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六边形循环图案探究

六边形循环图案探究
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2014-07-14 23:06
多边形变换
①本范例演示了通过旋转和拼接六边形进行平面铺砌，探索图案形成的规律；②涉及的核心数学概念包括多边形构造、旋转变换、参数方程、几何逻辑推理；③作图要点提炼自构造摘要，使用了Polygon（多边形）、Segment（线段）、Point（点）、Rotate（旋转）等GeoGebra命令及对象；④适合高中数学课堂，用于培养学生的空间想象能力、几何直观和探究精神。

GeoGebra 命令及参数

  1. Polygon(A_1, B_1, 6) → poly1_1, a_1, F_1, b_1, c_1, d_1, e_1, f_1, C_1, D_1, E_1
  2. Segment(E_1, D_1) → g_1
  3. Segment(D_1, C_1) → h_1
  4. Segment(C_1, B_1) → i_1
  5. Segment(B_1, A_1) → j_1
  6. Segment(A_1, F_1) → k_1
  7. Segment(F_1, E_1) → l_1
  8. Point(Circle[A, a_1]) → B
  9. Rotate(B + (b_1, 0), Angle[C_1 - B_1] + Angle[B - A] - Angle[B_1 - A_1], B) → C
  10. Rotate(C + (c_1, 0), Angle[D_1 - C_1] + Angle[C - B] - Angle[C_1 - B_1], C) → D
  11. Rotate(D + (d_1, 0), Angle[E_1 - D_1] + Angle[D - C] - Angle[D_1 - C_1], D) → E
  12. Rotate(E + (e_1, 0), Angle[F_1 - E_1] + Angle[E - D] - Angle[E_1 - D_1], E) → F
  13. Polygon(A, B, C, D, E, F) → poly1, a, b, c, d, e, f
  14. Point(Circle[G, a_1]) → H
  15. Rotate(H + (b_1, 0), Angle[C_1 - B_1] + Angle[H - G] - Angle[B_1 - A_1], H) → I
  16. Rotate(I + (c_1, 0), Angle[D_1 - C_1] + Angle[I - H] - Angle[C_1 - B_1], I) → J
  17. Rotate(J + (d_1, 0), Angle[E_1 - D_1] + Angle[J - I] - Angle[D_1 - C_1], J) → K
  18. Rotate(K + (e_1, 0), Angle[F_1 - E_1] + Angle[K - J] - Angle[E_1 - D_1], K) → L
  19. Polygon(G, H, I, J, K, L) → poly2, g, h, i, j, k, l
  20. Point(Circle[M, a_1]) → N
  21. Rotate(N + (b_1, 0), Angle[C_1 - B_1] + Angle[N - M] - Angle[B_1 - A_1], N) → O
  22. Rotate(O + (c_1, 0), Angle[D_1 - C_1] + Angle[O - N] - Angle[C_1 - B_1], O) → P
  23. Rotate(P + (d_1, 0), Angle[E_1 - D_1] + Angle[P - O] - Angle[D_1 - C_1], P) → Q
  24. Rotate(Q + (e_1, 0), Angle[F_1 - E_1] + Angle[Q - P] - Angle[E_1 - D_1], Q) → R
  25. Polygon(M, N, O, P, Q, R) → poly3, m, n, o, p, q, r
  26. Point(Circle[S, a_1]) → T
  27. Rotate(T + (b_1, 0), Angle[C_1 - B_1] + Angle[T - S] - Angle[B_1 - A_1], T) → U
  28. Rotate(U + (c_1, 0), Angle[D_1 - C_1] + Angle[U - T] - Angle[C_1 - B_1], U) → V
  29. Rotate(V + (d_1, 0), Angle[E_1 - D_1] + Angle[V - U] - Angle[D_1 - C_1], V) → W
