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Klein Bottle Non-Orientable Surface Demo

Klein Bottle Non-Orientable Surface Demo
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2022-09-06 11:58
Solid GeometryCalculusDerivativeTransformationParametric Equation
This example demonstrates the 3D visualization and properties of the Klein bottle, a non-orientable surface. Key math concepts include surfaces, parametric equations, derivatives, and transformations. GeoGebra commands and objects used in the construction are Surface, Derivative, If, Translate, Mod, with elements like Ay, Ax, c, b, a, size, H1, G1, F1, Klein1, etc. Suitable for high school students, university beginners, and enthusiasts interested in topology for educational purposes and concept comprehension.

GeoGebra Commands & Parameters

  1. Surface(F1(u, v), G1(u, v), H1(u, v), u, pi, (2 * pi), v, 0, (2 * pi)) → Klein1
  2. Surface(F0(u, v), G0(u, v), H0(u, v), u, 0, pi, v, 0, (2 * pi)) → Klein0
  3. Derivative(F0, x) → dF0x
  4. Derivative(F0, y) → dF0y
  5. Derivative(G0, x) → dG0x
  6. Derivative(G0, y) → dG0y
  7. Derivative(H0, x) → dH0x
  8. Derivative(H0, y) → dH0y
  9. Derivative(F1, x) → dF1x
  10. Derivative(F1, y) → dF1y
  11. Derivative(G1, x) → dG1x
  12. Derivative(G1, y) → dG1y
  13. Derivative(H1, x) → dH1x
  14. Derivative(H1, y) → dH1y
  15. If(Ax < pi, (uN0x(Ax, Ay), uN0y(Ax, Ay), uN0z(Ax, Ay)), If[Ax < (2 * pi), (uN1x(Ax, Ay), uN1y(Ax, Ay), uN1z(Ax, Ay)), If[Ax < (3 * pi), ((-uN0x(Ax, Ay)), (-uN0y(Ax, Ay)), (-uN0z(Ax, Ay))), ((-uN1x(Ax, Ay)), (-uN1y(Ax, Ay)), (-uN1z(Ax, Ay)))]]) → C
  16. If(Ax < pi, Klein0(Ax, Ay), Klein1(Ax, Ay)) → A
  17. Translate(Vector[(0, 0, 0), C], A) → u
  18. Mod(((1 * 4) * pi / n) + Ax, (4 * pi)) → A1
  19. Mod(((2 * 4) * pi / n) + Ax, (4 * pi)) → A2
  20. Mod(((3 * 4) * pi / n) + Ax, (4 * pi)) → A3
  21. Mod(((4 * 4) * pi / n) + Ax, (4 * pi)) → A4
  22. Mod(((5 * 4) * pi / n) + Ax, (4 * pi)) → A5
  23. If(A1 < pi, Klein0(A1, pi / 2), Klein1(A1, pi / 2)) → B1
  24. If(A2 < pi, Klein0(A2, pi / 2), Klein1(A2, pi / 2)) → B2
  25. If(A3 < pi, Klein0(A3, pi / 2), Klein1(A3, pi / 2)) → B3
  26. If(A4 < pi, Klein0(A4, pi / 2), Klein1(A4, pi / 2)) → B4
  27. If(A5 < pi, Klein0(A5, pi / 2), Klein1(A5, pi / 2)) → B5
