积极 AI Logo 积极 AI

分形树分支数量探索

分形树分支数量探索
几何画板及绘图内容加载中，请稍等……
2019-07-17 03:00
三角函数数列几何作图
①本范例演示如何利用 GeoGebra 的序列命令绘制分形树，并探讨其分支数量规律；②涉及的核心数学概念包括分形几何、递归序列、几何变换；③作图要点：使用了 Segment（线段）、Point（点）、Angle（角度）、Rotate（旋转）、Ray（射线）、Circle（圆）、Intersect（交点）、treebranch（树分支）、Textfield（文本框）等命令和对象；④适合高中及以上数学学习者，用于理解分形图案的生成与数学建模。

GeoGebra 命令及参数

  1. Segment(M_5, N_5) → g_7
  2. Point(g_7) → S_5
  3. Segment(A, B) → a
  4. Segment(B, C) → b
  5. Segment(B, D) → c
  6. Angle(C, B, A) → α
  7. Segment(B, D') → r
  8. Segment(E, F) → h
  9. Rotate(E, (-α), F) → E'
  10. Angle(E', F, E) → ε
  11. Ray(F, E') → i
  12. Circle(F, (b / a * Distance[E, F])) → k
  13. Intersect(k, i, 1) → G
  14. Angle(A, B, D) → β
  15. Rotate(E, β, F) → E'_1
  16. Angle(E, F, E'_1) → γ
  17. Ray(F, E'_1) → d
  18. Circle(F, (c / a * Distance[F, E])) → e
  19. Intersect(e, d, 1) → H
  20. Segment(F, G) → f
  21. Segment(F, H) → g
  22. treebranch(A, B, C, D, B', C') → I, J, j, l
  23. treebranch(A, B, C, D, B', D') → K, L, m, n_1
  24. treebranch(A, B, C, D, C', I) → M, N, p, q
  25. treebranch(A, B, C, D, Element[list1, 1], Element[list2, 2]) → O, P, s, t
  26. treebranch(A, B, C, D, Element[list1, 2], Element[list2, 3]) → Q, R, a_1, b_1
  27. treebranch(A, B, C, D, Element[list1, 2], Element[list2, 4]) → S, T, c_1, d_1
  28. treebranch(A, B, C, D, Element[list2, 1], Element[list3, 1]) → U, V, e_1, f_1
  29. treebranch(A, B, C, D, Element[list2, 1], Element[list3, 2]) → W, Z, g_1, h_1
  30. treebranch(A, B, C, D, Element[list2, 2], Element[list3, 3]) → A_1, B_1, i_1, j_1
  31. treebranch(A, B, C, D, Element[list2, 2], Element[list3, 4]) → C_1, D_1, k_1, l_1
  32. treebranch(A, B, C, D, Element[list2, 3], Element[list3, 5]) → E_1, F_1, m_1, p_1
  33. treebranch(A, B, C, D, Element[list2, 3], Element[list3, 6]) → G_1, H_1, q_1, r_1
  34. treebranch(A, B, C, D, Element[list2, 4], Element[list3, 7]) → I_1, J_1, s_1, t_1
  35. treebranch(A, B, C, D, Element[list2, 4], Element[list3, 8]) → K_1, L_1, a_2, b_2
  36. treebranch(A, B, C, D, Element[list3, 1], Element[list4, 1]) → M_1, N_1, c_2, d_2
  37. treebranch(A, B, C, D, Element[list3, 1], Element[list4, 2]) → O_1, P_1, e_2, f_2
  38. treebranch(A, B, C, D, Element[list3, 2], Element[list4, 3]) → Q_1, R_1, g_2, h_2
  39. treebranch(A, B, C, D, Element[list3, 2], Element[list4, 4]) → S_1, T_1, i_2, j_2
  40. treebranch(A, B, C, D, Element[list3, 3], Element[list4, 5]) → U_1, V_1, k_2, l_2
  41. treebranch(A, B, C, D, Element[list3, 3], Element[list4, 6]) → W_1, Z_1, m_2, n_2
  42. treebranch(A, B, C, D, Element[list3, 4], Element[list4, 7]) → A_2, B_2, p_2, q_2
  43. treebranch(A, B, C, D, Element[list3, 4], Element[list4, 8]) → C_2, D_2, r_2, s_2
  44. treebranch(A, B, C, D, Element[list3, 5], Element[list4, 9]) → E_2, F_2, t_2, a_3
  45. treebranch(A, B, C, D, Element[list3, 5], Element[list4, 10]) → G_2, H_2, b_3, c_3
  46. treebranch(A, B, C, D, Element[list3, 6], Element[list4, 11]) → I_2, J_2, d_3, e_3
  47. treebranch(A, B, C, D, Element[list3, 6], Element[list4, 12]) → K_2, L_2, f_3, g_3
  48. treebranch(A, B, C, D, Element[list3, 7], Element[list4, 13]) → M_2, N_2, h_3, i_3
  49. treebranch(A, B, C, D, Element[list3, 7], Element[list4, 14]) → O_2, P_2, j_3, k_3
  50. treebranch(A, B, C, D, Element[list3, 8], Element[list4, 15]) → Q_2, R_2, l_3, m_3
  51. treebranch(A, B, C, D, Element[list3, 8], Element[list4, 16]) → S_2, T_2, n_3, p_3
  52. treebranch(A, B, C, D, Element[list4, 1], Element[list5, 1]) → U_2, V_2, q_3, r_3
