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Dynamic Geometric Illustration by Tim Brzezinski (4)

Dynamic Geometric Illustration by Tim Brzezinski (4)
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2017-12-26 14:33
TrigonometryCoordinate GeometryComplex NumberGeometric Construction
This example dynamically demonstrates the connection between complex numbers and geometric transformations, visually presenting mathematical principles through changing graphics. Core concepts include complex numbers, linear equations, trigonometric functions, and geometric transformations. Construction highlights use of GeoGebra commands like polygon, segment, circle, intersect, midpoint, line, dilation, and rotation. Suitable for high school mathematics students and teachers for teaching demonstrations and exploration.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C, D) → q1, a, b, c, d
  2. Segment(D, A) → f
  3. Segment(A, B) → g
  4. Segment(B, C) → h
  5. Segment(C, D) → i
  6. Circle(D, f / 3) → e
  7. Intersect(e, f, 1) → E
  8. Circle(E, f / 3) → k
  9. Intersect(k, f) → F, G
  10. Circle(A, g / 3) → p
  11. Intersect(p, g) → H,
  12. Midpoint(H, B) → I
  13. Circle(B, h / 3) → q
  14. Intersect(q, h) → J,
  15. Midpoint(J, C) → K
  16. Circle(C, i / 3) → r
  17. Intersect(r, i) → L,
  18. Midpoint(L, D) → M
  19. Line(E, K) → j
  20. Line(G, J) → l
  21. Line(H, M) → m
  22. Line(I, L) → n
  23. Intersect(j, m) → N
  24. Intersect(l, m) → O
  25. Intersect(n, j) → P
  26. Intersect(l, q1) → Q, R, Intersect(l, q1)_1, Intersect(l, q1)_2
  27. Intersect(n, l) → S
  28. Dilate(A, If[0 < o ≤ 1, 1 - o, 0], H) → A'
  29. Dilate(D, If[0 < o ≤ 1, 1 - o, 0], E) → D'
  30. Dilate(B, If[0 < o ≤ 1, 1 - o, 0], R) → B'
  31. Dilate(C, If[0 < o ≤ 1, 1 - o, 0], L) → C'
  32. Segment(C, C') → s
  33. Segment(D, D') → t
  34. Segment(A, A') → f_1
  35. Segment(B, B') → g_1
  36. Rotate(B, If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], J) → B'_1
  37. Rotate(B', If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], J) → B''
  38. Segment(B'_1, B'') → h_1
  39. Rotate(C, If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], L) → C'_1
  40. Rotate(C', If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], L) → C''
  41. Segment(C'_1, C'') → s'
  42. Rotate(D, If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], E) → D'_1
  43. Rotate(D', If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], E) → D''
  44. Segment(D'_1, D'') → t'
  45. Rotate(A, If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], H) → A'_1
  46. Rotate(A', If[1 < o ≤ 2, ((-(o - 1)) * π), (-π)], H) → A''
  47. Segment(A'_1, A'') → i_1
  48. Rotate(C'_1, If[2 < o ≤ 3, ((-(o - 2)) * π), (-π)], M) → T
