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旋转与镶嵌

旋转与镶嵌
几何画板及绘图内容加载中，请稍等……
2020-07-22 18:29
多边形变换几何作图
本范例演示了如何通过旋转操作生成镶嵌图案。核心数学概念包括旋转对称性和几何变换。通过使用GeoGebra命令如Point, Polygon, Rotate, LineBisector等，可以构造出旋转对称的多边形。适合初中几何教学，帮助学生理解旋转对称性和镶嵌图案的生成过程。

GeoGebra 命令及参数

  1. Point(Circle[A, 1]) → B
  2. Polygon(A, B, 3) → poly1, f, g, h, C
  3. Point(Circle[D, 1]) → E
  4. Rotate(E, 90°, D) → E'
  5. Polygon(D, E, E') → t1, e', d, e
  6. Point(Circle[M, 1]) → N
  7. Rotate(N, 90°, M) → N'
  8. Rotate(M, 150°, N') → M'
  9. LineBisector(N, M') → i
  10. Midpoint(N, M') → F
  11. Circle(F, M') → c
  12. Intersect(c, i, 1) → G
  13. Polygon(M, N, G, M', N') → poly2, m, n, g_1, m', n'
  14. Point(Circle[H, 1]) → I
  15. Polygon(H, I, 4) → poly3, j, k, l, p, J, K
  16. Point(Circle[L, 1]) → O
  17. Polygon(L, O, 8) → poly4, q, Q, R, S, T, U, r, s, t, a, b, f_1, h_1, P
  18. Point(Circle[V, 1]) → W
  19. Polygon(V, W, 6) → poly5, i_1, C_1, j_1, k_1, l_1, m_1, n_1, Z, A_1, B_1
  20. Point(Circle[D_1, 1]) → E_1
  21. Rotate(E_1, 60°, D_1) → F_1
  22. Rotate(D_1, 120°, F_1) → G_1
  23. Polygon(D_1, E_1, G_1, F_1) → q1, d_1, e_1, p_1, q_1
  24. Point(Circle[A_2, 1]) → B_2
  25. Polygon(A_2, B_2, 3) → poly1_1, f_2, g_2, h_2, C_2
  26. Point(Circle[A_3, 1]) → B_3
  27. Polygon(A_3, B_3, 3) → poly1_2, f_3, g_3, h_3, C_3
  28. Point(Circle[A_4, 1]) → B_4
  29. Polygon(A_4, B_4, 3) → poly1_3, f_4, g_4, h_4, C_4
  30. Point(Circle[A_5, 1]) → B_5
  31. Polygon(A_5, B_5, 3) → poly1_4, f_5, g_5, h_5, C_5
  32. Point(Circle[A_6, 1]) → B_6
  33. Polygon(A_6, B_6, 3) → poly1_5, f_6, g_6, h_6, C_6
  34. Point(Circle[D_2, 1]) → E_2
  35. Rotate(E_2, 90°, D_2) → E'_1
  36. Polygon(D_2, E_2, E'_1) → t1_1, e'_1, d_2, e_2
  37. Point(Circle[D_3, 1]) → E_3
  38. Rotate(E_3, 90°, D_3) → E'_2
  39. Polygon(D_3, E_3, E'_2) → t1_2, e'_2, d_3, e_3
  40. Point(Circle[D_4, 1]) → E_4
  41. Rotate(E_4, 90°, D_4) → E'_3
  42. Polygon(D_4, E_4, E'_3) → t1_3, e'_3, d_4, e_4
  43. Point(Circle[D_5, 1]) → E_5
  44. Rotate(E_5, 90°, D_5) → E'_4
  45. Polygon(D_5, E_5, E'_4) → t1_4, e'_4, d_5, e_5
  46. Point(Circle[D_6, 1]) → E_6
  47. Rotate(E_6, 90°, D_6) → E'_5
  48. Polygon(D_6, E_6, E'_5) → t1_5, e'_5, d_6, e_6
  49. Point(Circle[H_1, 1]) → I_1
  50. Polygon(H_1, I_1, 4) → poly3_1, j_2, k_2, l_2, p_2, J_1, K_1
  51. Point(Circle[H_2, 1]) → I_2
  52. Polygon(H_2, I_2, 4) → poly3_2, j_3, k_3, l_3, p_3, J_2, K_2
  53. Point(Circle[H_3, 1]) → I_3
  54. Polygon(H_3, I_3, 4) → poly3_3, j_4, k_4, l_4, p_4, J_3, K_3
