity, and for n = −k it vanishes.It also does not have a reflection function (d = 0 in the definition (39)).
Volumes (63) and surfaces (64) of n-orthoplices inscribed in n-balls are shown in Figures 7 and 8.

n-Orthoplices Circumscribed about n-Balls
The diameter D BIO of an n-ball inscribed in an n-orthoplex (BIO) is known [26] to be where A is the edge length.Hence, the edge length A OCB of an n-orthoplex circumscribed about an n-ball (OCB) with a diameter D is so that its volume (31) becomes as shown in Figure 9.
The circumscribed n-orthoplex volume (67) is complex for n < 0. It oscillates and is initially convergent to achieve a modulus minimum of about 0.1181 at n ≈ −3.4976 (numerically computed) and then becomes divergent with n approaching negative infinity.For n = −k, it vanishes.For 0 < n < 1, it is smaller than the volume of an inscribed n-ball.
For n = −(2k + 1)/2, k ∈ N 0 the real part of the volume (67) equals the imaginary part in modulus, achieving a local minimum at n = −3.5 and amounts to The reflection function (38) of the volume (67) is and it equals the volume (67) for n = 2k, k ∈ N 0 .
Similarly, the surface (36) of a circumscribed n-orthoplex with the edge length A given by Equation (66) becomes as shown in Figure 10.The circumscribed n-orthoplex surface (70) is complex for n < 0. It oscillates and is initially convergent to achieve a modulus minimum of about 0.6244 at n ≈ −1.5 (numerical) and then becomes divergent with n approaching negative infinity.For n = −k, it vanishes.For 0 < n < 1, it is smaller than the surface of an inscribed n-ball.Furthermore, its real part equals the imaginary part in modulus for n = −(2k + 1)/2, k ∈ N 0 .It achieves a local minimum at n = −1.5 and amounts to (71) The reflection function (38) of the surface (70) is

n-Cubes Inscribed in and Circumscribed about n-Balls
The edge length A CCB of an n-cube circumscribed about an n-ball (CCB) corresponds to the diameter D of this n-ball.Thus, the volume of this cube is simply and the surface is However, the edge length A CIB of an n-cube inscribed in an n-ball (CIB) of diameter D is which is singular for n = 0 and complex for n < 0, rendering [18] the volume and the surface of an n-cube inscribed in an n-ball.
The volume (76) and the surface (77) are complex if n < 0 and divergent, with n approaching negative infinity.For 0 < n < 1, the volume (76) is larger than the volume of a circumscribing n-ball.

Discussion
The volumes and surfaces of n-balls, n-simplices, and n-orthoplices have been defined in any complex dimension.As these geometric objects occur, along with n-cubes, in all natural dimensions [19], this result makes them omnidimensional, i.e., present in any complex dimension.
Applications of these formulas to the omnidimensional polytopes inscribed in and circumscribed about n-balls revealed previously unknown properties of these geometric objects.In particular, for 0 < n < 1, the omnidimensional polytopes' volumes are larger than those of circumscribing n-balls, while their volumes and surfaces are smaller than those of inscribed n-balls.
Reflection functions for volumes and surfaces of these polytopes inscribed in and circumscribed about n-balls are proposed.
Symmetries of products ( 80)-( 85), ( 87)-(94) and quotients (96)-(98) of volumes of these circumscribed and inscribed omnidimensional polytopes and n-balls in complex dimensions n and −n and of surfaces in complex dimensions n and 2 − n are shown to be independent of the metric factor and the gamma function.
Specific symmetries also hold between volumes and surfaces in dimensions n = −1/2 and n = 1/2.
The results of this study could be applied in linguistic statistics, where the dimension in the distribution for frequency dictionaries is chosen to be negative [3], and in fog computing, where an n-simplex is related to a full mesh pattern, an n-orthoplex is linked to a quasi-full mesh structure, and an n-cube is referred to as a certain type of partial mesh layout [27].Further possible applications include molecular physics and crystallography.Perhaps the results of this study are also related to the 2-dimensional quantum hall effect.
One can also investigate the properties of the examined geometric objects extending these concepts to quaternions.This can help us discover many interesting mathematical properties and physical phenomena.One of the challenges certainly involves characterizing all types of symmetries introduced by Definition 1.
In the case of n-cubes, for example, the volume ( 8) is equal to

Theorem 4 .
The same branches of functions F(n) (37) and F(n) (38) are equal to each other for n ≤ 0 or for n ∈ Z such that sin c+dn 2 π = 0, with c, d defined by (39).

Table 1 .
Particular volumes of omnidimensional polytopes inscribed in and circumscribed about unit diameter n-balls (principal branch).

Table 2 .
Particular surfaces of omnidimensional polytopes inscribed in and circumscribed about unit diameter n-balls (principal branch).