# The Shape Entropy of Small Bodies

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## Abstract

**:**

## 1. Introduction

_{x}≤ I

_{y}≤ I

_{z}. According to Equation (1), the index is ρ ∈ [0, 1]. When ρ = 0, the shape of the small body is symmetric about the z-axis; when ρ = 1, the shape is symmetric about the x-axis. This shape index can describe the mass distribution characteristics of small bodies and reflect the shape of small bodies to a certain extent. However, when the shape of a small body is close to a sphere, that is, the three-axis inertias are very close, this index cannot accurately describe the shape characteristics of a small body, especially the approximation between the small body and the sphere.

_{20}, C

_{22}, and S

_{22}[18], we can find that these three coefficients still reflect the relationship between the inertia of small bodies.

## 2. Shape Entropy in the 2D Continuous Cases

#### 2.1. Definition

_{s}(θ) is a single-valued function, and the denominator part depicts the area of the plane geometry, making

_{s}(θ) ≡ a, thus

#### 2.2. Regular Polygons

_{n}} tends to the shape entropy of a circle log(2π) = 1.83788… when n tends to +∞. Note that entropy in statistical physics describes the concentration of states, and the value of entropy is the largest when the probabilities of all states are equal. Thus, when r

_{s}are equal, corresponding to the most regular case, the shape entropy in 2D cases is the largest.

#### 2.3. Rectangles and Ellipses

## 3. Shape Entropy in the 3D Continuous Cases

#### 3.1. Definition

_{s}(θ, ϕ) is a single-valued function, and the denominator part depicts the volume of the spatial geometry, making

_{s}(θ, φ) ≡ a, Equation (19) is transformed as

#### 3.2. Regular Polyhedrons

#### 3.3. Cuboids and Triaxial Ellipsoids

## 4. Shape Entropy Applied to Polyhedral Models of Small Bodies

#### 4.1. Definition

#### 4.2. Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## References

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**Figure 4.**The shape entropies of non-convex shapes are shown in Figure 3, provided that b = 1, a = 1, 1.5, $\sqrt{3}$, 2, and c ∈ [0.1, 1].

**Figure 9.**The shape entropies of cuboids with different edge length ratios of a and b, provided that c = 1. The blue area suggests a more irregular shape. The lower left corner corresponds to the regular hexahedron.

**Figure 10.**The shape entropies of ellipsoids with different axial length ratios of a and b, provided that c = 1. The blue area suggests a more irregular shape. The lower left corner corresponds to the sphere.

**Figure 11.**The differences in shape entropies of ellipsoids and cuboids with different axial length ratios. The blue area suggests less difference. Since it is assumed that c = 1 for all shapes shown, diagonal line a = b corresponds to flat shapes, and axis a or b corresponds to slender shapes.

**Figure 13.**Polyhedral models corresponding to points 1–4 in Figure 12. (

**a**) 52,760 (1998 ML

_{14}). (

**b**) 101,955 Bennu. (

**c**) 1998 KY

_{26}. (

**d**) 4 Vesta.

**Figure 14.**Polyhedral models corresponding to points 5–8 in Figure 12. (

**a**) 9P/Tempel. (

**b**) 6489 Golevka. (

**c**) 3103 Eger. (

**d**) 951 Gaspra.

**Figure 15.**Polyhedral models corresponding to points 9–12 in Figure 12. (

**a**) 433 Eros. (

**b**) 216 Kleopatra. (

**c**) 243 Ida. (

**d**) 103P/Hartley.

Number of Sides of Regular Polygons | Shape Entropy S |
---|---|

3 | 1.74557… |

4 | 1.81549… |

5 | 1.82964… |

6 | 1.83412… |

… | … |

∞ | log(2π) = 1.83788… |

a:b | Rectangle | Ellipse |

3:1 | 1.49387… | 1.55019… |

2:1 | 1.68228… | 1.72009… |

1.5:1 | 1.76905… | 1.79706… |

1:1 | 1.81549... | log(2π) = 1.83788… |

Number of Faces of Regular Polyhedrons | Shape Entropy S |
---|---|

4 | 2.60889… |

6 | 2.73379… |

8 | 2.82407... |

Spherical case | log(2π) + 1 = 2.83788... |

a:b:c | Cuboid | Ellipsoid |
---|---|---|

3:2:1 | 2.15642… | 2.29111… |

2:2:1 | 2.37094… | 2.48964… |

2:1.5:1 | 2.43230… | 2.55064… |

1:1:1 | 2.73379.... | log(2π) + 1 = 2.83788… |

Name of Small Bodies | Shape Entropy S | Vertices | Faces | ρ from Equation (1) | No. in Figure 12 |
---|---|---|---|---|---|

52760 (1998 ML_{14}) | −0.001118600 | 8162 | 16,320 | 0.877608 | 1 |

101955 Bennu | −0.001171272 | 1348 | 2692 | 0.320574 | 2 |

1998 KY_{26} | −0.001469927 | 2048 | 4092 | 0.823293 | 3 |

4 Vesta | −0.003386096 | 2522 | 5040 | 0.165797 | 4 |

9P/Tempel | −0.008419353 | 16,022 | 32,040 | 0.779807 | 5 |

6489 Golevka | −0.011491285 | 2048 | 4092 | 0.964472 | 6 |

3103 Eger | −0.013905315 | 997 | 1990 | 0.648360 | 7 |

951 Gaspra | −0.019957635 | 2522 | 5040 | 0.914083 | 8 |

4769 Castalia | −0.028763986 | 2048 | 4092 | 0.896695 | / |

2063 Bacchus | −0.034839183 | 2048 | 4092 | 0.986248 | / |

25143 Itokawa | −0.039069503 | 25,350 | 49,152 | 0.932418 | / |

1P/Halley | −0.039881773 | 2522 | 5040 | 0.934006 | / |

1620 Geographos | −0.042576975 | 8192 | 16,380 | 0.942497 | / |

4486 Mithra | −0.049464462 | 3000 | 5996 | 0.860466 | / |

1996 HW_{1} | −0.057551792 | 1392 | 2780 | 0.973871 | / |

433 Eros | −0.060992619 | 99,846 | 196,608 | 0.978736 | 9 |

216 Kleopatra | −0.074191101 | 2048 | 4092 | 0.990365 | 10 |

243 Ida | −0.085757437 | 2522 | 5040 | 0.883693 | 11 |

103P/Hartley | −0.098676873 | 16,022 | 32,040 | 0.975002 | 12 |

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**MDPI and ACS Style**

Ni, Y.; Zhang, H.; Li, J.; Baoyin, H.; Hu, J.
The Shape Entropy of Small Bodies. *Mathematics* **2023**, *11*, 878.
https://doi.org/10.3390/math11040878

**AMA Style**

Ni Y, Zhang H, Li J, Baoyin H, Hu J.
The Shape Entropy of Small Bodies. *Mathematics*. 2023; 11(4):878.
https://doi.org/10.3390/math11040878

**Chicago/Turabian Style**

Ni, Yanshuo, He Zhang, Junfeng Li, Hexi Baoyin, and Jiaye Hu.
2023. "The Shape Entropy of Small Bodies" *Mathematics* 11, no. 4: 878.
https://doi.org/10.3390/math11040878