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Jeener's Klein Surface

Jeener's Klein Surface
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2020-02-02 19:12
FunctionSolid GeometryCalculusParametric EquationGeometric Construction
In collections Color 3D surfaces
This example demonstrates the Jeener's Klein Surface, focusing on its continuity and the use of normal distribution, congruence, and piecewise functions. It involves constructing the surface using GeoGebra commands like Surface, parameters n, a, b, c, and objects X, Y, Z, O, A1-A26, B1-B4. Ideal for high school and advanced mathematics education, exploring parametric surfaces and calculus concepts.

GeoGebra Commands & Parameters

  1. Surface(X(u, v), Y(u, v), Z(u, v), u, 0, (2 * π), v, 0, (2 * π)) → O
  2. Surface(X(u, v), Y(u, v), Z(u, v), u, A1, A2, v, 0, (2 * π)) → B1
  3. Surface(X(u, v), Y(u, v), Z(u, v), u, A2, A3, v, 0, (2 * π)) → B2
  4. Surface(X(u, v), Y(u, v), Z(u, v), u, A3, A4, v, 0, (2 * π)) → B3
  5. Surface(X(u, v), Y(u, v), Z(u, v), u, A4, A5, v, 0, (2 * π)) → B4
  6. Surface(X(u, v), Y(u, v), Z(u, v), u, A5, A6, v, 0, (2 * π)) → B5
  7. Surface(X(u, v), Y(u, v), Z(u, v), u, A6, A7, v, 0, (2 * π)) → B6
  8. Surface(X(u, v), Y(u, v), Z(u, v), u, A7, A8, v, 0, (2 * π)) → B7
  9. Surface(X(u, v), Y(u, v), Z(u, v), u, A8, A9, v, 0, (2 * π)) → B8
  10. Surface(X(u, v), Y(u, v), Z(u, v), u, A9, A10, v, 0, (2 * π)) → B9
  11. Surface(X(u, v), Y(u, v), Z(u, v), u, A10, A11, v, 0, (2 * π)) → B10
  12. Surface(X(u, v), Y(u, v), Z(u, v), u, A11, A12, v, 0, (2 * π)) → B11
  13. Surface(X(u, v), Y(u, v), Z(u, v), u, A12, A13, v, 0, (2 * π)) → B12
  14. Surface(X(u, v), Y(u, v), Z(u, v), u, A13, A14, v, 0, (2 * π)) → B13
  15. Surface(X(u, v), Y(u, v), Z(u, v), u, A14, A15, v, 0, (2 * π)) → B14
  16. Surface(X(u, v), Y(u, v), Z(u, v), u, A15, A16, v, 0, (2 * π)) → B15
  17. Surface(X(u, v), Y(u, v), Z(u, v), u, A16, A17, v, 0, (2 * π)) → B16
  18. Surface(X(u, v), Y(u, v), Z(u, v), u, A17, A18, v, 0, (2 * π)) → B17
  19. Surface(X(u, v), Y(u, v), Z(u, v), u, A18, A19, v, 0, (2 * π)) → B18
  20. Surface(X(u, v), Y(u, v), Z(u, v), u, A19, A20, v, 0, (2 * π)) → B19
  21. Surface(X(u, v), Y(u, v), Z(u, v), u, A20, A21, v, 0, (2 * π)) → B20
  22. Surface(X(u, v), Y(u, v), Z(u, v), u, A21, A22, v, 0, (2 * π)) → B21
  23. Surface(X(u, v), Y(u, v), Z(u, v), u, A22, A23, v, 0, (2 * π)) → B22
  24. Surface(X(u, v), Y(u, v), Z(u, v), u, A23, A24, v, 0, (2 * π)) → B23
  25. Surface(X(u, v), Y(u, v), Z(u, v), u, A24, A25, v, 0, (2 * π)) → B24
  26. Surface(X(u, v), Y(u, v), Z(u, v), u, A25, A26, v, 0, (2 * π)) → B25
