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Gnomonic Growth of the Nautilus

Gnomonic Growth of the Nautilus
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2016-12-02 04:50
AngleVectorComplex NumberTransformationParametric Equation
In collections Logarithmic Spiral and Fibonacci Numbers
This example illustrates the gnomonic growth of a nautilus shell. Key mathematical concepts include similarity transformation, vectors, parametric curves, and complex numbers. Using GeoGebra commands such as Segment, Rotate, Angle, Ray, Circle, Intersect, and Polygon, the growth trajectory of the nautilus shell is constructed. Suitable for high school mathematics education and research.

GeoGebra Commands & Parameters

  1. Segment(A, B) → f
  2. Rotate(A, (-δ), B) → A'
  3. Angle(A', B, A) → α
  4. Rotate(B, ζ, A) → B'
  5. Angle(B, A, B') → β
  6. Ray(A, B') → g
  7. Ray(B, A') → h
  8. Circle(A, a) → c_2
  9. Intersect(c_2, g) → D,
  10. Circle(B, b) → d_1
  11. Intersect(d_1, h) → C,
  12. Rotate(C, β, D) → C'
  13. Ray(D, C') → j
  14. Polygon(A1) → C1
  15. Rotate(D, (-α), C) → D'
  16. Angle(D', C, D) → γ
  17. Rotate(C, β, D) → C'_1
  18. Angle(C, D, C'_1) → ε
  19. Ray(C, D') → i
  20. Ray(D, C'_1) → k
  21. Circle(C, (Distance[B, C] * B1)) → c
  22. Circle(D, (Distance[A, D] * B1)) → d
  23. Polygon(A2) → C2
  24. Polygon(A3) → C3
  25. Polygon(A4) → C4
  26. Polygon(A5) → C5
  27. Polygon(A6) → C6
  28. Polygon(A7) → C7
  29. Polygon(A8) → C8
  30. Polygon(A9) → C9
  31. Polygon(A10) → C10
  32. Polygon(A11) → C11
  33. Polygon(A12) → C12
  34. Polygon(A13) → C13
  35. Polygon(A14) → C14
  36. Polygon(A15) → C15
  37. Polygon(A16) → C16
  38. Polygon(A17) → C17
  39. Polygon(A18) → C18
  40. Polygon(A19) → C19
  41. Polygon(A20) → C20
  42. Polygon(A21) → C21
  43. Polygon(A22) → C22
  44. Polygon(A23) → C23
  45. Polygon(A24) → C24
  46. Polygon(A25) → C25
  47. Polygon(A26) → C26
  48. Polygon(A27) → C27
  49. Polygon(A28) → C28
  50. Polygon(A29) → C29
  51. Polygon(A30) → C30
  52. Polygon(A31) → C31
  53. Polygon(A32) → C32
  54. Polygon(A33) → C33
  55. Polygon(A34) → C34
  56. Polygon(A35) → C35
  57. Polygon(A36) → C36