  28. If(A1 < pi, (uN0x(A1, pi / 2), uN0y(A1, pi / 2), uN0z(A1, pi / 2)), If[A1 < (2 * pi), (uN1x(A1, pi / 2), uN1y(A1, pi / 2), uN1z(A1, pi / 2)), If[A1 < (3 * pi), ((-uN0x(A1, pi / 2)), (-uN0y(A1, pi / 2)), (-uN0z(A1, pi / 2))), ((-uN1x(A1, pi / 2)), (-uN1y(A1, pi / 2)), (-uN1z(A1, pi / 2)))]]) → C1
  29. If(A2 < pi, (uN0x(A2, pi / 2), uN0y(A2, pi / 2), uN0z(A2, pi / 2)), If[A2 < (2 * pi), (uN1x(A2, pi / 2), uN1y(A2, pi / 2), uN1z(A2, pi / 2)), If[A2 < (3 * pi), ((-uN0x(A2, pi / 2)), (-uN0y(A2, pi / 2)), (-uN0z(A2, pi / 2))), ((-uN1x(A2, pi / 2)), (-uN1y(A2, pi / 2)), (-uN1z(A2, pi / 2)))]]) → C2
  30. If(A3 < pi, (uN0x(A3, pi / 2), uN0y(A3, pi / 2), uN0z(A3, pi / 2)), If[A3 < (2 * pi), (uN1x(A3, pi / 2), uN1y(A3, pi / 2), uN1z(A3, pi / 2)), If[A3 < (3 * pi), ((-uN0x(A3, pi / 2)), (-uN0y(A3, pi / 2)), (-uN0z(A3, pi / 2))), ((-uN1x(A3, pi / 2)), (-uN1y(A3, pi / 2)), (-uN1z(A3, pi / 2)))]]) → C3
  31. If(A4 < pi, (uN0x(A4, pi / 2), uN0y(A4, pi / 2), uN0z(A4, pi / 2)), If[A4 < (2 * pi), (uN1x(A4, pi / 2), uN1y(A4, pi / 2), uN1z(A4, pi / 2)), If[A4 < (3 * pi), ((-uN0x(A4, pi / 2)), (-uN0y(A4, pi / 2)), (-uN0z(A4, pi / 2))), ((-uN1x(A4, pi / 2)), (-uN1y(A4, pi / 2)), (-uN1z(A4, pi / 2)))]]) → C4
  32. If(A5 < pi, (uN0x(A5, pi / 2), uN0y(A5, pi / 2), uN0z(A5, pi / 2)), If[A5 < (2 * pi), (uN1x(A5, pi / 2), uN1y(A5, pi / 2), uN1z(A5, pi / 2)), If[A5 < (3 * pi), ((-uN0x(A5, pi / 2)), (-uN0y(A5, pi / 2)), (-uN0z(A5, pi / 2))), ((-uN1x(A5, pi / 2)), (-uN1y(A5, pi / 2)), (-uN1z(A5, pi / 2)))]]) → C5
  33. Translate(Vector[(0, 0, 0), C1], B1) → D1
  34. Translate(Vector[(0, 0, 0), C2], B2) → D2
  35. Translate(Vector[(0, 0, 0), C3], B3) → D3
  36. Translate(Vector[(0, 0, 0), C4], B4) → D4
  37. Translate(Vector[(0, 0, 0), C5], B5) → D5
  38. Mod(((6 * 4) * pi / n) + Ax, (4 * pi)) → A6
  39. Mod(((7 * 4) * pi / n) + Ax, (4 * pi)) → A7
  40. Mod(((8 * 4) * pi / n) + Ax, (4 * pi)) → A8
  41. Mod(((9 * 4) * pi / n) + Ax, (4 * pi)) → A9
  42. Mod(((10 * 4) * pi / n) + Ax, (4 * pi)) → A10
  43. Mod(((11 * 4) * pi / n) + Ax, (4 * pi)) → A11
  44. Mod(((12 * 4) * pi / n) + Ax, (4 * pi)) → A12
  45. Mod(((13 * 4) * pi / n) + Ax, (4 * pi)) → A13
  46. Mod(((14 * 4) * pi / n) + Ax, (4 * pi)) → A14
  47. Mod(((15 * 4) * pi / n) + Ax, (4 * pi)) → A15
  48. If(A6 < pi, Klein0(A6, pi / 2), Klein1(A6, pi / 2)) → B6
  49. If(A7 < pi, Klein0(A7, pi / 2), Klein1(A7, pi / 2)) → B7
  50. If(A8 < pi, Klein0(A8, pi / 2), Klein1(A8, pi / 2)) → B8
  51. If(A9 < pi, Klein0(A9, pi / 2), Klein1(A9, pi / 2)) → B9