  55. Point(Circle[H_4, 1]) → I_4
  56. Polygon(H_4, I_4, 4) → poly3_4, j_5, k_5, l_5, p_5, J_4, K_4
  57. Point(Circle[H_5, 1]) → I_5
  58. Polygon(H_5, I_5, 4) → poly3_5, j_6, k_6, l_6, p_6, J_5, K_5
  59. Point(Circle[M_1, 1]) → N_1
  60. Rotate(N_1, 90°, M_1) → N'_1
  61. Rotate(M_1, 150°, N'_1) → M'_1
  62. LineBisector(N_1, M'_1) → i_2
  63. Midpoint(N_1, M'_1) → F_2
  64. Circle(F_2, M'_1) → c_1
  65. Intersect(c_1, i_2, 1) → G_2
  66. Polygon(M_1, N_1, G_2, M'_1, N'_1) → poly2_1, m_2, n_2, g_7, m'_1, n'_1
  67. Point(Circle[M_2, 1]) → N_2
  68. Rotate(N_2, 90°, M_2) → N'_2
  69. Rotate(M_2, 150°, N'_2) → M'_2
  70. LineBisector(N_2, M'_2) → i_3
  71. Midpoint(N_2, M'_2) → F_3
  72. Circle(F_3, M'_2) → c_2
  73. Intersect(c_2, i_3, 1) → G_3
  74. Polygon(M_2, N_2, G_3, M'_2, N'_2) → poly2_2, m_3, n_3, g_8, m'_2, n'_2
  75. Point(Circle[M_3, 1]) → N_3
  76. Rotate(N_3, 90°, M_3) → N'_3
  77. Rotate(M_3, 150°, N'_3) → M'_3
  78. LineBisector(N_3, M'_3) → i_4
  79. Midpoint(N_3, M'_3) → F_4
  80. Circle(F_4, M'_3) → c_3
  81. Intersect(c_3, i_4, 1) → G_4
  82. Polygon(M_3, N_3, G_4, M'_3, N'_3) → poly2_3, m_4, n_4, g_9, m'_3, n'_3
  83. Point(Circle[M_4, 1]) → N_4
  84. Rotate(N_4, 90°, M_4) → N'_4
  85. Rotate(M_4, 150°, N'_4) → M'_4
  86. LineBisector(N_4, M'_4) → i_5
  87. Midpoint(N_4, M'_4) → F_5
  88. Circle(F_5, M'_4) → c_4
  89. Intersect(c_4, i_5, 1) → G_5
  90. Polygon(M_4, N_4, G_5, M'_4, N'_4) → poly2_4, m_5, n_5, g_{10}, m'_4, n'_4
  91. Point(Circle[M_5, 1]) → N_5
  92. Rotate(N_5, 90°, M_5) → N'_5
  93. Rotate(M_5, 150°, N'_5) → M'_5
  94. LineBisector(N_5, M'_5) → i_6
  95. Midpoint(N_5, M'_5) → F_6
  96. Circle(F_6, M'_5) → c_5
  97. Intersect(c_5, i_6, 1) → G_6
  98. Polygon(M_5, N_5, G_6, M'_5, N'_5) → poly2_5, m_6, n_6, g_{11}, m'_5, n'_5
  99. Point(Circle[M_6, 1]) → N_6
  100. Rotate(N_6, 90°, M_6) → N'_6
  101. Rotate(M_6, 150°, N'_6) → M'_6
  102. LineBisector(N_6, M'_6) → i_7
  103. Midpoint(N_6, M'_6) → F_7
  104. Circle(F_7, M'_6) → c_6
  105. Intersect(c_6, i_7, 1) → G_7
  106. Polygon(M_6, N_6, G_7, M'_6, N'_6) → poly2_6, m_7, n_7, g_{12}, m'_6, n'_6
  107. Point(Circle[D_7, 1]) → E_7
  108. Rotate(E_7, 60°, D_7) → F_8
  109. Rotate(D_7, 120°, F_8) → G_8
  110. Polygon(D_7, E_7, G_8, F_8) → q1_1, d_7, e_7, p_7, q_2
  111. Point(Circle[D_8, 1]) → E_8
  112. Rotate(E_8, 60°, D_8) → F_9
  113. Rotate(D_8, 120°, F_9) → G_9
  114. Polygon(D_8, E_8, G_9, F_9) → q1_2, d_8, e_8, p_8, q_3
