# Novel Recurrence Relations for Volumes and Surfaces of n-Balls, Regular n-Simplices, and n-Orthoplices in Real Dimensions

## Abstract

**:**

## 1. Introduction

^{n}, where n = −1 is the dimension of the empty set, the void, having zero volume and undefined surface. Negatively dimensional spaces can be defined by analytic continuations from positive dimensions [3]. A spectrum, a topological generalization of the notion of space, allows for negative dimensions [2,4,5,6] that refer to densities, rather than to sizes as in the natural ones.

^{2}, there is a countably infinite number of regular, convex polygons; in ℝ

^{3}, there are five regular, convex Platonic solids; in ℝ

^{4}, there are six regular, convex polytopes. For n > 4, there are only three: self-dual n-simplex and n-cube dual to n-orthoplex [19]. Furthermore, ℝ

^{n}is also equipped with a perfectly regular, convex n-ball. The properties of these three regular, convex polytopes in natural dimensions are well known [20,21,22]. Fractal dimensions of hyperfractals based on these polytopes in natural dimensions were disclosed in [23].

## 2. Known Formulas

_{0}) and

^{+}[2,24].

_{0}(R)

_{B}= 1 and V

_{1}(R)

_{B}= 2R. It is also known [21] that the (n − 1)-dimensional surface of an n-ball can be expressed as

^{n}facets [21], being regular (n − 1)-simplices, its surface is

## 3. Novel Recurrence Relations

_{0}, where f

_{0}:= 1 and f

_{1}:= 2, allows for expressing the volumes and, using (5), surfaces of n-balls as

**Proof.**

_{0}, then by equating (2) with (12)

^{k}k!

^{k}

^{−1}(k − 1)!), which completes the proof. □

_{n}or nf

_{n}, the irrational factor π^⌊n/2⌋, and the metric (radius) factor R

^{n}or R

^{n}

^{−1}. The relation (11) can be extended into negative dimensions as

_{n}

_{−2}and assigning new n ∈ ℤ as old n − 2. Thus, it is sufficient to define f

_{−1}:= 1, f

_{0}:= 1 (for the empty set and point dimension) to initiate (11) and (16).

_{n}

_{−2}(R)

_{B}, yielding

**Proof.**

_{n}(16) is listed in Table 1 for n ∈ ℤ, and shown in Figure 1 along with the even algebraic form of f

_{n}(19), odd algebraic form of f

_{n}, (20) and the π^⌊n/2⌋ factor for n ∈ ℂ (for complex numbers ⌊a + bi⌋ = ⌊a⌋ + ⌊b⌋i). As shown, (19) and (20) bound the relation (16) for Re(n). Volumes and surfaces of n-balls calculated with (12) and (13) are shown in Figure 2.

**Proof.**

_{n}= 0 for negative, even n.

**Proof.**

^{k}π

^{−k−1}= π

^{−1}. Otherwise, set n = 2k ± ε, where 0 < ε ≤ 1, ε ∈ ℝ. For n = 2k + ε

**Proof.**

_{−2k}= 0 for k ∈ ℕ. Also Re(i

^{n}

^{−1}) = 0 and Im(i

^{n}

^{−1}) = ±1, as n is even.

_{−1}= 1). For instance, for k = {0, 2}

**Proof.**

^{k}π

^{1−k}= π. Otherwise, set n = 2k ± ε, where 0 < ε ≤ 1, ε ∈ ℝ. For n = 2k + ε

**Proof.**

_{−1}= 1

**Proof.**

^{k}π

^{−k}= 1. Otherwise, set n = 2k ± ε, where 0 < ε ≤ 1, ε ∈ ℝ. For n = 2k + ε

_{−1}:= 2 and g

_{0}:= 1. The diameter recurrence relation (31), (32) is related to radius recurrence relation (11), (16) by

**Proof.**

_{n}(32) is listed in Table 1 for n ∈ ℤ, and shown in Figure 3 along with the even algebraic form of g

_{n}((19) with (33)) the odd algebraic form of g

_{n}((20) with (33)), and the π^⌊n/2⌋ factor for n ∈ ℂ. Volumes and surfaces of n-balls calculated with relations (29) and (30) are shown in Figure 4.