  52. If(A10 < pi, Klein0(A10, pi / 2), Klein1(A10, pi / 2)) → B10
  53. If(A11 < pi, Klein0(A11, pi / 2), Klein1(A11, pi / 2)) → B11
  54. If(A12 < pi, Klein0(A12, pi / 2), Klein1(A12, pi / 2)) → B12
  55. If(A13 < pi, Klein0(A13, pi / 2), Klein1(A13, pi / 2)) → B13
  56. If(A14 < pi, Klein0(A14, pi / 2), Klein1(A14, pi / 2)) → B14
  57. If(A15 < pi, Klein0(A15, pi / 2), Klein1(A15, pi / 2)) → B15
  58. If(A6 < pi, (uN0x(A6, pi / 2), uN0y(A6, pi / 2), uN0z(A6, pi / 2)), If[A6 < (2 * pi), (uN1x(A6, pi / 2), uN1y(A6, pi / 2), uN1z(A6, pi / 2)), If[A6 < (3 * pi), ((-uN0x(A6, pi / 2)), (-uN0y(A6, pi / 2)), (-uN0z(A6, pi / 2))), ((-uN1x(A6, pi / 2)), (-uN1y(A6, pi / 2)), (-uN1z(A6, pi / 2)))]]) → C6
  59. If(A7 < pi, (uN0x(A7, pi / 2), uN0y(A7, pi / 2), uN0z(A7, pi / 2)), If[A7 < (2 * pi), (uN1x(A7, pi / 2), uN1y(A7, pi / 2), uN1z(A7, pi / 2)), If[A7 < (3 * pi), ((-uN0x(A7, pi / 2)), (-uN0y(A7, pi / 2)), (-uN0z(A7, pi / 2))), ((-uN1x(A7, pi / 2)), (-uN1y(A7, pi / 2)), (-uN1z(A7, pi / 2)))]]) → C7
  60. If(A8 < pi, (uN0x(A8, pi / 2), uN0y(A8, pi / 2), uN0z(A8, pi / 2)), If[A8 < (2 * pi), (uN1x(A8, pi / 2), uN1y(A8, pi / 2), uN1z(A8, pi / 2)), If[A8 < (3 * pi), ((-uN0x(A8, pi / 2)), (-uN0y(A8, pi / 2)), (-uN0z(A8, pi / 2))), ((-uN1x(A8, pi / 2)), (-uN1y(A8, pi / 2)), (-uN1z(A8, pi / 2)))]]) → C8
  61. If(A9 < pi, (uN0x(A9, pi / 2), uN0y(A9, pi / 2), uN0z(A9, pi / 2)), If[A9 < (2 * pi), (uN1x(A9, pi / 2), uN1y(A9, pi / 2), uN1z(A9, pi / 2)), If[A9 < (3 * pi), ((-uN0x(A9, pi / 2)), (-uN0y(A9, pi / 2)), (-uN0z(A9, pi / 2))), ((-uN1x(A9, pi / 2)), (-uN1y(A9, pi / 2)), (-uN1z(A9, pi / 2)))]]) → C9
  62. If(A10 < pi, (uN0x(A10, pi / 2), uN0y(A10, pi / 2), uN0z(A10, pi / 2)), If[A10 < (2 * pi), (uN1x(A10, pi / 2), uN1y(A10, pi / 2), uN1z(A10, pi / 2)), If[A10 < (3 * pi), ((-uN0x(A10, pi / 2)), (-uN0y(A10, pi / 2)), (-uN0z(A10, pi / 2))), ((-uN1x(A10, pi / 2)), (-uN1y(A10, pi / 2)), (-uN1z(A10, pi / 2)))]]) → C10
  63. If(A11 < pi, (uN0x(A11, pi / 2), uN0y(A11, pi / 2), uN0z(A11, pi / 2)), If[A11 < (2 * pi), (uN1x(A11, pi / 2), uN1y(A11, pi / 2), uN1z(A11, pi / 2)), If[A11 < (3 * pi), ((-uN0x(A11, pi / 2)), (-uN0y(A11, pi / 2)), (-uN0z(A11, pi / 2))), ((-uN1x(A11, pi / 2)), (-uN1y(A11, pi / 2)), (-uN1z(A11, pi / 2)))]]) → C11