  30. Rotate(W + (e_1, 0), Angle[F_1 - E_1] + Angle[W - V] - Angle[E_1 - D_1], W) → Z
  31. Polygon(S, T, U, V, W, Z) → poly4, s, t, u, v, w, z
  32. Point(Circle[G_1, a_1]) → H_1
  33. Rotate(H_1 + (b_1, 0), Angle[C_1 - B_1] + Angle[H_1 - G_1] - Angle[B_1 - A_1], H_1) → I_1
  34. Rotate(I_1 + (c_1, 0), Angle[D_1 - C_1] + Angle[I_1 - H_1] - Angle[C_1 - B_1], I_1) → J_1
  35. Rotate(J_1 + (d_1, 0), Angle[E_1 - D_1] + Angle[J_1 - I_1] - Angle[D_1 - C_1], J_1) → K_1
  36. Rotate(K_1 + (e_1, 0), Angle[F_1 - E_1] + Angle[K_1 - J_1] - Angle[E_1 - D_1], K_1) → L_1
  37. Polygon(G_1, H_1, I_1, J_1, K_1, L_1) → poly5, g_2, h_2, i_2, j_2, k_2, l_2
  38. Point(Circle[M_1, a_1]) → N_1
  39. Rotate(N_1 + (b_1, 0), Angle[C_1 - B_1] + Angle[N_1 - M_1] - Angle[B_1 - A_1], N_1) → O_1
  40. Rotate(O_1 + (c_1, 0), Angle[D_1 - C_1] + Angle[O_1 - N_1] - Angle[C_1 - B_1], O_1) → P_1
  41. Rotate(P_1 + (d_1, 0), Angle[E_1 - D_1] + Angle[P_1 - O_1] - Angle[D_1 - C_1], P_1) → Q_1
  42. Rotate(Q_1 + (e_1, 0), Angle[F_1 - E_1] + Angle[Q_1 - P_1] - Angle[E_1 - D_1], Q_1) → R_1
  43. Polygon(M_1, N_1, O_1, P_1, Q_1, R_1) → poly6, m_1, n_1, o_1, p_1, q_1, r_1
  44. Point(Circle[S_1, a_1]) → T_1
  45. Rotate(T_1 + (b_1, 0), Angle[C_1 - B_1] + Angle[T_1 - S_1] - Angle[B_1 - A_1], T_1) → U_1
  46. Rotate(U_1 + (c_1, 0), Angle[D_1 - C_1] + Angle[U_1 - T_1] - Angle[C_1 - B_1], U_1) → V_1
  47. Rotate(V_1 + (d_1, 0), Angle[E_1 - D_1] + Angle[V_1 - U_1] - Angle[D_1 - C_1], V_1) → W_1
  48. Rotate(W_1 + (e_1, 0), Angle[F_1 - E_1] + Angle[W_1 - V_1] - Angle[E_1 - D_1], W_1) → Z_1
  49. Polygon(S_1, T_1, U_1, V_1, W_1, Z_1) → poly7, s_1, t_1, u_1, v_1, w_1, z_1
  50. Point(Circle[A_2, a_1]) → B_2
  51. Rotate(B_2 + (b_1, 0), Angle[C_1 - B_1] + Angle[B_2 - A_2] - Angle[B_1 - A_1], B_2) → C_2
  52. Rotate(C_2 + (c_1, 0), Angle[D_1 - C_1] + Angle[C_2 - B_2] - Angle[C_1 - B_1], C_2) → D_2
  53. Rotate(D_2 + (d_1, 0), Angle[E_1 - D_1] + Angle[D_2 - C_2] - Angle[D_1 - C_1], D_2) → E_2
  54. Rotate(E_2 + (e_1, 0), Angle[F_1 - E_1] + Angle[E_2 - D_2] - Angle[E_1 - D_1], E_2) → F_2
  55. Polygon(A_2, B_2, C_2, D_2, E_2, F_2) → poly8, a_2, b_2, c_2, d_2, e_2, f_2
  56. Point(Circle[G_2, a_1]) → H_2
  57. Rotate(H_2 + (b_1, 0), Angle[C_1 - B_1] + Angle[H_2 - G_2] - Angle[B_1 - A_1], H_2) → I_2
