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Striped Klein Bottle

Striped Klein Bottle
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2020-11-19 00:34
TrigonometrySolid GeometryTransformation
This example demonstrates a striped Klein bottle, a non-orientable, non-Euclidean three-dimensional topological structure. It involves key mathematical concepts such as angles, trigonometric functions, and geometric transformations. Using the Surface command, multiple objects like c, a, b, etc., are constructed to achieve this complex geometric shape. Suitable for high school students and math enthusiasts.

GeoGebra Commands & Parameters

  1. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, 0, π) → klein
  2. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A1, A2) → B1
  3. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A2, A3) → B2
  4. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A3, A4) → B3
  5. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A4, A5) → B4
  6. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A5, A6) → B5
  7. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A6, A7) → B6
  8. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A7, A8) → B7
  9. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A8, A9) → B8
  10. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A9, A10) → B9
  11. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A10, A11) → B10
  12. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A11, A12) → B11
  13. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A12, A13) → B12
  14. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A13, A14) → B13
  15. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A14, A15) → B14
  16. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A15, A16) → B15
  17. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A16, A17) → B16
  18. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A17, A18) → B17
  19. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A18, A19) → B18
  20. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A19, A20) → B19
  21. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A20, A21) → B20
  22. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A21, A22) → B21
  23. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A22, A23) → B22
  24. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A23, A24) → B23
  25. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A24, A25) → B24
  26. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A25, A26) → B25
  27. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A26, A27) → B26
  28. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A27, A28) → B27
  29. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A28, A29) → B28
  30. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A29, A30) → B29
  31. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A30, A31) → B30
  32. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A31, A32) → B31
  33. Surface(If[0 ≤ u < π, ((c * cos(u)) * (a + sin(u))) + (((4 * (1 - cos(u) / b)) * cos(u)) * cos(v)), ((c * cos(u)) * (a + sin(u))) + ((4 * (a - cos(u) / b)) * cos(v + π))], If[0 ≤ u < π, (16 * sin(u)) + (((4 * (a - cos(u) / b)) * sin(u)) * cos(v)), (16 * sin(u))], ((4 * (a - cos(u) / b)) * sin(v)), u, 0, (2 * π), v, A32, A33) → B32

AI Drawing Prompt Reference

Draw a striped Klein bottle, a non-orientable, non-Euclidean three-dimensional topological structure. The shape consists of multiple surfaces, each generated using trigonometric functions and geometric transformations to create a continuous striped pattern. Start by drawing the base surface, then sequentially add multiple surfaces, adjusting parameters to ensure each connects smoothly with adjacent surfaces, forming a complete Klein bottle.
The prompt above was compiled from the original drawing commands and is for reference only.