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正方形打包：拼接与重叠关系的几何展示

正方形打包：拼接与重叠关系的几何展示
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2023-03-13 17:59
多边形向量变换几何作图
所属专题 geometriadinamica.es
该范例演示如何在平面中通过平移与构造，把若干正方形（或由正方形组合形成的多边形）进行“打包/拼接”，并观察相邻图形的边界与交叠结构。核心数学概念包括几何作图、正方形（多边形）的性质、向量平移变换（translation）、以及由多边形顶点与边的对应关系所形成的交接/重叠几何关系。作图要点上，利用 GeoGebra 的 Polygon 生成多边形顶点构成的形状，用 Vector 表示从一个位置到另一个位置的位移量，再通过 Translate 将已构造的图形整体平移到目标位置；对象方面，范例中出现了多组多边形顶点（如 A、B、C… 等）以及若干向量与辅助点（如 e、f、b、a、j、g 等）用来支撑不同位置的正方形/拼接单元。该示例适合中学到大学初级的几何建模与变换教学，用于说明“用向量做平移”“用多边形顶点定义形状”的建构思路，也适合作为数学实验课或几何探究活动的可视化素材。

GeoGebra 命令及参数

  1. Polygon(E, F, B, A) → P, e, f, b, a
  2. Polygon(J, G, F, I) → Q, j, g, f_1, i
  3. Polygon(L, K, E, G) → R, l, k, e_1, g_1
  4. Polygon(I, H, C, B) → S, i_1, h, c, b_1
  5. Polygon(N, M, D, C) → T, n, m, d, c_1
  6. Polygon(L, U, O, J) → V, l_1, u, o, j_1
  7. Polygon(W, Y, H, O) → Z, w, y, h_1, o_1
  8. Polygon(U, A_1, B_1, W) → P_1, u_1, a_1, b_2, w_1
  9. Polygon(B_1, Y, C_1, D_1) → Q_1, b_3, y_1, c_2, d_1
  10. Polygon(E_1, G_1, F_1, K) → R_1, e_2, g_2, f_2, k_1
  11. Polygon(G_1, H_1, I_1, J_1) → S_1, g_3, h_2, i_2, j_2
  12. Polygon(H_1, K_1, L_1, M_1) → T_1, h_3, k_2, l_2, m_1
  13. Polygon(M_1, N_1, O_1, I_1) → U_1, m_2, n_1, o_2, i_3
  14. Polygon(V_1, W_1, N_1, L_1) → Y_1, v_1, w_2, n_2, l_3
  15. Polygon(Z_1, A_2, M, V_1) → P_2, z_1, a_2, m_3, v_2
  16. Polygon(B_2, C_1, N, A_2) → Q_2, b_4, c_3, n_3, a_3
  17. Polygon(J_1, C_2, D_2, F_1) → R_2, j_3, c_4, d_2, f_3
  18. Polygon(A_1, D_2, E_2, D_1) → S_2, a_4, d_3, e_3, d_4
  19. Polygon(C_2, F_2, G_2, E_2) → T_2, c_5, f_4, g_4, e_4
  20. Polygon(H_2, Z_1, B_2, G_2) → U_2, h_4, z_2, b_5, g_5
  21. Polygon(H_2, F_2, O_1, W_1) → V_2, h_5, f_5, o_3, w_3
  22. Vector(I_2, J_2) → v
  23. Vector(K_2, L_2) → z
  24. Vector(O_2, W_2) → r
  25. Translate(E_1, Vector[(p * v)]) → E_1'
  26. Translate(G_1, Vector[(p * v)]) → G_1'
  27. Translate(F_1, Vector[(p * v)]) → F_1'
  28. Translate(K, Vector[(p * v)]) → K'
  29. Polygon(E_1', G_1', F_1', K') → R_1', e_1', g_1', f_1', k'
  30. Translate(L, Vector[(p * z)]) → L'
  31. Translate(K, Vector[(p * z)]) → K'_1
  32. Translate(E, Vector[(p * z)]) → E'
  33. Translate(G, Vector[(p * z)]) → G'
  34. Polygon(L', K'_1, E', G') → R', l', k'_1, e', g'
  35. Vector(K_2, N_2) → q
  36. Vector(K_2, M_2) → s
  37. Vector(K_2, B_3) → t
  38. Translate(E, Vector[(p * q)]) → E'_1
  39. Translate(F, Vector[(p * q)]) → F'
  40. Translate(B, Vector[(p * q)]) → B'
  41. Translate(A, Vector[(p * q)]) → A'
