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毕达哥拉斯分形树

毕达哥拉斯分形树
几何画板及绘图内容加载中，请稍等……
2017-09-28 23:20
测试改名0三角函数几何作图
所属专题 GCSE Higher Ch5 Angles & Trigonometry
①本范例演示了如何通过递归旋转和拼接直角三角形构造一个毕达哥拉斯分形树；②涉及的核心数学概念包括勾股定理、三角函数、角度旋转和几何作图；③作图要点——从下方『构造摘要』提炼用到了 Rotate（旋转）、Angle（角度）、Segment（线段）、Intersect（交点）、Polygon（多边形）等 GeoGebra 命令/对象；④适合高中数学教学，用于理解分形几何和几何变换的应用。

GeoGebra 命令及参数

  1. Rotate(A, (-α), B) → A'
  2. Angle(A', B, A) → β
  3. Rotate(B, 90° - α, A) → B'
  4. Angle(B, A, B') → γ
  5. Segment(A, B') → f
  6. Segment(B, A') → g
  7. Intersect(f, g) → C
  8. Polygon(A, C, B) → poly1, b, a, c
  9. Polygon(B, A, 4) → poly2, h, i, j, k, D, E
  10. Polygon(C, B, 4) → poly3, l, m, n, p, F, G
  11. Polygon(A, C, 4) → poly4, q, r, s, t, H, I
  12. Tool1(I, H, α) → poly5, poly6, poly7, J, K, L, M
  13. Tool1(G, F, α) → poly8, poly9, poly10, N, O, P, Q
  14. Tool1(L, M, α) → poly11, poly12, poly13, R, S, T, U
  15. Tool1(J, K, α) → poly14, poly15, poly16, V, W, Z, A_1
  16. Tool1(V, W, α) → poly17, poly18, poly19, B_1, C_1, D_1, E_1
  17. Tool1(B_1, C_1, α) → poly20, poly21, poly22, F_1, G_1, H_1, I_1
  18. Tool1(F_1, G_1, α) → poly23, poly24, poly25, J_1, K_1, L_1, M_1
  19. Tool1(J_1, K_1, α) → poly26, poly27, poly28, N_1, O_1, P_1, Q_1
  20. Tool1(Z, A_1, α) → poly29, poly30, poly31, R_1, S_1, T_1, U_1
  21. Tool1(D_1, E_1, α) → poly32, poly33, poly34, V_1, W_1, Z_1, A_2
  22. Tool1(H_1, I_1, α) → poly35, poly36, poly37, B_2, C_2, D_2, E_2
  23. Tool1(L_1, M_1, α) → poly38, poly39, poly40, F_2, G_2, H_2, I_2
  24. Tool1(P_1, Q_1, α) → poly41, poly42, poly43, J_2, K_2, L_2, M_2
  25. Tool1(R, S, α) → poly44, poly45, poly46, N_2, O_2, P_2, Q_2
  26. Tool1(N_2, O_2, α) → poly47, poly48, poly49, R_2, S_2, T_2, U_2
  27. Tool1(R_2, S_2, α) → poly50, poly51, poly52, V_2, W_2, Z_2, A_3
  28. Tool1(R_1, S_1, α) → poly53, poly54, poly55, B_3, C_3, D_3, E_3
  29. Tool1(B_3, C_3, α) → poly56, poly57, poly58, F_3, G_3, H_3, I_3
  30. Tool1(V_1, W_1, α) → poly59, poly60, poly61, J_3, K_3, L_3, M_3
  31. Tool1(J_3, K_3, α) → poly62, poly63, poly64, N_3, O_3, P_3, Q_3
  32. Tool1(B_2, C_2, α) → poly65, poly66, poly67, R_3, S_3, T_3, U_3
  33. Tool1(R_3, S_3, α) → poly68, poly69, poly70, V_3, W_3, Z_3, A_4
  34. Tool1(F_2, G_2, α) → poly71, poly72, poly73, B_4, C_4, D_4, E_4
  35. Tool1(B_4, C_4, α) → poly74, poly75, poly76, F_4, G_4, H_4, I_4
  36. Tool1(J_2, K_2, α) → poly77, poly78, poly79, J_4, K_4, L_4, M_4