  64. If(A12 < pi, (uN0x(A12, pi / 2), uN0y(A12, pi / 2), uN0z(A12, pi / 2)), If[A12 < (2 * pi), (uN1x(A12, pi / 2), uN1y(A12, pi / 2), uN1z(A12, pi / 2)), If[A12 < (3 * pi), ((-uN0x(A12, pi / 2)), (-uN0y(A12, pi / 2)), (-uN0z(A12, pi / 2))), ((-uN1x(A12, pi / 2)), (-uN1y(A12, pi / 2)), (-uN1z(A12, pi / 2)))]]) → C12
  65. If(A13 < pi, (uN0x(A13, pi / 2), uN0y(A13, pi / 2), uN0z(A13, pi / 2)), If[A13 < (2 * pi), (uN1x(A13, pi / 2), uN1y(A13, pi / 2), uN1z(A13, pi / 2)), If[A13 < (3 * pi), ((-uN0x(A13, pi / 2)), (-uN0y(A13, pi / 2)), (-uN0z(A13, pi / 2))), ((-uN1x(A13, pi / 2)), (-uN1y(A13, pi / 2)), (-uN1z(A13, pi / 2)))]]) → C13
  66. If(A14 < pi, (uN0x(A14, pi / 2), uN0y(A14, pi / 2), uN0z(A14, pi / 2)), If[A14 < (2 * pi), (uN1x(A14, pi / 2), uN1y(A14, pi / 2), uN1z(A14, pi / 2)), If[A14 < (3 * pi), ((-uN0x(A14, pi / 2)), (-uN0y(A14, pi / 2)), (-uN0z(A14, pi / 2))), ((-uN1x(A14, pi / 2)), (-uN1y(A14, pi / 2)), (-uN1z(A14, pi / 2)))]]) → C14
  67. If(A15 < pi, (uN0x(A15, pi / 2), uN0y(A15, pi / 2), uN0z(A15, pi / 2)), If[A15 < (2 * pi), (uN1x(A15, pi / 2), uN1y(A15, pi / 2), uN1z(A15, pi / 2)), If[A15 < (3 * pi), ((-uN0x(A15, pi / 2)), (-uN0y(A15, pi / 2)), (-uN0z(A15, pi / 2))), ((-uN1x(A15, pi / 2)), (-uN1y(A15, pi / 2)), (-uN1z(A15, pi / 2)))]]) → C15
  68. Translate(Vector[(0, 0, 0), C6], B6) → D6
  69. Translate(Vector[(0, 0, 0), C7], B7) → D7
  70. Translate(Vector[(0, 0, 0), C8], B8) → D8
  71. Translate(Vector[(0, 0, 0), C9], B9) → D9
  72. Translate(Vector[(0, 0, 0), C10], B10) → D10
  73. Translate(Vector[(0, 0, 0), C11], B11) → D11
  74. Translate(Vector[(0, 0, 0), C12], B12) → D12
  75. Translate(Vector[(0, 0, 0), C13], B13) → D13
  76. Translate(Vector[(0, 0, 0), C14], B14) → D14
  77. Translate(Vector[(0, 0, 0), C15], B15) → D15
  78. Surface((1.01 * F0(u, v)), (1.01 * G0(u, v)), (1.01 * H0(u, v)), u, 0, pi, v, pi / 2 - 0.2, pi / 2 + 0.2) → Klein00
  79. Surface((1.01 * F1(u, v)), (1.01 * G1(u, v)), (1.01 * H1(u, v)), u, pi, (2 * pi), v, pi / 2 - 0.2, pi / 2 + 0.2) → Klein01

AI Drawing Prompt Reference

Draw a 3D Klein bottle, a non-orientable surface. The key objects include parametric equations, derivatives, and transformations. The construction order is: first define parametric equations, then compute derivatives, then use conditional logic and modular arithmetic to generate different parts, and finally translate. This presents a topological solid suitable for understanding surface transformations.
The prompt above was compiled from the original drawing commands and is for reference only.