_{0}(A)

_{S}:= 1

**Proof.**

_{n}

_{−1}and assigning new n ∈ ℤ as old n − 1, yields

_{−1}(A)

_{S}= 0, while its surface S

_{−1}(A)

_{S}(8) is undefined, as for the void itself.

_{0}(A)

_{O}:= 1.

**Proof.**

_{n}

_{−1}and assigning new n ∈ ℤ as old n − 1, yields

## 4. n-Balls Circumscribed about and Inscribed in n-Cubes

_{CC}of an n-cube circumscribed (CC) about an n-ball corresponds to the diameter D of this n-ball. Thus, the volume of this cube is simply V

_{n}(D)

_{CC}= D

^{n}, and the surface is S

_{n}(D)

_{CC}= 2nD

^{n}

^{−1}.

_{CI}of an n-cube inscribed (CI) inside an n-ball of diameter D is A

_{CI}= D/√n, which is singular for n = 0 and complex for n < 0. Thus, the volume of an n-cube inscribed in an n-ball is

^{0}:= 1), and complex if n < 0, n ∈ ℝ. To examine reflection relations we set m = −n in (39) and (40). This yields volume

**Proof.**

**Proof.**

## 5. Summary

_{n}(11) enables us to express the known recurrence relation (4) for n-ball volume and the known relation (5) for n-ball surface as a function of π^⌊n/2⌋, showing that the value of π as n-ball volume and surface irrational factor appears only for n < 0 and n ≥ 2 (π^⌊n/2⌋ = 1 for 0 ≤ n < 2).

_{−2}= 0, in negative, even dimensions, n-balls have zero (void-like) volumes and zero (point-like) surfaces and become divergent with decreasing n. Curiously, the double factorial n‼ can be extended to negative, odd integers by inverting its recurrence relation and is not defined for negative even integers.

_{n}> f

_{n}

_{−1}, while n = 7 (the largest unit radius n-ball surface) is the smallest odd n where f

_{n}< f

_{n}

_{−1}. The diameter recurrence relation g

_{n}(32) is related with (16) by f

_{n}= 2

^{n}g

_{n}.

^{n}[27], and by researching Euclidean space ℝ

^{n}as a simplicial n-manifold, topological (metric-independent) and geometrical (metric-dependent) content of the modeled quantities are disentangled [27]. Therefore, the lack of n-simplices in the negative, integer dimensions excludes the notion of negatively dimensional Euclidean space ℝ

^{n}for n < −1. Volumes and surfaces of regular n-simplices are imaginary in negative, fractional dimensions for n < −1 (surfaces also for n < 0) and are divergent with decreasing n.

_{1}(A)

_{O}= A√2 not A, as in the case of 1-simplex and 1-cube.

_{n}(A)

_{C}= A

^{n}and surface S

_{n}(A)

_{C}= 2nA

^{n}

^{−1}= 2dV

_{n}(A)

_{C}/dA of an n-cube are defined for any n ∈ ℝ, and are real if A ∈ ℝ. Interestingly, in ℝ

^{3}, the fractal dimension of the Sierpiński 3-simplex is 2, of the Sierpiński 3-orthoplex is 2.585, while only the Sierpiński 3-cube retains its regular dimension [28].

## 6. Discussion

_{0}A

^{1/3}, where A is the atomic number and r

_{0}= 1.25 ± 0.2 fm.

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**n-ball radius recurrence relation f

_{n}for n ∈ ℤ (blue); even (yellow) and odd (black) algebraic forms of f

_{n}, and the π^⌊n/2⌋ factor (green); for −7 ≤ n ≤ 7, n ∈ ℂ.

**Figure 2.**Graphs of volumes (V) and surface areas (S) of n-balls of unit radius for n = −25, −24, …, 15.

**Figure 3.**n-ball diameter recurrence relation g

_{n}for n ∈ ℤ (blue); even (yellow) and odd (black) algebraic forms of g

_{n}, and the π^⌊n/2⌋ factor (green) for −4 ≤ n ≤ 4, n ∈ ℂ.