  37. Tool1(J_4, K_4, α) → poly80, poly81, poly82, N_4, O_4, P_4, Q_4
  38. Tool1(F_4, G_4, α) → poly83, poly84, poly85, R_4, S_4, T_4, U_4
  39. Tool1(R_4, S_4, α) → poly86, poly87, poly88, V_4, W_4, Z_4, A_5
  40. Tool1(N_4, O_4, α) → poly89, poly90, poly91, B_5, C_5, D_5, E_5
  41. Tool1(B_5, C_5, α) → poly92, poly93, poly94, F_5, G_5, H_5, I_5
  42. Tool1(F_5, G_5, α) → poly95, poly96, poly97, J_5, K_5, L_5, M_5
  43. Tool1(J_5, K_5, α) → poly98, poly99, poly100, N_5, O_5, P_5, Q_5
  44. Tool1(V_3, W_3, α) → poly101, poly102, poly103, R_5, S_5, T_5, U_5
  45. Tool1(R_5, S_5, α) → poly104, poly105, poly106, V_5, W_5, Z_5, A_6
  46. Tool1(N_3, O_3, α) → poly107, poly108, poly109, B_6, C_6, D_6, E_6
  47. Tool1(B_6, C_6, α) → poly110, poly111, poly112, F_6, G_6, H_6, I_6
  48. Tool1(F_3, G_3, α) → poly113, poly114, poly115, J_6, K_6, L_6, M_6
  49. Tool1(J_6, K_6, α) → poly116, poly117, poly118, N_6, O_6, P_6, Q_6
  50. Tool1(V_2, W_2, α) → poly119, poly120, poly121, R_6, S_6, T_6, U_6
  51. Tool1(N, O, α) → poly122, poly123, poly124, V_6, W_6, Z_6, A_7
  52. Tool1(V_6, W_6, α) → poly125, poly126, poly127, B_7, C_7, D_7, E_7
  53. Tool1(B_7, C_7, α) → poly128, poly129, poly130, F_7, G_7, H_7, I_7
  54. Tool1(F_7, G_7, α) → poly131, poly132, poly133, J_7, K_7, L_7, M_7
  55. Tool1(V_4, W_4, α) → poly134, poly135, poly136, N_7, O_7, P_7, Q_7
  56. Tool1(N_7, O_7, α) → poly137, poly138, poly139, R_7, S_7, T_7, U_7
  57. Tool1(V_5, W_5, α) → poly140, poly141, poly142, V_7, W_7, Z_7, A_8
  58. Tool1(V_7, W_7, α) → poly143, poly144, poly145, B_8, C_8, D_8, E_8
  59. Tool1(F_6, G_6, α) → poly146, poly147, poly148, F_8, G_8, H_8, I_8
  60. Tool1(F_8, G_8, α) → poly149, poly150, poly151, J_8, K_8, L_8, M_8
  61. Tool1(N_6, O_6, α) → poly152, poly153, poly154, N_8, O_8, P_8, Q_8
  62. Tool1(N_8, O_8, α) → poly155, poly156, poly157, R_8, S_8, T_8, U_8
  63. Tool1(R_6, S_6, α) → poly158, poly159, poly160, V_8, W_8, Z_8, A_9
  64. Tool1(V_8, W_8, α) → poly161, poly162, poly163, B_9, C_9, D_9, E_9
  65. Tool1(J_7, K_7, α) → poly164, poly165, poly166, F_9, G_9, H_9, I_9
  66. Tool1(F_9, G_9, α) → poly167, poly168, poly169, J_9, K_9, L_9, M_9
  67. Tool1(J_9, K_9, α) → poly170, poly171, poly172, N_9, O_9, P_9, Q_9
  68. Tool1(N_9, O_9, α) → poly173, poly174, poly175, R_9, S_9, T_9, U_9
  69. Tool1(R_9, S_9, α) → poly176, poly177, poly178, V_9, W_9, Z_9, A_{10}
