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General Solution for Symmetry Numbers on Sphere

General Solution for Symmetry Numbers on Sphere
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2019-05-25 20:08
Geometric Construction
This example demonstrates the general solution for symmetry numbers on spheres. It involves the geometric properties and symmetry analysis of spheres. Using GeoGebra commands such as Circle, Point, Barycenter, Sphere, PointIn, and Segment, the construction of symmetric figures on the sphere is illustrated. Suitable for high school mathematics education and research.

GeoGebra Commands & Parameters

  1. Circle(Sun, a) → c
  2. Point(c) → N
  3. Point(c) → C
  4. Point(c) → L
  5. Point(c) → K
  6. Point(c) → J
  7. Point(c) → I
  8. Point(c) → H
  9. Point(c) → G
  10. Point(c) → F
  11. Point(c) → E
  12. Point(c) → D
  13. Point(c) → B
  14. Point(c) → A
  15. Point(c) → M
  16. Barycenter({A, B, C, D, E, F, G, H, I, J, K, L}, {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}) → Earth
  17. Sphere(Earth, a) → b
  18. PointIn(b) → Moon
  19. Segment(Moon, E) → f
  20. Segment(Moon, D) → g
  21. Segment(Moon, F) → h
  22. Segment(Moon, G) → i
  23. Segment(Moon, H) → j
  24. Segment(Moon, I) → k
  25. Segment(Moon, J) → l
  26. Segment(Moon, K) → m
  27. Segment(Moon, L) → n
  28. Segment(Moon, A) → p
  29. Segment(Moon, B) → r
  30. Segment(Moon, C) → q
  31. Point(c) → O

AI Drawing Prompt Reference

In 3D space, first draw a circle centered at the Sun with radius a, and place multiple points on its circumference, including the north pole, south pole, and twelve equally spaced points. Compute the barycenter of these points as the Earth's center, then construct a sphere with this center and the same radius a. Pick a random point inside the sphere as the Moon, and draw twelve line segments from the Moon to each of the twelve points on the circle. Finally, add one more point on the circle as a reference point to complete the symmetric structure.
The prompt above was compiled from the original drawing commands and is for reference only.