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正方形螺旋：内接正方形旋转构成的四重螺旋图案

正方形螺旋：内接正方形旋转构成的四重螺旋图案
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2020-03-01 10:10
多边形函数数列变换几何作图
该范例通过将正方形依次内接于更大的正方形中，并旋转一定角度，动态生成四条相互交织的螺旋线，直观展示了几何变换与序列构造的视觉效果。核心数学概念包括正方形的内接关系、旋转对称、比例缩放以及分段函数（用于描述螺旋路径），同时涉及反射与相关性等拓展思想。作图要点：使用GeoGebra的Point、Segment、Circle、Intersect、Polygon等命令创建基础图形，通过定义多个点（A、B、C、D等）、线段（f、g、h、i等）、圆（c、d等）和多边形（poly1、poly2等）对象，逐步构造出嵌套正方形并提取螺旋顶点。该范例适合中学高年级或大学低年级学生，用于探索几何变换、序列规律以及数学与艺术结合的美感，也可作为编程或动态几何软件教学的示例。

GeoGebra 命令及参数

  1. Point(xAxis) → A
  2. Point(xAxis) → B
  3. Point(yAxis) → C
  4. Segment(C, A) → f
  5. Segment(A, B) → g
  6. Segment(B, D) → h
  7. Segment(D, C) → i
  8. Point(g) → E
  9. Circle(B, d1) → c
  10. Intersect(c, h, 1) → F
  11. Polygon(E, F, 4) → poly1, j, k, l, m, G, H
  12. Segment(H, E) → n
  13. Segment(H, G) → p
  14. Segment(G, F) → q
  15. Segment(E, F) → r
  16. Circle(E, d2) → d
  17. Intersect(d, r, 1) → I
  18. Circle(F, d2) → e
  19. Intersect(e, q, 1) → J
  20. Polygon(I, J, 4) → poly2, s, t, a, b, K, L
  21. Segment(L, I) → f_1
  22. Segment(I, J) → g_1
  23. Segment(J, K) → h_1
  24. Segment(K, L) → i_1
  25. Circle(I, d3) → c_1
  26. Intersect(c_1, g_1, 1) → M
  27. Circle(J, d3) → d_1
  28. Intersect(d_1, h_1, 1) → N
  29. Polygon(M, N, 4) → poly3, j_1, k_1, l_1, m_1, O, P
  30. Segment(P, M) → n_1
  31. Segment(M, N) → p_1
  32. Segment(N, O) → q_1
  33. Segment(O, P) → r_1
  34. Circle(M, d4) → e_1
  35. Intersect(e_1, p_1, 1) → Q
  36. Circle(N, d4) → s_1
  37. Intersect(s_1, q_1, 1) → R
  38. Polygon(Q, R, 4) → poly4, t_1, a_1, b_1, f_2, S, T
  39. Segment(S, T) → g_2
  40. Segment(T, Q) → h_2
  41. Segment(Q, R) → i_2
  42. Segment(R, S) → j_2
  43. Circle(Q, d5) → c_2
  44. Intersect(c_2, i_2, 1) → U
  45. Circle(R, d5) → d_2
  46. Intersect(d_2, j_2, 1) → V
  47. Polygon(U, V, 4) → poly5, k_2, l_2, m_2, n_2, W, Z
  48. Segment(Z, U) → p_2
  49. Segment(V, U) → q_2
  50. Segment(V, W) → r_2
  51. Segment(W, Z) → s_2
  52. Circle(U, d6) → e_2
  53. Intersect(e_2, q_2, 1) → A_1
  54. Circle(V, d6) → t_2
  55. Intersect(t_2, r_2, 1) → B_1
  56. Polygon(A_1, B_1, 4) → poly6, a_2, b_2, f_3, g_3, C_1, D_1
  57. Segment(D_1, A_1) → h_3
  58. Segment(A_1, B_1) → i_3
  59. Segment(B_1, C_1) → j_3
  60. Segment(C_1, D_1) → k_3
  61. Circle(A_1, d7) → c_3
  62. Intersect(c_3, i_3, 1) → E_1
  63. Circle(B_1, d7) → d_3
  64. Intersect(d_3, j_3, 1) → F_1
  65. Polygon(E_1, F_1, 4) → poly7, l_3, m_3, n_3, p_3, G_1, H_1
  66. Segment(F_1, E_1) → q_3
  67. Segment(E_1, H_1) → r_3
  68. Segment(H_1, G_1) → s_3