  70. Tool1(V_9, W_9, α) → poly179, poly180, poly181, B_{10}, C_{10}, D_{10}, E_{10}
  71. Tool1(B_{10}, C_{10}, α) → poly182, poly183, poly184, F_{10}, G_{10}, H_{10}, I_{10}
  72. Tool1(F_{10}, G_{10}, α) → poly185, poly186, poly187, J_{10}, K_{10}, L_{10}, M_{10}
  73. Tool1(B_9, C_9, α) → poly188, poly189, poly190, N_{10}, O_{10}, P_{10}, Q_{10}
  74. Tool1(N_{10}, O_{10}, α) → poly191, poly192, poly193, R_{10}, S_{10}, T_{10}, U_{10}
  75. Tool1(R_{10}, S_{10}, α) → poly194, poly195, poly196, V_{10}, W_{10}, Z_{10}, A_{11}
  76. Tool1(V_{10}, W_{10}, α) → poly197, poly198, poly199, B_{11}, C_{11}, D_{11}, E_{11}
  77. Tool1(B_{11}, C_{11}, α) → poly200, poly201, poly202, F_{11}, G_{11}, H_{11}, I_{11}
  78. Tool1(F_{11}, G_{11}, α) → poly203, poly204, poly205, J_{11}, K_{11}, L_{11}, M_{11}
  79. Tool1(J_{11}, K_{11}, α) → poly206, poly207, poly208, N_{11}, O_{11}, P_{11}, Q_{11}
  80. Tool1(N_{11}, O_{11}, α) → poly209, poly210, poly211, R_{11}, S_{11}, T_{11}, U_{11}
  81. Tool1(R_{11}, S_{11}, α) → poly212, poly213, poly214, V_{11}, W_{11}, Z_{11}, A_{12}
  82. Tool1(V_{11}, W_{11}, α) → poly215, poly216, poly217, B_{12}, C_{12}, D_{12}, E_{12}
  83. Tool1(B_{12}, C_{12}, α) → poly218, poly219, poly220, F_{12}, G_{12}, H_{12}, I_{12}
  84. Tool1(P, Q, α) → poly221, poly222, poly223, J_{12}, K_{12}, L_{12}, M_{12}
  85. Tool1(Z_6, A_7, α) → poly224, poly225, poly226, N_{12}, O_{12}, P_{12}, Q_{12}
  86. Tool1(D_7, E_7, α) → poly227, poly228, poly229, R_{12}, S_{12}, T_{12}, U_{12}
  87. Tool1(H_7, I_7, α) → poly230, poly231, poly232, V_{12}, W_{12}, Z_{12}, A_{13}
  88. Tool1(L_7, M_7, α) → poly233, poly234, poly235, B_{13}, C_{13}, D_{13}, E_{13}
  89. Tool1(H_9, I_9, α) → poly236, poly237, poly238, F_{13}, G_{13}, H_{13}, I_{13}
  90. Tool1(L_9, M_9, α) → poly239, poly240, poly241, J_{13}, K_{13}, L_{13}, M_{13}
  91. Tool1(P_9, Q_9, α) → poly242, poly243, poly244, N_{13}, O_{13}, P_{13}, Q_{13}
  92. Tool1(T_9, U_9, α) → poly245, poly246, poly247, R_{13}, S_{13}, T_{13}, U_{13}
  93. Tool1(Z_9, A_{10}, α) → poly248, poly249, poly250, V_{13}, W_{13}, Z_{13}, A_{14}
  94. Tool1(D_{10}, E_{10}, α) → poly251, poly252, poly253, B_{14}, C_{14}, D_{14}, E_{14}
  95. Tool1(H_{10}, I_{10}, α) → poly254, poly255, poly256, F_{14}, G_{14}, H_{14}, I_{14}
  96. Tool1(L_{10}, M_{10}, α) → poly257, poly258, poly259, J_{14}, K_{14}, L_{14}, M_{14}