  42. Polygon(E'_1, F', B', A') → P', e'_1, f', b', a'
  43. Translate(J, Vector[(p * r)]) → J'
  44. Translate(G, Vector[(p * r)]) → G'_1
  45. Translate(F, Vector[(p * r)]) → F'_1
  46. Translate(I, Vector[(p * r)]) → I'
  47. Polygon(J', G'_1, F'_1, I') → Q', j', g'_1, f'_1, i'
  48. Translate(I, Vector[(p * t)]) → I'_1
  49. Translate(H, Vector[(p * t)]) → H'
  50. Translate(C, Vector[(p * t)]) → C'
  51. Translate(B, Vector[(p * t)]) → B'_1
  52. Polygon(I'_1, H', C', B'_1) → S', i'_1, h', c', b'_1
  53. Translate(L, Vector[(p * s)]) → L'_1
  54. Translate(U, Vector[(p * s)]) → U'
  55. Translate(O, Vector[(p * s)]) → O'
  56. Translate(J, Vector[(p * s)]) → J'_1
  57. Polygon(L'_1, U', O', J'_1) → V', l'_1, u', o', j'_1
  58. Vector(K_1, Y_2) → p_1
  59. Translate(U, Vector[(p * p_1)]) → U'_1
  60. Translate(A_1, Vector[(p * p_1)]) → A_1'
  61. Translate(B_1, Vector[(p * p_1)]) → B_1'
  62. Translate(W, Vector[(p * p_1)]) → W'
  63. Polygon(U'_1, A_1', B_1', W') → P_1', u'_1, a_1', b_1', w'
  64. Vector(K_1, Z_2) → q_1
  65. Translate(G_1, Vector[(p * q_1)]) → G_3
  66. Translate(H_1, Vector[(p * q_1)]) → H_1'
  67. Translate(I_1, Vector[(p * q_1)]) → I_1'
  68. Translate(J_1, Vector[(p * q_1)]) → J_1'
  69. Polygon(G_3, H_1', I_1', J_1') → S_1', g_6, h_1', i_1', j_1'
  70. Vector(A, A_3) → r_1
  71. Translate(J_1, Vector[(p * r_1)]) → J_3
  72. Translate(C_2, Vector[(p * r_1)]) → C_2'
  73. Translate(D_2, Vector[(p * r_1)]) → D_2'
  74. Translate(F_1, Vector[(p * r_1)]) → F_3
  75. Polygon(J_3, C_2', D_2', F_3) → R_2', j_4, c_2', d_2', f_6
  76. Vector(K_1, C_3) → s_1
  77. Translate(W, Vector[(p * s_1)]) → W'_1
  78. Translate(Y, Vector[(p * s_1)]) → Y'
  79. Translate(H, Vector[(p * s_1)]) → H'_1
  80. Translate(O, Vector[(p * s_1)]) → O'_1
  81. Polygon(W'_1, Y', H'_1, O'_1) → Z', w'_1, y', h'_1, o'_1
  82. Vector(K_1, D_3) → t_1
  83. Translate(B_1, Vector[(p * t_1)]) → B_4
  84. Translate(Y, Vector[(p * t_1)]) → Y'_1
  85. Translate(C_1, Vector[(p * t_1)]) → C_1'
  86. Translate(D_1, Vector[(p * t_1)]) → D_1'
  87. Polygon(B_4, Y'_1, C_1', D_1') → Q_1', b_6, y'_1, c_1', d_1'
  88. Vector(D, E_3) → u_2
  89. Translate(C_2, Vector[(p * u_2)]) → C_4
  90. Translate(F_2, Vector[(p * u_2)]) → F_2'
  91. Translate(G_2, Vector[(p * u_2)]) → G_2'
  92. Translate(E_2, Vector[(p * u_2)]) → E_2'
  93. Polygon(C_4, F_2', G_2', E_2') → T_2', c_6, f_2', g_2', e_2'
  94. Vector(D, H_3) → p_2
  95. Translate(A_1, Vector[(p * p_2)]) → A_4
  96. Translate(D_2, Vector[(p * p_2)]) → D_4
  97. Translate(E_2, Vector[(p * p_2)]) → E_4