  27. Surface(X(u, v), Y(u, v), Z(u, v), u, A26, A27, v, 0, (2 * π)) → B26
  28. Surface(X(u, v), Y(u, v), Z(u, v), u, A27, A28, v, 0, (2 * π)) → B27
  29. Surface(X(u, v), Y(u, v), Z(u, v), u, A28, A29, v, 0, (2 * π)) → B28
  30. Surface(X(u, v), Y(u, v), Z(u, v), u, A29, A30, v, 0, (2 * π)) → B29
  31. Surface(X(u, v), Y(u, v), Z(u, v), u, A30, A31, v, 0, (2 * π)) → B30
  32. Surface(X(u, v), Y(u, v), Z(u, v), u, A31, A32, v, 0, (2 * π)) → B31
  33. Surface(X(u, v), Y(u, v), Z(u, v), u, A32, A33, v, 0, (2 * π)) → B32
  34. Surface(X(u, v), Y(u, v), Z(u, v), u, A33, A34, v, 0, (2 * π)) → B33
  35. Surface(X(u, v), Y(u, v), Z(u, v), u, A34, A35, v, 0, (2 * π)) → B34
  36. Surface(X(u, v), Y(u, v), Z(u, v), u, A35, A36, v, 0, (2 * π)) → B35
  37. Surface(X(u, v), Y(u, v), Z(u, v), u, A36, A37, v, 0, (2 * π)) → B36
  38. Surface(X(u, v), Y(u, v), Z(u, v), u, A37, A38, v, 0, (2 * π)) → B37
  39. Surface(X(u, v), Y(u, v), Z(u, v), u, A38, A39, v, 0, (2 * π)) → B38
  40. Surface(X(u, v), Y(u, v), Z(u, v), u, A39, A40, v, 0, (2 * π)) → B39
  41. Surface(X(u, v), Y(u, v), Z(u, v), u, A40, A41, v, 0, (2 * π)) → B40
  42. Surface(X(u, v), Y(u, v), Z(u, v), u, A41, A42, v, 0, (2 * π)) → B41
  43. Surface(X(u, v), Y(u, v), Z(u, v), u, A42, A43, v, 0, (2 * π)) → B42
  44. Surface(X(u, v), Y(u, v), Z(u, v), u, A43, A44, v, 0, (2 * π)) → B43
  45. Surface(X(u, v), Y(u, v), Z(u, v), u, A44, A45, v, 0, (2 * π)) → B44
  46. Surface(X(u, v), Y(u, v), Z(u, v), u, A45, A46, v, 0, (2 * π)) → B45
  47. Surface(X(u, v), Y(u, v), Z(u, v), u, A46, A47, v, 0, (2 * π)) → B46
  48. Surface(X(u, v), Y(u, v), Z(u, v), u, A47, A48, v, 0, (2 * π)) → B47
  49. Surface(X(u, v), Y(u, v), Z(u, v), u, A48, A49, v, 0, (2 * π)) → B48
  50. Surface(X(u, v), Y(u, v), Z(u, v), u, A49, A50, v, 0, (2 * π)) → B49
  51. Surface(X(u, v), Y(u, v), Z(u, v), u, A50, A51, v, 0, (2 * π)) → B50
  52. Surface(X(u, v), Y(u, v), Z(u, v), u, A51, A52, v, 0, (2 * π)) → B51
  53. Surface(X(u, v), Y(u, v), Z(u, v), u, A52, A53, v, 0, (2 * π)) → B52

AI Drawing Prompt Reference

Draw a closed parametric surface defined by continuous functions, incorporating normal distribution and piecewise definitions. The surface is divided into multiple patches along one parameter direction, each covering a subinterval, while maintaining continuity across boundaries. The overall shape resembles a Klein surface variant with intricate folds and symmetry. Construct the full surface first, then subdivide it into segments to reveal local structure and smooth connections.
The prompt above was compiled from the original drawing commands and is for reference only.