  97. Tool1(P_2, Q_2, α) → poly260, poly261, poly262, N_{14}, O_{14}, P_{14}, Q_{14}
  98. Tool1(T_2, U_2, α) → poly263, poly264, poly265, R_{14}, S_{14}, T_{14}, U_{14}
  99. Tool1(Z_2, A_3, α) → poly266, poly267, poly268, V_{14}, W_{14}, Z_{14}, A_{15}
  100. Tool1(T_6, U_6, α) → poly269, poly270, poly271, B_{15}, C_{15}, D_{15}, E_{15}
  101. Tool1(D_3, E_3, α) → poly272, poly273, poly274, F_{15}, G_{15}, H_{15}, I_{15}
  102. Tool1(H_3, I_3, α) → poly275, poly276, poly277, J_{15}, K_{15}, L_{15}, M_{15}
  103. Tool1(L_6, M_6, α) → poly278, poly279, poly280, N_{15}, O_{15}, P_{15}, Q_{15}
  104. Tool1(P_6, Q_6, α) → poly281, poly282, poly283, R_{15}, S_{15}, T_{15}, U_{15}
  105. Tool1(L_3, M_3, α) → poly284, poly285, poly286, V_{15}, W_{15}, Z_{15}, A_{16}
  106. Tool1(P_3, Q_3, α) → poly287, poly288, poly289, B_{16}, C_{16}, D_{16}, E_{16}
  107. Tool1(D_6, E_6, α) → poly290, poly291, poly292, F_{16}, G_{16}, H_{16}, I_{16}
  108. Tool1(H_6, I_6, α) → poly293, poly294, poly295, J_{16}, K_{16}, L_{16}, M_{16}
  109. Tool1(T_3, U_3, α) → poly296, poly297, poly298, N_{16}, O_{16}, P_{16}, Q_{16}
  110. Tool1(Z_3, A_4, α) → poly299, poly300, poly301, R_{16}, S_{16}, T_{16}, U_{16}
  111. Tool1(T_5, U_5, α) → poly302, poly303, poly304, V_{16}, W_{16}, Z_{16}, A_{17}
  112. Tool1(Z_5, A_6, α) → poly305, poly306, poly307, B_{17}, C_{17}, D_{17}, E_{17}
  113. Tool1(D_4, E_4, α) → poly308, poly309, poly310, F_{17}, G_{17}, H_{17}, I_{17}
  114. Tool1(H_4, I_4, α) → poly311, poly312, poly313, J_{17}, K_{17}, L_{17}, M_{17}
  115. Tool1(T_4, U_4, α) → poly314, poly315, poly316, N_{17}, O_{17}, P_{17}, Q_{17}
  116. Tool1(Z_4, A_5, α) → poly317, poly318, poly319, R_{17}, S_{17}, T_{17}, U_{17}
  117. Tool1(L_4, M_4, α) → poly320, poly321, poly322, V_{17}, W_{17}, Z_{17}, A_{18}
  118. Tool1(P_4, Q_4, α) → poly323, poly324, poly325, B_{18}, C_{18}, D_{18}, E_{18}
  119. Tool1(D_5, E_5, α) → poly326, poly327, poly328, F_{18}, G_{18}, H_{18}, I_{18}
  120. Tool1(H_5, I_5, α) → poly329, poly330, poly331, J_{18}, K_{18}, L_{18}, M_{18}
  121. Tool1(P_5, Q_5, α) → poly332, poly333, poly334, N_{18}, O_{18}, P_{18}, Q_{18}
  122. Tool1(L_5, M_5, α) → poly335, poly336, poly337, R_{18}, S_{18}, T_{18}, U_{18}
  123. Tool1(P_7, Q_7, α) → poly338, poly339, poly340, V_{18}, W_{18}, Z_{18}, A_{19}
  124. Tool1(T_7, U_7, α) → poly341, poly342, poly343, B_{19}, C_{19}, D_{19}, E_{19}