  58. Rotate(I_2 + (c_1, 0), Angle[D_1 - C_1] + Angle[I_2 - H_2] - Angle[C_1 - B_1], I_2) → J_2
  59. Rotate(J_2 + (d_1, 0), Angle[E_1 - D_1] + Angle[J_2 - I_2] - Angle[D_1 - C_1], J_2) → K_2
  60. Rotate(K_2 + (e_1, 0), Angle[F_1 - E_1] + Angle[K_2 - J_2] - Angle[E_1 - D_1], K_2) → L_2
  61. Polygon(G_2, H_2, I_2, J_2, K_2, L_2) → poly9, g_3, h_3, i_3, j_3, k_3, l_3
  62. Point(Circle[M_2, a_1]) → N_2
  63. Rotate(N_2 + (b_1, 0), Angle[C_1 - B_1] + Angle[N_2 - M_2] - Angle[B_1 - A_1], N_2) → O_2
  64. Rotate(O_2 + (c_1, 0), Angle[D_1 - C_1] + Angle[O_2 - N_2] - Angle[C_1 - B_1], O_2) → P_2
  65. Rotate(P_2 + (d_1, 0), Angle[E_1 - D_1] + Angle[P_2 - O_2] - Angle[D_1 - C_1], P_2) → Q_2
  66. Rotate(Q_2 + (e_1, 0), Angle[F_1 - E_1] + Angle[Q_2 - P_2] - Angle[E_1 - D_1], Q_2) → R_2
  67. Polygon(M_2, N_2, O_2, P_2, Q_2, R_2) → poly10, m_2, n_2, o_2, p_2, q_2, r_2
  68. Point(Circle[S_2, a_1]) → T_2
  69. Rotate(T_2 + (b_1, 0), Angle[C_1 - B_1] + Angle[T_2 - S_2] - Angle[B_1 - A_1], T_2) → U_2
  70. Rotate(U_2 + (c_1, 0), Angle[D_1 - C_1] + Angle[U_2 - T_2] - Angle[C_1 - B_1], U_2) → V_2
  71. Rotate(V_2 + (d_1, 0), Angle[E_1 - D_1] + Angle[V_2 - U_2] - Angle[D_1 - C_1], V_2) → W_2
  72. Rotate(W_2 + (e_1, 0), Angle[F_1 - E_1] + Angle[W_2 - V_2] - Angle[E_1 - D_1], W_2) → Z_2
  73. Polygon(S_2, T_2, U_2, V_2, W_2, Z_2) → poly11, s_2, t_2, u_2, v_2, w_2, z_2
  74. Point(Circle[A_3, a_1]) → B_3
  75. Rotate(B_3 + (b_1, 0), Angle[C_1 - B_1] + Angle[B_3 - A_3] - Angle[B_1 - A_1], B_3) → C_3
  76. Rotate(C_3 + (c_1, 0), Angle[D_1 - C_1] + Angle[C_3 - B_3] - Angle[C_1 - B_1], C_3) → D_3
  77. Rotate(D_3 + (d_1, 0), Angle[E_1 - D_1] + Angle[D_3 - C_3] - Angle[D_1 - C_1], D_3) → E_3
  78. Rotate(E_3 + (e_1, 0), Angle[F_1 - E_1] + Angle[E_3 - D_3] - Angle[E_1 - D_1], E_3) → F_3
  79. Polygon(A_3, B_3, C_3, D_3, E_3, F_3) → poly12, a_3, b_3, c_3, d_3, e_3, f_3
  80. Point(Circle[G_3, a_1]) → H_3
  81. Rotate(H_3 + (b_1, 0), Angle[C_1 - B_1] + Angle[H_3 - G_3] - Angle[B_1 - A_1], H_3) → I_3
  82. Rotate(I_3 + (c_1, 0), Angle[D_1 - C_1] + Angle[I_3 - H_3] - Angle[C_1 - B_1], I_3) → J_3
  83. Rotate(J_3 + (d_1, 0), Angle[E_1 - D_1] + Angle[J_3 - I_3] - Angle[D_1 - C_1], J_3) → K_3