  49. Rotate(C'', If[2 < o ≤ 3, ((-(o - 2)) * π), (-π)], M) → C'''
  50. Segment(T, C''') → s''
  51. Rotate(D'_1, If[2 < o ≤ 3, ((-(o - 2)) * π), (-π)], G) → U
  52. Rotate(D'', If[2 < o ≤ 3, ((-(o - 2)) * π), (-π)], G) → D'''
  53. Segment(U, D''') → t''
  54. Rotate(A'_1, If[2 < o ≤ 3, ((-(o - 2)) * π), (-π)], I) → V
  55. Rotate(A'', If[2 < o ≤ 3, ((-(o - 2)) * π), (-π)], I) → A'''
  56. Segment(V, A''') → j_1
  57. Rotate(B'_1, If[2 < o ≤ 3, ((-(o - 2)) * π), (-π)], K) → W
  58. Rotate(B'', If[2 < o ≤ 3, ((-(o - 2)) * π), (-π)], K) → B'''
  59. Segment(W, B''') → k_1
  60. Dilate(E, If[4 < o ≤ 5, 5 - o, 0], N) → E'
  61. Dilate(M, If[4 < o ≤ 5, 5 - o, 0], N) → M'
  62. Dilate(G, If[4 < o ≤ 5, 5 - o, 0], O) → G'
  63. Dilate(L, If[4 < o ≤ 5, 5 - o, 0], P) → L'
  64. Segment(E, E') → l_1
  65. Segment(G, G') → m_1
  66. Segment(M, M') → n_1
  67. Segment(L, L') → p_1
  68. Dilate(N, If[5 < o ≤ 6, 6 - o, 0], P) → N'
  69. Dilate(O, If[5 < o ≤ 6, 6 - o, 0], S) → O'
  70. Dilate(N, If[5 < o ≤ 6, 6 - o, 0], O) → N'_1
  71. Dilate(P, If[5 < o ≤ 6, 6 - o, 0], S) → P'
  72. Segment(N, N'_1) → q_1
  73. Segment(N, N') → r_1
  74. Segment(P, P') → s_1
  75. Segment(O, O') → t_1
  76. Dilate(P, If[6 < o ≤ 7, 7 - o, 0], K) → P'_1
  77. Dilate(S, If[6 < o ≤ 7, 7 - o, 0], J) → S'
  78. Dilate(O, If[6 < o ≤ 7, 7 - o, 0], H) → O'_1
  79. Dilate(S, If[6 < o ≤ 7, 7 - o, 0], I) → S'_1
  80. Segment(P, P'_1) → a_1
  81. Segment(S, S') → b_1
  82. Segment(S, S'_1) → c_1
  83. Segment(O, O'_1) → d_1
  84. Rotate(E, If[8 < o ≤ 9, ((o - 8) * π), π], N) → E'_1
  85. Rotate(E', If[8 < o ≤ 9, ((o - 8) * π), π], N) → E''
  86. Segment(E'_1, E'') → e_1
  87. Rotate(E'_1, If[9 < o ≤ 10, ((o - 9) * π), π], P) → Z
  88. Rotate(E'', If[9 < o ≤ 10, ((o - 9) * π), π], P) → E'''
  89. Segment(Z, E''') → f_2
  90. Rotate(G, If[10 < o ≤ 11, ((o - 10) * π), π], O) → G'_1
  91. Rotate(G', If[10 < o ≤ 11, ((o - 10) * π), π], O) → G''
  92. Segment(G'_1, G'') → g_2
  93. Rotate(G'_1, If[11 < o ≤ 12, ((o - 11) * π), π], S) → A_1
  94. Rotate(G'', If[11 < o ≤ 12, ((o - 11) * π), π], S) → G'''
  95. Segment(A_1, G''') → h_2
  96. Rotate(M, If[12 < o ≤ 13, ((o - 12) * π), π], N) → M'_1
  97. Rotate(M', If[12 < o ≤ 13, ((o - 12) * π), π], N) → M''
  98. Segment(M'_1, M'') → i_2
  99. Rotate(M'_1, If[13 < o ≤ 14, ((o - 13) * π), π], O) → B_1
  100. Rotate(M'', If[13 < o ≤ 14, ((o - 13) * π), π], O) → M'''
  101. Segment(B_1, M''') → j_2
  102. Rotate(L, If[14 < o ≤ 15, ((o - 14) * π), π], P) → L'_1
  103. Rotate(L', If[14 < o ≤ 15, ((o - 14) * π), π], P) → L''
  104. Segment(L'_1, L'') → k_2
  105. Rotate(L'_1, If[15 < o ≤ 16, ((o - 15) * π), π], S) → C_1
  106. Rotate(L'', If[15 < o ≤ 16, ((o - 15) * π), π], S) → L'''
  107. Segment(C_1, L''') → l_2