  125. Tool1(L_2, M_2, α) → poly344, poly345, poly346, F_{19}, G_{19}, H_{19}, I_{19}
  126. Tool1(Z_7, A_8, α) → poly347, poly348, poly349, J_{19}, K_{19}, L_{19}, M_{19}
  127. Tool1(D_8, E_8, α) → poly350, poly351, poly352, N_{19}, O_{19}, P_{19}, Q_{19}
  128. Tool1(H_2, I_2, α) → poly353, poly354, poly355, R_{19}, S_{19}, T_{19}, U_{19}
  129. Tool1(D_2, E_2, α) → poly356, poly357, poly358, V_{19}, W_{19}, Z_{19}, A_{20}
  130. Tool1(H_8, I_8, α) → poly359, poly360, poly361, B_{20}, C_{20}, D_{20}, E_{20}
  131. Tool1(L_8, M_8, α) → poly362, poly363, poly364, F_{20}, G_{20}, H_{20}, I_{20}
  132. Tool1(Z_1, A_2, α) → poly365, poly366, poly367, J_{20}, K_{20}, L_{20}, M_{20}
  133. Tool1(T_8, U_8, α) → poly368, poly369, poly370, N_{20}, O_{20}, P_{20}, Q_{20}
  134. Tool1(P_8, Q_8, α) → poly371, poly372, poly373, R_{20}, S_{20}, T_{20}, U_{20}
  135. Tool1(T_{10}, U_{10}, α) → poly374, poly375, poly376, V_{20}, W_{20}, Z_{20}, A_{21}
  136. Tool1(H_{11}, I_{11}, α) → poly377, poly378, poly379, B_{21}, C_{21}, D_{21}, E_{21}
  137. Tool1(D_{11}, E_{11}, α) → poly380, poly381, poly382, F_{21}, G_{21}, H_{21}, I_{21}
  138. Tool1(P_{10}, Q_{10}, α) → poly383, poly384, poly385, J_{21}, K_{21}, L_{21}, M_{21}
  139. Tool1(D_9, E_9, α) → poly386, poly387, poly388, N_{21}, O_{21}, P_{21}, Q_{21}
  140. Tool1(L_{11}, M_{11}, α) → poly389, poly390, poly391, R_{21}, S_{21}, T_{21}, U_{21}
  141. Tool1(P_{11}, Q_{11}, α) → poly392, poly393, poly394, V_{21}, W_{21}, Z_{21}, A_{22}
  142. Tool1(T_{11}, U_{11}, α) → poly395, poly396, poly397, B_{22}, C_{22}, D_{22}, E_{22}
  143. Tool1(Z_{11}, A_{12}, α) → poly398, poly399, poly400, F_{22}, G_{22}, H_{22}, I_{22}
  144. Tool1(T_1, U_1, α) → poly401, poly402, poly403, J_{22}, K_{22}, L_{22}, M_{22}
  145. Tool1(D_{12}, E_{12}, α) → poly404, poly405, poly406, N_{22}, O_{22}, P_{22}, Q_{22}
  146. Tool1(H_{12}, I_{12}, α) → poly407, poly408, poly409, R_{22}, S_{22}, T_{22}, U_{22}
  147. Tool1(Z_{10}, A_{11}, α) → poly410, poly411, poly412, V_{22}, W_{22}, Z_{22}, A_{23}
  148. Tool1(Z_8, A_9, α) → poly413, poly414, poly415, B_{23}, C_{23}, D_{23}, E_{23}
  149. Tool1(T, U, α) → poly416, poly417, poly418, F_{23}, G_{23}, H_{23}, I_{23}

AI 绘图提示语参考

绘制一棵毕达哥拉斯分形树。先画一个直角三角形，以其斜边为底边向外作一个正方形，再以该正方形的两条相邻边为斜边分别构造两个更小的直角三角形，并在这两个小三角形的斜边上继续作正方形，如此递归迭代多层，形成左右对称的树状分形结构。所有三角形均为直角三角形，正方形边与对应三角形的斜边重合，整体呈现从底部主干向上逐层分叉的几何图案，角度参数保持一致。
以上提示语为根据原始绘图命令整理而来，仅供参考。