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Iterated Golden Rectangle Construction

Iterated Golden Rectangle Construction
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2017-01-07 08:35
CirclePolygonGeometric Construction
This example demonstrates the iterative construction of a Golden Rectangle through successive inscribed circles and divisions, highlighting the geometric properties of the Golden Ratio. Key math concepts include Golden Rectangle, incircle, similarity ratio, and geometric construction. Construction involves commands like Segment, Polygon, Midpoint, Circle, Ray, Intersect, OrthogonalLine, CircleArc, and PointIn to build the rectangle and its inscribed circles iteratively. Suitable for high school mathematics education to understand the application and construction methods of the Golden Ratio in geometry.

GeoGebra Commands & Parameters

  1. Segment(A, B) → f
  2. Polygon(A, B, 4) → poly1, g, h, i, j, C, D
  3. Midpoint(h) → E
  4. Circle(E, D) → c
  5. Ray(B, C) → k
  6. Intersect(c, k, 1) → F
  7. OrthogonalLine(F, k) → l
  8. OrthogonalLine(D, i) → m
  9. Intersect(l, m) → G
  10. Polygon(G, D, 4) → poly2, n, p, q, r, H, I
  11. Segment(C, F) → s
  12. Segment(F, I) → t
  13. Polygon(F, I, 4) → poly3, a_{14}, b_{14}, d, e, J, K
  14. Polygon(C, K, 4) → poly4, f_1, g_1, h_1, i_1, L, M
  15. Polygon(H, M, 4) → poly5, j_1, k_1, l_1, m_1, N, O
  16. Polygon(J, O, 4) → poly6, n_1, p_1, q_1, r_1, P, Q
  17. Polygon(L, Q, 4) → poly7, s_1, t_1, a_1, b_1, R, S
  18. Polygon(N, S, 4) → poly8, c_1, d_1, e_1, f_2, T, U
  19. Polygon(P, U, 4) → poly9, g_2, h_2, i_2, j_2, V, W
  20. Polygon(R, W, 4) → poly10, k_2, l_2, m_2, n_2, Z, A_1
  21. Polygon(T, A_1, 4) → poly11, p_2, q_2, r_2, s_2, B_1, C_1
  22. Polygon(V, C_1, 4) → poly12, t_2, a_2, b_2, c_2, D_1, E_1
  23. Polygon(Z, E_1, 4) → poly13, d_2, e_2, f_3, g_3, F_1, G_1
  24. Polygon(B_1, G_1, 4) → poly14, h_3, i_3, j_3, k_3, H_1, I_1
  25. Polygon(D_1, I_1, 4) → poly15, l_3, m_3, n_3, p_3, J_1, K_1
  26. Polygon(F_1, K_1, 4) → poly16, q_3, r_3, s_3, t_3, L_1, M_1
  27. Polygon(H_1, M_1, 4) → poly17, a_3, b_3, c_3, d_3, N_1, O_1
  28. Polygon(J_1, O_1, 4) → poly18, e_3, f_4, g_4, h_4, P_1, Q_1
  29. Polygon(L_1, Q_1, 4) → poly19, i_4, j_4, k_4, l_4, R_1, S_1
  30. Polygon(N_1, S_1, 4) → poly20, m_4, n_4, p_4, q_4, T_1, U_1
  31. Polygon(P_1, U_1, 4) → poly21, r_4, s_4, t_4, a_4, V_1, W_1
