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Constructing Pi from the Geometry of Imaginary Unit i

Constructing Pi from the Geometry of Imaginary Unit i
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2015-08-03 08:57
CircleAngleTrigonometryComplex NumberGeometric Construction
In collections Complex Math
This GeoGebra example demonstrates a geometric construction that approximates the value of pi using the geometry of the imaginary unit i. It connects complex-number geometry with circle and angle relationships, including semicircles, chords, slopes, mirror symmetry, and polygonal constraints. The construction uses commands such as Semicircle, Circle, Segment, Polygon, Vector, Angle, Point, Distance, Intersect, Mirror, Textfield, and Slope, with labeled points A through L and objects such as i, r, alpha_u, beta_v, and Delta_v to define the figure. It is suitable for high school geometry, introductory college mathematics, or mathematical visualization activities focused on pi, complex numbers, and geometric construction.

GeoGebra Commands & Parameters

  1. Semicircle(C, D) → c
  2. Semicircle(D, C) → d
  3. Segment(A, D) → a
  4. Circle(A, 1) → f
  5. Semicircle(E, F) → r
  6. Semicircle(F, E) → g
  7. Polygon(B, A, C, G) → poly1, b_1, a_1, c_1, g_1
  8. Vector((1.414213562373095, 1.414213562373095)) → u_3
  9. Vector((1.79, 0.89)) → v
  10. Segment(G, B) → e
  11. Segment(G, J) → h
  12. Segment(G, K) → i
  13. Angle(B, G, K) → α_u
  14. Angle(B, G, J) → β_v
  15. Vector((0.89, 1.79)) → w
  16. Segment(G, L) → j
  17. Angle(B, G, L) → δ_w
  18. Angle(J, G, L) → Δ_v
  19. Circle(G, K) → k
  20. Point(r) → N
  21. Point(g) → M
  22. Segment(N, A) → p
  23. Segment(A, M) → q
  24. Segment(M, N) → s
  25. Polygon(N, C, G, B, M) → poly2, n_1, c_2, g_2, b_2, m_1
  26. Vector((2, 1)) → t
  27. Vector((1, 2)) → u
  28. Segment(G, N) → d_1
  29. Segment(G, M) → e_1
  30. Angle(M, G, N) → α_t
  31. Segment(A, R) → k_1
  32. Segment(A, Q) → l_1
  33. Segment(Q, R) → p_1
  34. Circle(A, S) → q_1
  35. Distance(G, A) → distanceGA
  36. Point(Circle[A, 3]) → T
  37. Segment(A, T) → r_1
  38. Point(Circle[G, 3]) → U
  39. Segment(G, U) → s_1
  40. Point(Circle[T, 2.8284271247461903]) → V
  41. Segment(T, V) → t_1
  42. Distance(G, Q) → distanceGQ
  43. Distance(G, M) → distanceGM
  44. Segment(G, M) → a_2
  45. Segment(W, Z) → d_2
  46. Vector(A_1, B_1) → u_2
  47. Vector(C_1, D_1) → v_2
  48. Distance(G, N) → distanceGN
  49. Segment(G, N) → f_2
  50. Distance(G, Q) → j_2
  51. Point(k) → E_1
  52. Point(k) → F_1
  53. Distance(F_1, E_1) → distanceF1E1
  54. Segment(E_1, F_1) → k_2
  55. Point(k_2) → G_1
  56. Point(s) → H_1
  57. Distance(G_1, H_1) → distanceG1H1
  58. Segment(G_1, H_1) → l_2
  59. Segment(I_1, J_1) → m_2
  60. Segment(K_1, L_1) → n_2
  61. Segment(K_2, L_2) → n_3
  62. Segment(K_3, L_3) → n_4
  63. Segment(K_4, L_4) → n_5
  64. Segment(K_5, L_5) → n_6
  65. Segment(K_8, L_8) → n_9
  66. Polygon(M_1, N_1, O_1, P_1) → poly3, m_3, n_{13}, o_2, p_2
  67. Intersect(k, q_1, 1) → Q_1
  68. Point(f_2) → R_1
  69. Circle(Q_1, R_1) → q_2
  70. Mirror(q_2, i) → r_2
  71. Segment(S_1, T_1) → a_3
  72. Segment(U_1, V_1) → b_3
  73. Intersect(f_3, i, 1) → W_1
  74. Point(f_3) → Z_1
  75. Segment(W_1, Z_1) → d_3
  76. Textfield(σ) → inputBox1
  77. Point(f_3) → C_2
  78. Point(f_3) → D_2
  79. Vector(G, D_2) → v_3
  80. Distance(G, D_2) → distanceGD2
  81. Vector(E_2, F_2) → w_3
  82. Mirror(N, p_1) → N'
  83. Mirror(A, p_1) → A'
  84. Segment(N', A') → p'
  85. Mirror(A, p_1) → A'_1
  86. Mirror(R, p_1) → R'
  87. Segment(A'_1, R') → g_3
  88. Mirror(A, p_1) → A'_2
  89. Mirror(M, p_1) → M'
  90. Segment(A'_2, M') → q'
  91. Mirror(A, p_1) → A'_3
  92. Mirror(Q, p_1) → Q'
  93. Segment(A'_3, Q') → h_3
  94. Segment(G, D_2) → i_3
  95. Slope(i_3) → m_4
  96. Point(p_3) → G_2
  97. Distance(G, G_2) → distanceGG2
  98. Vector(G, G_2) → j_3
  99. Segment(G, G_2) → l_3
  100. Angle(K, G, G_2) → α
  101. Point(q_3) → H_2
  102. Point(s_3) → I_2
  103. Point(r_3) → J_2
  104. Polygon(H_2, I_2, J_2, G) → poly4, h_4, i_4, j_4, g_4
  105. Point(h_4) → M_2
  106. Point(j_4) → N_2
  107. Point(i_4) → O_2
  108. Point(g_4) → P_2
  109. Segment(O_2, M_2) → t_3
  110. Segment(N_2, P_2) → a_4
  111. Point(a_4) → Q_2
  112. Point(t_3) → R_2
  113. Segment(Q_2, R_2) → b_4
  114. Point(i_4) → T_2

AI Drawing Prompt Reference

Draw a geometric construction approximating pi using the imaginary unit i. Start with two intersecting semicircles, a unit circle, and a horizontal segment. Form a quadrilateral and add several vectors and segments from a central point to create multiple angles. Then draw a larger circle, place points on the semicircles, and connect them to form triangles. Add auxiliary circles and segments, measure distances from the center to various points, and place additional points to create more triangles and segments, illustrating angle and distance relationships.
The prompt above was compiled from the original drawing commands and is for reference only.