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几何图案——伊西斯项目：构造与谜题

几何图案——伊西斯项目：构造与谜题
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2019-10-15 03:02
圆多边形三角函数变换几何作图
① 本范例演示了通过一系列几何作图命令生成复杂几何图案的过程，并配有谜题式探索，展示几何变换与构造的视觉美感。② 核心数学概念包括几何变换（平移、相似三角形）、切线函数、中线、正交线以及多边形与圆的组合关系，涉及解析几何与三角函数的交叉应用。③ 作图要点：从构造摘要可见，使用了GeoGebra的Intersect（交点）、Point（点）、Circle（圆）、OrthogonalLine（正交线）、Segment（线段）、Line（直线）、Polygon（多边形）等命令，创建了A、B、C、D、E、F、G、H、I、J、K、L、M、N、O、P、Q、R、S等大量对象，以及辅助线c、d、e、f、g、h、i、j、k、l、m、n、p、q、r、s、t、a、b、f_1、g_1，通过交点与正交关系构建出对称且富有规律的图案。④ 适合中学高年级至大学低年级的几何课程，也可用于数学艺术设计或几何谜题挑战，帮助学生理解几何构造逻辑与变换性质。

GeoGebra 命令及参数

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Circle(A, B) → c
  4. Circle(B, Segment[A, B]) → d
  5. Intersect(c, xAxis, 2) → C
  6. Circle(C, Segment[A, B]) → e
  7. OrthogonalLine(A, xAxis) → f
  8. Intersect(c, f, 1) → D
  9. Intersect(c, f, 2) → E
  10. Circle(D, Segment[A, B]) → g
  11. Circle(E, Segment[A, B]) → h
  12. Intersect(e, h, 1) → F
  13. Intersect(d, g, 1) → G
  14. Intersect(d, h, 2) → H
  15. Intersect(e, g, 2) → I
  16. Segment(I, H) → i
  17. Segment(F, G) → j
  18. Segment(F, H) → k
  19. Segment(H, G) → l
  20. Segment(G, I) → m
  21. Segment(I, F) → n
  22. Intersect(c, j, 1) → J
  23. Intersect(c, i, 2) → K
  24. Intersect(c, j, 2) → L
  25. Intersect(c, i, 1) → M
  26. Line(J, K) → p
  27. Line(M, L) → q
  28. Line(L, K) → r
  29. Line(J, M) → s
  30. Segment(D, C) → t
  31. Segment(C, E) → a
  32. Segment(E, B) → b
  33. Segment(B, D) → f_1
  34. Intersect(h, r, 1) → N
  35. Intersect(d, p, 1) → O
  36. Intersect(e, a, 1) → P
  37. Intersect(h, a, 1) → Q
  38. Intersect(g, t, 1) → R
  39. Intersect(d, q, 1) → S
  40. Line(P, A) → g_1
  41. Line(O, A) → h_1
  42. Line(N, A) → i_1
  43. Intersect(g, f_1, 1) → T
  44. Line(T, A) → j_1
  45. Intersect(e, h_1, 2) → U
  46. Intersect(g_1, m) → V
  47. Circle(V, Segment[V, D]) → c_1
  48. Intersect(c_1, g_1, 1) → W
  49. Circle(A, Segment[A, W]) → d_1
  50. Intersect(d_1, g_1, 1) → Z
  51. Intersect(d_1, h_1, 1) → A_1
  52. Intersect(d_1, i_1, 1) → B_1
  53. Line(A, B) → k_1
  54. Intersect(d_1, j_1, 1) → C_1
  55. Intersect(d_1, h_1, 2) → D_1
  56. Intersect(d_1, i_1, 2) → E_1
  57. Intersect(d_1, j_1, 2) → F_1
  58. Line(Z, D_1) → l_1
  59. Line(W, A_1) → m_1