  53. treebranch(A, B, C, D, Element[list4, 1], Element[list5, 2]) → W_2, Z_2, s_3, t_3
  54. treebranch(A, B, C, D, Element[list4, 2], Element[list5, 3]) → A_3, B_3, a_4, b_4
  55. treebranch(A, B, C, D, Element[list4, 2], Element[list5, 4]) → C_3, D_3, c_4, d_4
  56. treebranch(A, B, C, D, Element[list4, 3], Element[list5, 5]) → E_3, F_3, e_4, f_4
  57. treebranch(A, B, C, D, Element[list4, 3], Element[list5, 6]) → G_3, H_3, g_4, h_4
  58. treebranch(A, B, C, D, Element[list4, 4], Element[list5, 7]) → I_3, J_3, i_4, j_4
  59. treebranch(A, B, C, D, Element[list4, 4], Element[list5, 8]) → K_3, L_3, k_4, l_4
  60. treebranch(A, B, C, D, Element[list4, 5], Element[list5, 9]) → M_3, N_3, m_4, n_4
  61. treebranch(A, B, C, D, Element[list4, 5], Element[list5, 10]) → O_3, P_3, p_4, q_4
  62. treebranch(A, B, C, D, Element[list4, 6], Element[list5, 11]) → Q_3, R_3, r_4, s_4
  63. treebranch(A, B, C, D, Element[list4, 6], Element[list5, 12]) → S_3, T_3, t_4, a_5
  64. treebranch(A, B, C, D, Element[list4, 7], Element[list5, 13]) → U_3, V_3, b_5, c_5
  65. treebranch(A, B, C, D, Element[list4, 7], Element[list5, 14]) → W_3, Z_3, d_5, e_5
  66. treebranch(A, B, C, D, Element[list4, 8], Element[list5, 15]) → A_4, B_4, f_5, g_5
  67. treebranch(A, B, C, D, Element[list4, 8], Element[list5, 16]) → C_4, D_4, h_5, i_5
  68. treebranch(A, B, C, D, Element[list4, 9], Element[list5, 17]) → E_4, F_4, j_5, k_5
  69. treebranch(A, B, C, D, Element[list4, 9], Element[list5, 18]) → G_4, H_4, l_5, m_5
  70. treebranch(A, B, C, D, Element[list4, 10], Element[list5, 19]) → I_4, J_4, n_5, p_5
  71. treebranch(A, B, C, D, Element[list4, 10], Element[list5, 20]) → K_4, L_4, q_5, r_5
  72. treebranch(A, B, C, D, Element[list4, 11], Element[list5, 21]) → M_4, N_4, s_5, t_5
  73. treebranch(A, B, C, D, Element[list4, 11], Element[list5, 22]) → O_4, P_4, a_6, b_6
  74. treebranch(A, B, C, D, Element[list4, 12], Element[list5, 23]) → Q_4, R_4, c_6, d_6
  75. treebranch(A, B, C, D, Element[list4, 12], Element[list5, 24]) → S_4, T_4, e_6, f_6
  76. treebranch(A, B, C, D, Element[list4, 13], Element[list5, 25]) → U_4, V_4, g_6, h_6
  77. treebranch(A, B, C, D, Element[list4, 13], Element[list5, 26]) → W_4, Z_4, i_6, j_6
  78. treebranch(A, B, C, D, Element[list4, 14], Element[list5, 27]) → A_5, B_5, k_6, l_6
  79. treebranch(A, B, C, D, Element[list4, 14], Element[list5, 28]) → C_5, D_5, m_6, n_6
  80. treebranch(A, B, C, D, Element[list4, 15], Element[list5, 29]) → E_5, F_5, p_6, q_6
  81. treebranch(A, B, C, D, Element[list4, 15], Element[list5, 30]) → G_5, H_5, r_6, s_6
  82. treebranch(A, B, C, D, Element[list4, 16], Element[list5, 31]) → I_5, J_5, t_6, a_7
  83. treebranch(A, B, C, D, Element[list4, 16], Element[list5, 32]) → K_5, L_5, b_7, c_7
  84. Segment(A, Base) → f_7
  85. Segment(O_5, N_5) → h_7
  86. Segment(M_5, P_5) → i_7
  87. Segment(P_5, O_5) → j_7
  88. Point(i_7) → Q_5
  89. Segment(Q_5, R_5) → k_7
  90. Segment(U_5, V_5) → m_7
  91. Segment(W_5, Z_5) → l_7
  92. Segment(A_6, B_6) → n_7
  93. Segment(D_6, C_6) → p_7
  94. Segment(S_5, T_5) → q_7
  95. Segment(E_6, F_6) → r_7
  96. Textfield(ans1) → InputBox1
  97. Textfield(ans2) → InputBox2
  98. Textfield(ans3) → InputBox3
  99. Textfield(ans4) → InputBox4
  100. Textfield(ans5) → InputBox5

AI 绘图提示语参考

绘制一棵分形树，从点A出发，通过旋转和缩放生成分支。首先画出初始的两条分支AB和AC，然后围绕点B旋转并缩放生成新的分支BD和BE，接着画出新的角度和圆，找到新的交点G和H，连接这些点形成新的分支。重复此过程，生成多级分支，形成复杂的分形树结构。
以上提示语为根据原始绘图命令整理而来，仅供参考。