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Exploring Fractal Tree Branches

Exploring Fractal Tree Branches
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2019-07-17 03:00
TrigonometrySequenceGeometric Construction
This example demonstrates how to create a fractal tree using GeoGebra's sequence command and investigates the number of branches. Key math concepts include fractal geometry, recursive sequences, and geometric transformations. Construction highlights the use of commands like Segment, Point, Angle, Rotate, Ray, Circle, Intersect, and objects such as treebranch and Textfield. Suitable for high school and above students learning about fractal patterns and mathematical modeling.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

Draw a fractal tree starting from point A, generating branches through rotation and scaling. First, draw the initial branches AB and AC, then rotate and scale around point B to generate new branches BD and BE. Draw new angles and circles, find new intersection points G and H, and connect these points to form new branches. Repeat this process to generate multiple levels of branches, forming a complex fractal tree structure.
The prompt above was compiled from the original drawing commands and is for reference only.