  60. Line(F_1, B_1) → n_1
  61. Line(C_1, E_1) → p_1
  62. Line(F_1, W) → q_1
  63. Line(C_1, Z) → r_1
  64. Line(A_1, E_1) → s_1
  65. Line(D_1, B_1) → t_1
  66. Line(B, K) → a_1
  67. Line(K, E) → b_1
  68. Line(E, J) → e_1
  69. Line(J, C) → f_2
  70. Line(C, M) → g_2
  71. Line(M, D) → h_2
  72. Line(D, L) → i_2
  73. Line(L, B) → j_2
  74. Intersect(l_1, h_2) → G_1
  75. Intersect(j_2, h_2) → H_1
  76. Segment(G_1, H_1) → k_2
  77. Intersect(i_1, f_2) → I_1
  78. Intersect(h_2, t_1) → J_1
  79. Segment(J_1, I_1) → l_2
  80. Intersect(h_1, g_2) → K_1
  81. Intersect(j_1, a_1) → L_1
  82. Intersect(i_2, q_1) → M_1
  83. Segment(M_1, L_1) → m_2
  84. Intersect(m_1, i_2) → N_1
  85. Segment(N_1, K_1) → n_2
  86. Intersect(i_1, b_1) → O_1
  87. Intersect(p_1, j_2) → P_1
  88. Segment(O_1, P_1) → p_2
  89. Intersect(r_1, j_2) → Q_1
  90. Segment(Q_1, H_1) → q_2
  91. Intersect(s_1, b_1) → R_1
  92. Segment(R_1, O_1) → r_2
  93. Intersect(m_1, b_1) → S_1
  94. Intersect(g_1, b_1) → T_1
  95. Segment(T_1, S_1) → s_2
  96. Intersect(l_1, e_1) → U_1
  97. Intersect(h_1, e_1) → V_1
  98. Segment(V_1, U_1) → t_2
  99. Intersect(r_1, e_1) → W_1
  100. Intersect(j_1, e_1) → Z_1
  101. Segment(W_1, Z_1) → a_2
  102. Intersect(n_1, a_1) → A_2
  103. Segment(A_2, L_1) → b_2
  104. Intersect(t_1, a_1) → B_2
  105. Segment(B_2, V_1) → c_2
  106. Intersect(n_1, f_2) → C_2
  107. Intersect(f_2, q_1) → D_2
  108. Segment(T_1, D_2) → d_2
  109. Segment(C_2, I_1) → e_2
  110. Intersect(s_1, g_2) → E_2
  111. Segment(K_1, E_2) → f_3
  112. Intersect(p_1, g_2) → F_2
  113. Segment(Z_1, F_2) → g_3
  114. Intersect(h_1, p_1) → G_2
  115. Intersect(g_1, p_1) → H_2
  116. Segment(F_2, G_2) → h_3
  117. Segment(H_2, P_1) → i_3
  118. Intersect(s_1, g_1) → I_2
  119. Intersect(h_1, r_1) → J_2
  120. Segment(C_2, I_2) → j_3
  121. Segment(J_2, A_2) → k_3
  122. Intersect(m_1, r_1) → K_2
  123. Intersect(j_1, m_1) → L_2
  124. Segment(S_1, K_2) → l_3
  125. Segment(L_2, N_1) → m_3
  126. Intersect(j_1, l_1) → M_2
  127. Intersect(i_1, l_1) → N_2
  128. Segment(U_1, M_2) → n_3
  129. Segment(N_2, G_1) → p_3
  130. Segment(W_1, J_2) → q_3
  131. Segment(D_2, N_2) → r_3
  132. Segment(K_2, Q_1) → s_3
  133. Segment(M_1, G_2) → t_3
  134. Segment(R_1, I_2) → a_3
  135. Segment(B_2, L_2) → b_3