  115. Point(Circle[D_9, 1]) → E_9
  116. Rotate(E_9, 60°, D_9) → F_{10}
  117. Rotate(D_9, 120°, F_{10}) → G_{10}
  118. Polygon(D_9, E_9, G_{10}, F_{10}) → q1_3, d_9, e_9, p_9, q_4
  119. Point(Circle[D_{10}, 1]) → E_{10}
  120. Rotate(E_{10}, 60°, D_{10}) → F_{11}
  121. Rotate(D_{10}, 120°, F_{11}) → G_{11}
  122. Polygon(D_{10}, E_{10}, G_{11}, F_{11}) → q1_4, d_{10}, e_{10}, p_{10}, q_5
  123. Point(Circle[D_{11}, 1]) → E_{11}
  124. Rotate(E_{11}, 60°, D_{11}) → F_{12}
  125. Rotate(D_{11}, 120°, F_{12}) → G_{12}
  126. Polygon(D_{11}, E_{11}, G_{12}, F_{12}) → q1_5, d_{11}, e_{11}, p_{11}, q_6
  127. Point(Circle[V_1, 1]) → W_1
  128. Polygon(V_1, W_1, 6) → poly5_1, i_8, C_7, j_7, k_7, l_7, m_8, n_8, Z_1, A_7, B_7
  129. Point(Circle[V_2, 1]) → W_2
  130. Polygon(V_2, W_2, 6) → poly5_2, i_9, C_8, j_8, k_8, l_8, m_9, n_9, Z_2, A_8, B_8
  131. Point(Circle[V_3, 1]) → W_3
  132. Polygon(V_3, W_3, 6) → poly5_3, i_{10}, C_9, j_9, k_9, l_9, m_{10}, n_{10}, Z_3, A_9, B_9
  133. Point(Circle[V_4, 1]) → W_4
  134. Polygon(V_4, W_4, 6) → poly5_4, i_{11}, C_{10}, j_{10}, k_{10}, l_{10}, m_{11}, n_{11}, Z_4, A_{10}, B_{10}
  135. Point(Circle[V_5, 1]) → W_5
  136. Polygon(V_5, W_5, 6) → poly5_5, i_{12}, C_{11}, j_{11}, k_{11}, l_{11}, m_{12}, n_{12}, Z_5, A_{11}, B_{11}
  137. Point(Circle[L_1, 1]) → O_1
  138. Polygon(L_1, O_1, 8) → poly4_1, q_7, Q_1, R_1, S_1, T_1, U_1, r_1, s_1, t_1, a_1, b_1, f_7, h_7, P_1
  139. Point(Circle[L_2, 1]) → O_2
  140. Polygon(L_2, O_2, 8) → poly4_2, q_8, Q_2, R_2, S_2, T_2, U_2, r_2, s_2, t_2, a_2, b_2, f_8, h_8, P_2
  141. Point(Circle[L_3, 1]) → O_3
  142. Polygon(L_3, O_3, 8) → poly4_3, q_9, Q_3, R_3, S_3, T_3, U_3, r_3, s_3, t_3, a_3, b_3, f_9, h_9, P_3
  143. Point(Circle[L_4, 1]) → O_4
  144. Polygon(L_4, O_4, 8) → poly4_4, q_{10}, Q_4, R_4, S_4, T_4, U_4, r_4, s_4, t_4, a_4, b_4, f_{10}, h_{10}, P_4
  145. Point(Circle[L_5, 1]) → O_5
  146. Polygon(L_5, O_5, 8) → poly4_5, q_{11}, Q_5, R_5, S_5, T_5, U_5, r_5, s_5, t_5, a_5, b_5, f_{11}, h_{11}, P_5

AI 绘图提示语参考

绘制一个旋转对称的多边形图案，首先在圆上取点A和D，分别以A和D为顶点画三个和四个边的多边形，然后围绕D旋转E点90度得到E'，再以D、E和E'为顶点画一个三角形。接着围绕M旋转N点90度得到N'，再以M、N和N'为顶点画一个三角形。继续围绕N'旋转M点150度得到M'，再以M、N'和M'为顶点画一个三角形。画出一个以N和M'为对角线中点的圆，从中点F画一条与圆的交线i，再画一个以F为圆心、M'为半径的圆，从中点G画一个以M、N和G为顶点的三角形。接着在圆上取点H和L，分别以H和L为顶点画一个四边形和八边形，再在圆上取点V，以V为顶点画一个六边形。最后围绕D_1旋转E_1点60度得到F_1，再围绕F_1旋转E_1点120度得到G_1，以D_1、E_1和G_1为顶点画一个三角形。重复上述步骤，围绕A_2、A_3、A_4、A_5和A_6分别画三个三角形，围绕D_2、D_3、D_4、D_5和D_6分别画三个三角形。
以上提示语为根据原始绘图命令整理而来，仅供参考。