**Figure 4.**Graphs of volumes (V) and surface areas (S) of n-balls of unit diameter for n = −10, −9, …, 8.

**Figure 5.**Graphs of volumes (V) and surface areas (S) of regular n-simplices of unit edge length for n = −1, …, 7.

**Figure 6.**Graphs of volumes (V) and surface areas (S) of n-orthoplices of unit edge length for n = −1, …, 7.

**Figure 7.**Graphs of volumes ((

**a**,

**b**), pink) and surfaces ((

**c**,

**d**), blue) of unit radius n-balls, along with volumes and surface areas of n-cubes circumscribed about (yellow) and inscribed in (green, black) these n-balls.

n | f_{n} | g_{n} | V_{n} (R = 1)_{B} | S_{n} (R = 1)_{B} | V_{n} (D = 1)_{B} | S_{n} (D = 1)_{B} |
---|---|---|---|---|---|---|

−11 | −945/32 | −60,480 | −0.031 | 0.338 | −62.909 | 1383.997 |

−9 | 105/16 | 3360 | 0.021 | −0.193 | 10.980 | −197.634 |

−7 | −15/8 | −240 | −0.019 | 0.135 | −2.464 | 34.494 |

−5 | 3/4 | 24 | 0.024 | −0.121 | 0.774 | −7.7404 |

−3 | −1/2 | −4 | −0.051 | 0.152 | −0.405 | 2.432 |

−1 | 1 | 2 | 0.318 | −0.318 | 0.637 | −1.273 |

0 | 1 | 1 | 1 | 0 | 1 | 0 |

1 | 2/1 | 1 | 2 | 2 | 1 | 2 |

2 | 1/1 | 1/4 | 3.142 | 6.283 | 0.785 | 3.142 |

3 | 4/3 | 1/6 | 4.189 | 12.566 | 0.524 | 3.142 |

4 | ½ | 1/32 | 4.935 | 19.739 | 0.308 | 2.467 |

5 | 8/15 | 1/60 | 5.264 | 26.319 | 0.164 | 1.645 |

6 | 1/6 | 1/384 | 5.168 | 31.006 | 0.081 | 0.969 |

7 | 16/105 | 1/840 | 4.725 | 33.073 | 0.037 | 0.517 |

8 | 1/24 | 1/6144 | 4.059 | 32.470 | 0.016 | 0.254 |

9 | 32/945 | 1/15,120 | 3.299 | 29.687 | 0.006 | 0.116 |

**Table 2.**Volumes and surfaces of n-cubes inscribed in n-balls of unit radius and diameter for −8 ≤ n ≤ 3 (rational fraction approximation using Matlab rats function).

n | V_{n}(R = 1)_{CI} | S_{n} (R = 1)_{CI} | V_{n} (D = 1)_{CI} | S_{n} (D = 1)_{CI} |
---|---|---|---|---|

−8 | 16 | −362.0387i | 4096 | −185,363.8i |

−7 | −7.0898i | −16,807/128 | −907.4927i | −33,614 |

−6 | −27/8 | 49.6022i | −216 | 6349.077i |

−5 | 1.7469i | 625/32 | 55.9017i | 1250 |

−4 | 1 | −8i | 16 | −256i |

−3 | −0.6495i | −27/8 | −5.1961i | −54 |

−2 | −1/2 | i√2 | −2 | 8i√2 |

−1 | i/2 | 1/2 | i | 2 |

0 | 1 | 0 | 1 | 0 |

1 | 2 | 2 | 1 | 2 |

2 | 2 | 4√2 | ½ | 2√2 |

3 | 8 × 3^{−3/2} | 8 | 3^{−3/2} | 2 |

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**MDPI and ACS Style**

Łukaszyk, S. Novel Recurrence Relations for Volumes and Surfaces of *n*-Balls, Regular *n*-Simplices, and *n*-Orthoplices in Real Dimensions. *Mathematics* **2022**, *10*, 2212.
https://doi.org/10.3390/math10132212

**AMA Style**

Łukaszyk S. Novel Recurrence Relations for Volumes and Surfaces of *n*-Balls, Regular *n*-Simplices, and *n*-Orthoplices in Real Dimensions. *Mathematics*. 2022; 10(13):2212.
https://doi.org/10.3390/math10132212

**Chicago/Turabian Style**

Łukaszyk, Szymon. 2022. "Novel Recurrence Relations for Volumes and Surfaces of *n*-Balls, Regular *n*-Simplices, and *n*-Orthoplices in Real Dimensions" *Mathematics* 10, no. 13: 2212.
https://doi.org/10.3390/math10132212