  136. Segment(J_1, H_2) → c_3
  137. Segment(E_2, M_2) → d_3
  138. Polygon(E, S_1, A_1, R_1, K, V_1) → pol1, e_3, s_4, a_4, r_4, k_4, v_1
  139. Polygon(K, O_1, B, A_2, B_1, B_2) → pol2, k_5, o_1, b_4, a_5, b_5, b_6
  140. Polygon(B, P_1, C_1, Q_1, L, L_1) → pol3, b_7, p_4, c_4, q_4, l_4, l_5
  141. Polygon(D, H_1, L, M_1, W, N_1) → pol4, d_4, h_4, l_6, m_4, w, n_4
  142. Polygon(D, K_1, M, J_1, D_1, G_1) → pol5, d_5, k_6, m_5, j_4, d_6, g_4
  143. Polygon(M, E_2, E_1, F_2, C, I_1) → pol6, m_6, e_4, e_5, f_4, c_5, i_4
  144. Polygon(J, Z_1, C, C_2, F_1, D_2) → pol7, j_5, z_1, c_6, c_7, f_5, d_7
  145. Polygon(J, T_1, E, U_1, Z, W_1) → pol8, j_6, t_4, e_6, u_1, z_2, w_1
  146. Intersect(j_3, n_3) → O_2
  147. Intersect(q_3, a_3) → P_2
  148. Intersect(k_3, l_3) → Q_2
  149. Intersect(s_3, b_3) → R_2
  150. Intersect(i_3, m_3) → S_2
  151. Intersect(t_3, c_3) → T_2
  152. Intersect(h_3, p_3) → U_2
  153. Intersect(r_3, d_3) → V_2
  154. Polygon(Z, U_1, E, S_1, A_1, P_2) → pol9, z_3, u_2, e_7, s_5, a_6, p_5
  155. Polygon(F_1, O_2, Z, W_1, J, D_2) → pol10, f_6, o_2, z_4, w_2, j_7, d_8
  156. Polygon(C_2, F_1, V_2, E_1, F_2, C) → pol11, c_8, f_7, v_2, e_8, f_8, c_9
  157. Polygon(E_1, U_2, D_1, J_1, M, E_2) → pol12, e_9, u_3, d_9, j_8, m_7, e_{10}
  158. Polygon(T_2, D_1, G_1, D, N_1, W) → pol13, t_5, d_{10}, g_5, d_{11}, n_5, w_3
  159. Polygon(S_2, C_1, Q_1, L, M_1, W) → pol14, s_6, c_{10}, q_5, l_7, m_8, w_4
  160. Polygon(R_2, C_1, P_1, B, A_2, B_1) → pol15, r_5, c_{11}, p_6, b_8, a_7, b_9
  161. Polygon(A_1, R_1, K, B_2, B_1, Q_2) → pol16, a_8, r_6, k_7, b_{10}, b_{11}, q_6
  162. Polygon(O_2, I_2, P_2, Z) → q1, o_3, i_5, p_7, z_5
  163. Polygon(F_1, V_2, M_2, O_2) → q2, f_9, v_3, m_9, o_4
  164. Polygon(V_2, N_2, U_2, E_1) → q3, v_4, n_6, u_4, e_{11}
  165. Polygon(T_2, D_1, U_2, G_2) → q4, t_6, d_{12}, u_5, g_6
  166. Polygon(T_2, W, S_2, H_2) → q5, t_7, w_5, s_7, h_5
  167. Polygon(R_2, L_2, S_2, C_1) → q6, r_7, l_8, s_8, c_{12}
  168. Polygon(R_2, B_1, Q_2, K_2) → q7, r_8, b_{12}, q_7, k_8
  169. Polygon(P_2, A_1, Q_2, J_2) → q8, p_8, a_9, q_8, j_9
  170. Polygon(M_2, O_2, H_2, T_2, G_2, U_2, N_2, V_2, I_2, P_2, J_2, Q_2, K_2, R_2, L_2, S_2) → pol17, m_{10}, s_9, h_6, t_8, g_7, u_6, n_7, v_5, o_5, i_6, p_9, j_{10}, q_9, k_9, r_9, l_9