  32. Polygon(R_1, W_1, 4) → poly22, b_4, c_4, d_4, e_4, Z_1, A_2
  33. Polygon(T_1, A_2, 4) → poly23, f_5, g_5, h_5, i_5, B_2, C_2
  34. Polygon(V_1, C_2, 4) → poly24, j_5, k_5, l_5, m_5, D_2, E_2
  35. Polygon(Z_1, E_2, 4) → poly25, n_5, p_5, q_5, r_5, F_2, G_2
  36. Polygon(B_2, G_2, 4) → poly26, s_5, t_5, a_5, b_5, H_2, I_2
  37. Polygon(D_2, I_2, 4) → poly27, c_5, d_5, e_5, f_6, J_2, K_2
  38. Polygon(F_2, K_2, 4) → poly28, g_6, h_6, i_6, j_6, L_2, M_2
  39. Polygon(H_2, M_2, 4) → poly29, k_6, l_6, m_6, n_6, N_2, O_2
  40. Polygon(J_2, O_2, 4) → poly30, p_6, q_6, r_6, s_6, P_2, Q_2
  41. Polygon(L_2, Q_2, 4) → poly31, t_6, a_6, b_6, c_6, R_2, S_2
  42. Polygon(N_2, S_2, 4) → poly32, d_6, e_6, f_7, g_7, T_2, U_2
  43. CircleArc(C, D, B) → c_7
  44. CircleArc(H, I, D) → d_7
  45. CircleArc(J, K, I) → e_7
  46. CircleArc(L, M, K) → q_7
  47. CircleArc(N, O, M) → r_7
  48. CircleArc(P, Q, O) → s_7
  49. CircleArc(R, S, Q) → t_7
  50. CircleArc(T, U, S) → c_8
  51. CircleArc(V, W, U) → d_8
  52. CircleArc(Z, A_1, W) → e_8
  53. CircleArc(B_1, C_1, A_1) → f_8
  54. CircleArc(D_1, E_1, C_1) → g_8
  55. CircleArc(F_1, G_1, E_1) → h_8
  56. CircleArc(H_1, I_1, G_1) → k_8
  57. CircleArc(J_1, K_1, I_1) → p_8
  58. CircleArc(L_1, M_1, K_1) → q_8
  59. CircleArc(N_1, O_1, M_1) → r_8
  60. CircleArc(P_1, Q_1, O_1) → s_8
  61. CircleArc(R_1, S_1, Q_1) → t_8
  62. CircleArc(T_1, U_1, S_1) → c_9
  63. CircleArc(V_1, W_1, U_1) → d_9
  64. CircleArc(Z_1, A_2, W_1) → e_9
  65. CircleArc(B_2, C_2, A_2) → f_9
  66. CircleArc(D_2, E_2, C_2) → g_9
  67. CircleArc(F_2, G_2, E_2) → h_9
  68. CircleArc(H_2, I_2, G_2) → k_9
  69. CircleArc(J_2, K_2, I_2) → p_9
  70. CircleArc(L_2, M_2, K_2) → q_9
  71. CircleArc(N_2, O_2, M_2) → r_9
  72. CircleArc(P_2, Q_2, O_2) → h_7
  73. CircleArc(R_2, S_2, Q_2) → k_7
  74. CircleArc(T_2, U_2, S_2) → p_7
  75. Polygon(P_2, U_2, 4) → poly33, i_7, j_7, l_7, m_7, V_2, W_2
  76. Polygon(R_2, W_2, 4) → poly34, n_7, a_7, b_7, i_8, Z_2, A_3
  77. CircleArc(V_2, W_2, U_2) → s_9
  78. CircleArc(Z_2, A_3, W_2) → t_9
  79. Segment(Z_2, T_2) → j_8
  80. Segment(V_2, A_3) → l_8
  81. Intersect(j_8, l_8) → B_3
  82. Segment(B, F) → m_8
  83. Segment(F, G) → n_8
  84. Segment(G, A) → a_8
  85. Polygon(F, B, 4) → poly35, b_8, i_9, j_9, l_9, C_3, D_3
  86. CircleArc(F, B, D_3) → c_{10}
  87. Polygon(G, D_3, 4) → poly36, m_9, n_9, a_9, b_9, E_3, F_3
  88. Polygon(A, F_3, 4) → poly37, f_{10}, g_{10}, h_{10}, i_{10}, G_3, H_3