  69. Segment(G_1, F_1) → t_3
  70. Circle(E_1, d8) → e_3
  71. Intersect(e_3, q_3, 1) → I_1
  72. Circle(F_1, d8) → c_4
  73. Intersect(c_4, t_3, 1) → J_1
  74. Polygon(I_1, J_1, 4) → poly8, a_3, b_3, f_4, g_4, K_1, L_1
  75. Segment(K_1, L_1) → h_4
  76. Segment(L_1, I_1) → i_4
  77. Segment(I_1, J_1) → j_4
  78. Segment(K_1, J_1) → k_4
  79. Circle(I_1, d9) → d_4
  80. Intersect(d_4, j_4, 1) → M_1
  81. Circle(J_1, d9) → e_4
  82. Intersect(e_4, k_4, 1) → N_1
  83. Polygon(M_1, N_1, 4) → poly9, l_4, m_4, n_4, p_4, O_1, P_1
  84. Segment(O_1, N_1) → q_4
  85. Segment(N_1, M_1) → r_4
  86. Segment(M_1, P_1) → s_4
  87. Segment(P_1, O_1) → t_4
  88. Circle(M_1, d10) → c_5
  89. Circle(N_1, d10) → d_5
  90. Intersect(c_5, r_4, 1) → Q_1
  91. Intersect(d_5, q_4, 1) → R_1
  92. Polygon(Q_1, R_1, 4) → poly10, a_4, b_4, f_5, g_5, S_1, T_1
  93. Segment(R_1, S_1) → h_5
  94. Segment(S_1, T_1) → i_5
  95. Segment(T_1, Q_1) → j_5
  96. Segment(Q_1, R_1) → k_5
  97. Circle(R_1, d11) → e_5
  98. Circle(S_1, d11) → p_5
  99. Intersect(e_5, h_5, 1) → U_1
  100. Intersect(p_5, i_5, 1) → V_1
  101. Polygon(U_1, V_1, 4) → poly11, l_5, m_5, n_5, q_5, W_1, Z_1
  102. Segment(U_1, V_1) → r_5
  103. Segment(V_1, W_1) → s_5
  104. Segment(W_1, Z_1) → t_5
  105. Segment(Z_1, U_1) → a_5
  106. Circle(Z_1, d12) → c_6
  107. Circle(U_1, d12) → d_6
  108. Intersect(c_6, a_5, 1) → A_2
  109. Intersect(d_6, r_5, 1) → B_2
  110. Polygon(A_2, B_2, 4) → poly12, b_5, f_6, g_6, h_6, C_2, D_2
  111. Segment(B_2, C_2) → i_6
  112. Segment(C_2, D_2) → j_6
  113. Segment(D_2, A_2) → k_6
  114. Segment(A_2, B_2) → l_6
  115. Circle(A_2, d13) → e_6
  116. Circle(B_2, d13) → p_6
  117. Intersect(p_6, i_6, 1) → E_2
  118. Intersect(e_6, l_6, 1) → F_2
  119. Polygon(F_2, E_2, 4) → poly13, m_6, n_6, q_6, r_6, G_2, H_2
  120. Segment(G_2, H_2) → s_6
  121. Segment(G_2, E_2) → t_6
  122. Segment(E_2, F_2) → a_6
  123. Segment(F_2, H_2) → b_6
  124. Circle(F_2, d13) → c_7
  125. Circle(E_2, d13) → d_7
  126. Intersect(c_7, a_6, 1) → I_2
  127. Intersect(d_7, t_6, 1) → J_2
  128. Polygon(I_2, J_2, 4) → poly14, f_7, g_7, h_7, i_7, K_2, L_2
  129. Segment(J_2, K_2) → j_7
  130. Segment(K_2, L_2) → k_7
  131. Segment(L_2, I_2) → l_7
  132. Segment(I_2, J_2) → m_7
  133. Circle(L_2, d15) → e_7
  134. Circle(I_2, d15) → p_7
  135. Intersect(e_7, l_7, 1) → M_2
  136. Intersect(p_7, m_7, 1) → N_2
  137. Polygon(M_2, N_2, 4) → poly15, n_7, q_7, r_7, s_7, O_2, P_2
  138. Segment(O_2, P_2) → t_7
  139. Segment(P_2, M_2) → a_7
  140. Segment(M_2, N_2) → b_7
  141. Segment(N_2, O_2) → f_8