  84. Rotate(K_3 + (e_1, 0), Angle[F_1 - E_1] + Angle[K_3 - J_3] - Angle[E_1 - D_1], K_3) → L_3
  85. Polygon(G_3, H_3, I_3, J_3, K_3, L_3) → poly13, g_4, h_4, i_4, j_4, k_4, l_4
  86. Point(Circle[M_3, a_1]) → N_3
  87. Rotate(N_3 + (b_1, 0), Angle[C_1 - B_1] + Angle[N_3 - M_3] - Angle[B_1 - A_1], N_3) → O_3
  88. Rotate(O_3 + (c_1, 0), Angle[D_1 - C_1] + Angle[O_3 - N_3] - Angle[C_1 - B_1], O_3) → P_3
  89. Rotate(P_3 + (d_1, 0), Angle[E_1 - D_1] + Angle[P_3 - O_3] - Angle[D_1 - C_1], P_3) → Q_3
  90. Rotate(Q_3 + (e_1, 0), Angle[F_1 - E_1] + Angle[Q_3 - P_3] - Angle[E_1 - D_1], Q_3) → R_3
  91. Polygon(M_3, N_3, O_3, P_3, Q_3, R_3) → poly14, m_3, n_3, o_3, p_3, q_3, r_3
  92. Point(Circle[S_3, a_1]) → T_3
  93. Rotate(T_3 + (b_1, 0), Angle[C_1 - B_1] + Angle[T_3 - S_3] - Angle[B_1 - A_1], T_3) → U_3
  94. Rotate(U_3 + (c_1, 0), Angle[D_1 - C_1] + Angle[U_3 - T_3] - Angle[C_1 - B_1], U_3) → V_3
  95. Rotate(V_3 + (d_1, 0), Angle[E_1 - D_1] + Angle[V_3 - U_3] - Angle[D_1 - C_1], V_3) → W_3
  96. Rotate(W_3 + (e_1, 0), Angle[F_1 - E_1] + Angle[W_3 - V_3] - Angle[E_1 - D_1], W_3) → Z_3
  97. Polygon(S_3, T_3, U_3, V_3, W_3, Z_3) → poly15, s_3, t_3, u_3, v_3, w_3, z_3
  98. Point(Circle[A_4, a_1]) → B_4
  99. Rotate(B_4 + (b_1, 0), Angle[C_1 - B_1] + Angle[B_4 - A_4] - Angle[B_1 - A_1], B_4) → C_4
  100. Rotate(C_4 + (c_1, 0), Angle[D_1 - C_1] + Angle[C_4 - B_4] - Angle[C_1 - B_1], C_4) → D_4
  101. Rotate(D_4 + (d_1, 0), Angle[E_1 - D_1] + Angle[D_4 - C_4] - Angle[D_1 - C_1], D_4) → E_4
  102. Rotate(E_4 + (e_1, 0), Angle[F_1 - E_1] + Angle[E_4 - D_4] - Angle[E_1 - D_1], E_4) → F_4
  103. Polygon(A_4, B_4, C_4, D_4, E_4, F_4) → poly16, a_4, b_4, c_4, d_4, e_4, f_4
  104. Point(Circle[G_4, a_1]) → H_4
  105. Rotate(H_4 + (b_1, 0), Angle[C_1 - B_1] + Angle[H_4 - G_4] - Angle[B_1 - A_1], H_4) → I_4
  106. Rotate(I_4 + (c_1, 0), Angle[D_1 - C_1] + Angle[I_4 - H_4] - Angle[C_1 - B_1], I_4) → J_4
  107. Rotate(J_4 + (d_1, 0), Angle[E_1 - D_1] + Angle[J_4 - I_4] - Angle[D_1 - C_1], J_4) → K_4
  108. Rotate(K_4 + (e_1, 0), Angle[F_1 - E_1] + Angle[K_4 - J_4] - Angle[E_1 - D_1], K_4) → L_4
  109. Polygon(G_4, H_4, I_4, J_4, K_4, L_4) → poly17, g_5, h_5, i_5, j_5, k_5, l_5