  89. Polygon(C_3, H_3, 4) → poly38, j_{10}, k_{10}, l_{10}, m_{10}, I_3, J_3
  90. Polygon(E_3, J_3, 4) → poly39, n_{10}, p_{10}, q_{10}, r_{10}, K_3, L_3
  91. Polygon(G_3, L_3, 4) → poly40, s_{10}, t_{10}, a_{10}, b_{10}, M_3, N_3
  92. Polygon(I_3, N_3, 4) → poly41, d_{10}, e_{10}, f_{11}, g_{11}, O_3, P_3
  93. Polygon(K_3, P_3, 4) → poly42, h_{11}, i_{11}, j_{11}, k_{11}, Q_3, R_3
  94. Polygon(M_3, R_3, 4) → poly43, l_{11}, m_{11}, n_{11}, p_{11}, S_3, T_3
  95. Polygon(O_3, T_3, 4) → poly44, q_{11}, r_{11}, s_{11}, t_{11}, U_3, V_3
  96. Polygon(Q_3, V_3, 4) → poly45, a_{11}, b_{11}, c_{11}, d_{11}, W_3, Z_3
  97. Polygon(S_3, Z_3, 4) → poly46, e_{11}, f_{12}, g_{12}, h_{12}, A_4, B_4
  98. Polygon(U_3, B_4, 4) → poly47, i_{12}, j_{12}, k_{12}, l_{12}, C_4, D_4
  99. Polygon(W_3, D_4, 4) → poly48, m_{12}, n_{12}, p_{12}, q_{12}, E_4, F_4
  100. Polygon(A_4, F_4, 4) → poly49, r_{12}, s_{12}, t_{12}, a_{12}, G_4, H_4
  101. Polygon(C_4, H_4, 4) → poly50, b_{12}, c_{12}, d_{12}, e_{12}, I_4, J_4
  102. Polygon(E_4, J_4, 4) → poly51, f_{13}, g_{13}, h_{13}, i_{13}, K_4, L_4
  103. Polygon(G_4, L_4, 4) → poly52, j_{13}, k_{13}, l_{13}, m_{13}, M_4, N_4
  104. Polygon(I_4, N_4, 4) → poly53, n_{13}, p_{13}, q_{13}, r_{13}, O_4, P_4
  105. Polygon(K_4, P_4, 4) → poly54, s_{13}, t_{13}, a_{13}, b_{13}, Q_4, R_4
  106. CircleArc(G, D_3, F_3) → c_{13}
  107. CircleArc(A, F_3, H_3) → d_{13}
  108. CircleArc(C_3, H_3, J_3) → e_{13}
  109. CircleArc(E_3, J_3, L_3) → c_{14}
  110. CircleArc(G_3, L_3, N_3) → d_{14}
  111. CircleArc(I_3, N_3, P_3) → e_{14}
  112. CircleArc(K_3, P_3, R_3) → f_{14}
  113. CircleArc(M_3, R_3, T_3) → g_{14}
  114. CircleArc(O_3, T_3, V_3) → h_{14}
  115. CircleArc(Q_3, V_3, Z_3) → k_{14}
  116. CircleArc(S_3, Z_3, B_4) → p_{14}
  117. CircleArc(U_3, B_4, D_4) → q_{14}
  118. CircleArc(W_3, D_4, F_4) → r_{14}
  119. CircleArc(A_4, F_4, H_4) → s_{14}
  120. CircleArc(C_4, H_4, J_4) → t_{14}
  121. CircleArc(E_4, J_4, L_4) → c_{15}
  122. CircleArc(G_4, L_4, N_4) → d_{15}
  123. CircleArc(I_4, N_4, P_4) → e_{15}
  124. CircleArc(K_4, P_4, R_4) → f_{15}
  125. PointIn(poly54) → T_4
  126. PointIn(poly54) → S_4
  127. PointIn(poly53) → U_4
  128. PointIn(poly52) → V_4
  129. Segment(T_4, U_4) → i_{14}
  130. Segment(U_4, V_4) → j_{14}
  131. Segment(V_4, S_4) → l_{14}
  132. Segment(S_4, T_4) → m_{14}
  133. Midpoint(l_{14}) → W_4
  134. Midpoint(j_{14}) → Z_4