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Approximating π with Euler's Zigzag Numbers

Approximating π with Euler's Zigzag Numbers
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2014-10-18 21:07
SequenceLimitGeometric Construction
In collections Pi Approximations
This example demonstrates how to approximate π using Euler's zigzag numbers. Key mathematical concepts include sequence approximation, geometric construction, and limit ideas. Construction involves GeoGebra commands like Sum, Intersect, Point, and Segment, using objects such as B1 to L1 and A2 to A12. Suitable for high school students learning geometry and sequences, or for mathematical modeling education.

GeoGebra Commands & Parameters

  1. Sum(D5:E5) → D6
  2. Sum(C5:E5) → E6
  3. Sum(B5:E5) → F6
  4. Sum(E5:E5) → C6
  5. Sum(B3:C3) → D4
  6. Sum(C4:D4) → D5
  7. Sum(B4:D4) → E5
  8. Sum(E6:F6) → D7
  9. Sum(D6:F6) → E7
  10. Sum(C6:F6) → F7
  11. Sum(B6:F6) → G7
  12. Sum(F7:G7) → D8
  13. Sum(E7:G7) → E8
  14. Sum(D7:G7) → F8
  15. Sum(C7:G7) → G8
  16. Sum(B7:G7) → H8
  17. Sum(G8:H8) → D9
  18. Sum(F8:H8) → E9
  19. Sum(E8:H8) → F9
  20. Sum(D8:H8) → G9
  21. Sum(C8:H8) → H9
  22. Sum(B8:H8) → I9
  23. Sum(H9:I9) → D10
  24. Sum(G9:I9) → E10
  25. Sum(F9:I9) → F10
  26. Sum(E9:I9) → G10
  27. Sum(D9:I9) → H10
  28. Sum(C9:I9) → I10
  29. Sum(B9:I9) → J10
  30. Sum(I10:J10) → D11
  31. Sum(H10:J10) → E11
  32. Sum(G10:J10) → F11
  33. Sum(F10:J10) → G11
  34. Sum(E10:J10) → H11
  35. Sum(D10:J10) → I11
  36. Sum(C10:J10) → J11
  37. Sum(B10:J10) → K11
  38. Sum(J11:K11) → D12
  39. Sum(I11:K11) → E12
  40. Sum(H11:K11) → F12
  41. Sum(G11:K11) → G12
  42. Sum(F11:K11) → H12
  43. Sum(E11:K11) → I12
  44. Sum(D11:K11) → J12
  45. Sum(C11:K11) → K12
  46. Sum(B11:K11) → L12
  47. Intersect(xAxis, yAxis) → A
  48. Point(yAxis) → B
  49. Point(xAxis) → C
  50. Point(xAxis) → G
  51. Segment(A, B) → a
  52. Segment(C, F) → b
  53. Segment(G, I) → c
  54. Segment(B, F) → d
  55. Segment(F, I) → e
  56. Point(xAxis) → J
  57. Point(xAxis) → M
  58. Point(xAxis) → Q
  59. Segment(J, L) → f
  60. Segment(M, P) → g
  61. Segment(Q, R) → h
  62. Segment(L, P) → i
  63. Segment(P, R) → j
  64. Intersect(xAxis, yAxis) → S
  65. Point(yAxis) → T
  66. Point(xAxis) → U
  67. Point(xAxis) → A_1
  68. Point(xAxis) → D_1
  69. Point(xAxis) → I_1
  70. Point(xAxis) → K_1
  71. Point(xAxis) → P_1
  72. Point(xAxis) → S_1
  73. Point(xAxis) → W_1
  74. Point(xAxis) → B_2
  75. Point(xAxis) → F_2
  76. Point(xAxis) → H_2
  77. Point(yAxis) → M_2
  78. Point(yAxis) → N_2
  79. Point(yAxis) → O_2
  80. Segment(S, T) → m
  81. Segment(U, Z) → n
  82. Segment(A_1, C_1) → p
  83. Segment(D_1, H_1) → q
  84. Segment(I_1, J_1) → r
  85. Segment(K_1, O_1) → s
  86. Segment(P_1, R_1) → t
  87. Segment(S_1, V_1) → a_1
  88. Segment(W_1, A_2) → b_1
  89. Segment(B_2, E_2) → c_1
  90. Segment(F_2, G_2) → d_1
  91. Segment(H_2, L_2) → e_1
  92. Segment(M_2, O_2) → f_1
  93. Segment(P_2, B_3) → g_1
  94. Segment(T_2, U_2) → h_1
  95. Segment(V_2, A_3) → i_1
  96. Segment(C_3, P_3) → j_1
  97. Segment(F_3, J_3) → k_1
  98. Segment(K_3, L_3) → l_1
  99. Segment(M_3, O_3) → m_1
  100. Segment(T, Z) → n_1
  101. Segment(Z, C_1) → p_1
  102. Segment(C_1, H_1) → q_1
  103. Segment(J_1, O_1) → r_1
  104. Segment(O_1, R_1) → s_1
  105. Segment(R_1, V_1) → t_1
  106. Segment(A_2, E_2) → a_2
  107. Segment(E_2, G_2) → b_2
  108. Segment(G_2, L_2) → c_2
  109. Segment(O_2, B_3) → d_2
  110. Segment(B_3, U_2) → e_2
  111. Segment(U_2, A_3) → f_2
  112. Segment(P_3, J_3) → g_2
  113. Segment(J_3, L_3) → h_2
  114. Segment(L_3, O_3) → i_2

AI Drawing Prompt Reference

Draw a geometric illustration showing how Euler's zigzag numbers approximate π, constructed through a series of summations and intersections starting from basic geometric units, gradually building complex patterns that resemble a circle, demonstrating sequence approximation, geometric construction, and limit concepts.
The prompt above was compiled from the original drawing commands and is for reference only.