  110. Point(Circle[M_4, a_1]) → N_4
  111. Rotate(N_4 + (b_1, 0), Angle[C_1 - B_1] + Angle[N_4 - M_4] - Angle[B_1 - A_1], N_4) → O_4
  112. Rotate(O_4 + (c_1, 0), Angle[D_1 - C_1] + Angle[O_4 - N_4] - Angle[C_1 - B_1], O_4) → P_4
  113. Rotate(P_4 + (d_1, 0), Angle[E_1 - D_1] + Angle[P_4 - O_4] - Angle[D_1 - C_1], P_4) → Q_4
  114. Rotate(Q_4 + (e_1, 0), Angle[F_1 - E_1] + Angle[Q_4 - P_4] - Angle[E_1 - D_1], Q_4) → R_4
  115. Polygon(M_4, N_4, O_4, P_4, Q_4, R_4) → poly18, m_4, n_4, o_4, p_4, q_4, r_4
  116. Point(Circle[S_4, a_1]) → T_4
  117. Rotate(T_4 + (b_1, 0), Angle[C_1 - B_1] + Angle[T_4 - S_4] - Angle[B_1 - A_1], T_4) → U_4
  118. Rotate(U_4 + (c_1, 0), Angle[D_1 - C_1] + Angle[U_4 - T_4] - Angle[C_1 - B_1], U_4) → V_4
  119. Rotate(V_4 + (d_1, 0), Angle[E_1 - D_1] + Angle[V_4 - U_4] - Angle[D_1 - C_1], V_4) → W_4
  120. Rotate(W_4 + (e_1, 0), Angle[F_1 - E_1] + Angle[W_4 - V_4] - Angle[E_1 - D_1], W_4) → Z_4
  121. Polygon(S_4, T_4, U_4, V_4, W_4, Z_4) → poly19, s_4, t_4, u_4, v_4, w_4, z_4
  122. Point(Circle[A_5, a_1]) → B_5
  123. Rotate(B_5 + (b_1, 0), Angle[C_1 - B_1] + Angle[B_5 - A_5] - Angle[B_1 - A_1], B_5) → C_5
  124. Rotate(C_5 + (c_1, 0), Angle[D_1 - C_1] + Angle[C_5 - B_5] - Angle[C_1 - B_1], C_5) → D_5
  125. Rotate(D_5 + (d_1, 0), Angle[E_1 - D_1] + Angle[D_5 - C_5] - Angle[D_1 - C_1], D_5) → E_5
  126. Rotate(E_5 + (e_1, 0), Angle[F_1 - E_1] + Angle[E_5 - D_5] - Angle[E_1 - D_1], E_5) → F_5
  127. Polygon(A_5, B_5, C_5, D_5, E_5, F_5) → poly20, a_5, b_5, c_5, d_5, e_5, f_5
  128. Point(Circle[G_5, a_5]) → H_5
  129. Rotate(H_5 + (b_5, 0), Angle[C_5 - B_5] + Angle[H_5 - G_5] - Angle[B_5 - A_5], H_5) → I_5
  130. Rotate(I_5 + (c_5, 0), Angle[D_5 - C_5] + Angle[I_5 - H_5] - Angle[C_5 - B_5], I_5) → J_5
  131. Rotate(J_5 + (d_5, 0), Angle[E_5 - D_5] + Angle[J_5 - I_5] - Angle[D_5 - C_5], J_5) → K_5
  132. Rotate(K_5 + (e_5, 0), Angle[F_5 - E_5] + Angle[K_5 - J_5] - Angle[E_5 - D_5], K_5) → L_5
  133. Polygon(G_5, H_5, I_5, J_5, K_5, L_5) → poly21, g_6, h_6, i_6, j_6, k_6, l_6

AI 绘图提示语参考

绘制一个由六个正六边形组成的图案，每个六边形通过旋转和组合形成。首先创建一个正六边形，然后依次连接其顶点形成边。接着，以其中一个顶点为中心，将正六边形旋转一定角度，并重复此步骤，生成多个旋转后的六边形。最后，将这些六边形组合在一起，形成复杂的六边形环状图案，展示旋转和组合的几何关系。
以上提示语为根据原始绘图命令整理而来，仅供参考。