  98. Translate(D_1, Vector[(p * p_2)]) → D_5
  99. Polygon(A_4, D_4, E_4, D_5) → S_2', a_5, d_5, e_5, d_6
  100. Vector(D, I_3) → q_2
  101. Vector(D, K_3) → r_2
  102. Vector(D, L_3) → s_2
  103. Vector(D, M_3) → t_2
  104. Vector(D, N_3) → u_3
  105. Vector(D, O_3) → v_3
  106. Vector(D, P_3) → z_3
  107. Translate(H_1, Vector[(p * q_2)]) → H_4
  108. Translate(K_1, Vector[(p * q_2)]) → K_1'
  109. Translate(L_1, Vector[(p * q_2)]) → L_1'
  110. Translate(M_1, Vector[(p * q_2)]) → M_1'
  111. Polygon(H_4, K_1', L_1', M_1') → T_1', h_6, k_1', l_1', m_1'
  112. Translate(M_1, Vector[(p * r_2)]) → M_4
  113. Translate(N_1, Vector[(p * r_2)]) → N_1'
  114. Translate(O_1, Vector[(p * r_2)]) → O_1'
  115. Translate(I_1, Vector[(p * r_2)]) → I_4
  116. Polygon(M_4, N_1', O_1', I_4) → U_1', m_4, n_1', o_1', i_4
  117. Translate(H_2, Vector[(p * s_2)]) → H_2'
  118. Translate(F_2, Vector[(p * s_2)]) → F_4
  119. Translate(O_1, Vector[(p * s_2)]) → O_4
  120. Translate(W_1, Vector[(p * s_2)]) → W_1'
  121. Polygon(H_2', F_4, O_4, W_1') → V_2', h_2', f_7, o_4, w_1'
  122. Translate(H_2, Vector[(p * t_2)]) → H_5
  123. Translate(Z_1, Vector[(p * t_2)]) → Z_1'
  124. Translate(B_2, Vector[(p * t_2)]) → B_2'
  125. Translate(G_2, Vector[(p * t_2)]) → G_4
  126. Polygon(H_5, Z_1', B_2', G_4) → U_2', h_7, z_1', b_2', g_7
  127. Translate(V_1, Vector[(p * u_3)]) → V_1'
  128. Translate(W_1, Vector[(p * u_3)]) → W_3
  129. Translate(N_1, Vector[(p * u_3)]) → N_4
  130. Translate(L_1, Vector[(p * u_3)]) → L_4
  131. Polygon(V_1', W_3, N_4, L_4) → Y_1', v_1', w_4, n_4, l_4
  132. Translate(N, Vector[(p * z_3)]) → N'
  133. Translate(M, Vector[(p * z_3)]) → M'
  134. Translate(D, Vector[(p * z_3)]) → D'
  135. Translate(C, Vector[(p * z_3)]) → C'_1
  136. Polygon(N', M', D', C'_1) → T', n', m', d', c'_1
  137. Vector(D, Q_3) → k_3
  138. Translate(Z_1, Vector[(p * k_3)]) → Z_3
  139. Translate(A_2, Vector[(p * k_3)]) → A_2'
  140. Translate(M, Vector[(p * k_3)]) → M'_1
  141. Translate(V_1, Vector[(p * k_3)]) → V_3
  142. Polygon(Z_3, A_2', M'_1, V_3) → P_2', z_4, a_2', m'_1, v_4
  143. Translate(B_2, Vector[(p * v_3)]) → B_5
  144. Translate(C_1, Vector[(p * v_3)]) → C_5
  145. Translate(N, Vector[(p * v_3)]) → N'_1
  146. Translate(A_2, Vector[(p * v_3)]) → A_5
  147. Polygon(B_5, C_5, N'_1, A_5) → Q_2', b_7, c_7, n'_1, a_6
  148. Polygon(A, E_1, K_1, D) → R_3, a_7, e_6, k_4, d_7

AI 绘图提示语参考

绘制一个由多个正方形组成的图形，通过平移和多边形构造。首先创建多个正方形，然后通过平移这些正方形来构建更大的图形。使用向量定义平移方向，确保相邻正方形正确对齐。逐步构建多个层叠的正方形，形成复杂的图案。
以上提示语为根据原始绘图命令整理而来，仅供参考。