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Mapping Diagram for Moebius Function

Mapping Diagram for Moebius Function
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2016-01-03 13:22
FunctionCoordinate GeometryComplex Number
In collections Mapping Diagrams to Visualize Complex Analysis
This example visualizes the Moebius function with lines and circles. The Moebius function is a significant function in number theory, used to determine the prime factorization of integers. Using commands like Sequence, PointIn, Vector, Zip, f_mabc, CurveCartesian, Textfield, Line, Point in GeoGebra, the construction demonstrates the mapping of the function. Suitable for high school mathematics education, particularly in coordinate geometry and number theory courses.

GeoGebra Commands & Parameters

  1. Sequence((r * (cos(t) + (sin(t) * ί))), t, 0, 6.28319, 0.5236) → z_*
  2. PointIn(p_0) → A
  3. PointIn(p_h) → B
  4. Vector(A, B) → u
  5. Zip(f_mabc[t, a, b, c], t, z_*) → list2
  6. f_mabc(z_#, a, b, c) → f_{z#}
  7. Vector(a, fz_{z#^n}) → v
  8. Sequence((x(Element[list2, k]), y(Element[list2, k]), h), k, 1, 12) → list4
  9. Sequence(Vector[Element[z_*, j], Element[list4, j]], j, 1, 12) → list5
  10. CurveCartesian((m * cos(t)), (m * sin(t)), h, t, 0, 6.283185307179586) → d
  11. Textfield(list2) → inputBox1
  12. Vector(z_#, fz_{z#^n}) → w
  13. Line((0, 0), (1, m)) → f
  14. Point(f) → C
  15. Vector(C) → z_x
  16. ToComplex(C) → z_w
  17. f_mabc(z_w, a, b, c) → z_L
  18. ToPoint(z_L) → Z_L
  19. Locus(Z_L, C) → g
  20. Locus(Z3_L, C) → G
  21. Sequence(t + ((m * t) * ί), t, -4, 4, 0.5) → Linelist
  22. Zip(f_mabc[t, a, b, c], t, Linelist) → Linelist2
  23. Sequence((x(Element[Linelist2, k]), y(Element[Linelist2, k]), h), k, 1, 12) → Linelist4
  24. Sequence(Vector[Element[Linelist, j], Element[Linelist4, j]], j, 1, 12) → Linelist5
  25. Circle((0, 0), r) → k
  26. Vector(C, Z3_L) → j
  27. Point(k) → D
  28. ToComplex(D) → z_D
  29. f_mabc(z_D, a, b, c) → f_{zD}
  30. Locus(Z3_D, D) → H
  31. Vector(D, Z3_D) → l