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Girih 瓷砖：伊斯兰几何图案的构造与对称性

Girih 瓷砖：伊斯兰几何图案的构造与对称性
几何画板及绘图内容加载中，请稍等……
2019-03-06 18:21
多边形数列变换几何作图轨迹
该范例演示了 Girih 瓷砖（一种伊斯兰传统几何镶嵌图案）的构造过程，通过旋转、镜像和序列等操作生成具有高度对称性的多边形组合。核心数学概念包括多边形、对称变换（旋转与镜像）、全等、切线以及分形几何中的自相似性。作图要点：利用 GeoGebra 命令如 Point、Rotate、Polygon、Midpoint、Segment、Intersect、Mirror、Centroid、Line、PolyLine 和 Sequence，创建点 O、A 等基础对象，并通过旋转和镜像生成多个副本（如 P'、O'、V'、U' 等），最终形成复杂的瓷砖图案。该范例适合中学高年级或大学低年级的几何课程，用于探索对称变换、镶嵌原理以及几何作图的逻辑思维。

GeoGebra 命令及参数

  1. Point(Circle[O, 4]) → N
  2. Rotate(N, 72°, O) → P
  3. Rotate(O, 72°, P) → Q
  4. Rotate(P, (-(144°)), Q) → P'
  5. Rotate(O, 144°, N) → O'
  6. Polygon(P, O, N, O', P', Q) → veelhoek1, p, o, n, o', p', q
  7. Midpoint(o) → R
  8. Midpoint(p) → S
  9. Midpoint(q) → T
  10. Midpoint(n) → U
  11. Midpoint(o') → V
  12. Midpoint(p') → W
  13. Segment(S, R) → f
  14. Segment(S, T) → g
  15. Segment(W, V) → h
  16. Segment(V, U) → i
  17. Rotate(V, 72°, U) → V'
  18. Rotate(U, 72°, U) → U'
  19. Segment(V', U') → i'
  20. Rotate(W, (-(72°)), W) → W'
  21. Rotate(V, (-(72°)), W) → V'_1
  22. Segment(W', V'_1) → h'
  23. Intersect(i', h') → Z
  24. Segment(Q, N) → j
  25. Mirror(Z, j) → Z'
  26. Segment(T, Z') → k
  27. Segment(Z', R) → l
  28. Segment(W, Z) → m
  29. Segment(Z, U) → r
  30. Point(Circle[I, 4]) → J
  31. Rotate(J, 72°, I) → J'
  32. Rotate(I, 108°, J') → H
  33. Polygon(I, J, H, J') → vierhoek1, i_1, j_1, h_1, j'
  34. Midpoint(h_1) → K
  35. Midpoint(j') → L
  36. Midpoint(i_1) → M
  37. Midpoint(j_1) → A_1
  38. Segment(K, A_1) → s
  39. Segment(M, L) → t
  40. Rotate(M, 72°, L) → M'
  41. Rotate(L, 72°, L) → L'
  42. Segment(M', L') → t'
  43. Rotate(K, (-(72°)), K) → K'
  44. Rotate(A_1, (-(72°)), K) → B_1
  45. Segment(K', B_1) → s'
  46. Intersect(t', s') → C_1
  47. Segment(L, C_1) → a
  48. Segment(C_1, K) → b
  49. Segment(I, H) → c
  50. Mirror(C_1, c) → D_1
  51. Segment(M, D_1) → d
  52. Segment(D_1, A_1) → e
  53. Point(Circle[E_1, 4]) → F
  54. Rotate(F, 72°, E_1) → FF
  55. Rotate(E_1, (-(144°)), F) → K_1
  56. Rotate(F, (-(144°)), K_1) → F'
  57. Rotate(E_1, 144°, FF) → C
  58. Polygon(E_1, F, K_1, F', C, FF) → veelhoek2, e_1, f_1, k_1, f', c_1, ff
  59. Midpoint(ff) → D
  60. Midpoint(c_1) → E
  61. Midpoint(f') → G
  62. Midpoint(k_1) → F_1
  63. Midpoint(f_1) → G_1
  64. Midpoint(e_1) → H_1
  65. Segment(D, H_1) → g_1
  66. Segment(G, F_1) → l_1
  67. Rotate(D, 72°, D) → D'
  68. Rotate(H_1, 72°, D) → I_1
  69. Segment(D', I_1) → m_1
  70. Segment(I_1, E) → n_1
  71. Rotate(I_1, 72°, E) → J_1
  72. Rotate(E, 72°, E) → E'
  73. Segment(J_1, E') → p_1
  74. Segment(J_1, G) → q_1
  75. Segment(E_1, F') → r_1
  76. Mirror(J_1, r_1) → L_1
  77. Mirror(I_1, r_1) → M_1
  78. Segment(H_1, M_1) → s_1
  79. Segment(M_1, G_1) → t_1
  80. Segment(G_1, L_1) → a_1
  81. Segment(L_1, F_1) → b_1
  82. Point(Circle[N_1, 4]) → O_1
  83. Polygon(N_1, O_1, 5) → veelhoek3, d_1, f_2, g_2, h_2, i_2, P_1, Q_1, R_1
  84. Centroid(veelhoek3) → S_1
  85. Midpoint(g_2) → T_1
  86. Midpoint(d_1) → U_1
  87. Midpoint(f_2) → V_1
  88. Rotate(O_1, 54°, U_1) → W_1
  89. Segment(U_1, W_1) → j_2
  90. Rotate(O_1, (-(54°)), V_1) → Z_1
  91. Segment(V_1, Z_1) → k_2
  92. Intersect(j_2, k_2) → A_2
  93. Segment(V_1, A_2) → l_2
  94. Segment(A_2, U_1) → m_2
  95. Midpoint(i_2) → C_2
  96. Midpoint(h_2) → D_2
  97. Rotate(V_1, 72°, S_1) → E_2
  98. Rotate(A_2, 72°, S_1) → F_2
  99. Segment(E_2, F_2) → n_2
  100. Segment(V_1, F_2) → p_2
  101. Segment(T_1, N_1) → q_2
  102. Rotate(A_2, (-(72°)), S_1) → G_2
  103. Segment(U_1, G_2) → r_2
  104. Segment(G_2, C_2) → s_2
  105. Rotate(A_2, (-(144°)), S_1) → B_2
  106. Segment(C_2, B_2) → t_2
  107. Segment(B_2, D_2) → a_2
  108. Rotate(A_2, 144°, S_1) → H_2
  109. Segment(T_1, H_2) → b_2
  110. Segment(H_2, D_2) → c_2
  111. Polygon(I_2, J_2, 10) → veelhoek4, d_2, m_3, K_2, L_2, M_2, N_2, O_2, P_2, Q_2, R_2, e_2, f_3, g_3, h_3, i_3, j_3, k_3, l_3
  112. Centroid(veelhoek4) → S_2
  113. Midpoint(f_3) → T_2
  114. Rotate(K_2, (-(54°)), T_2) → U_2
  115. Rotate(L_2, 54°, T_2) → V_2
  116. Line(T_2, V_2) → n_3
  117. Line(T_2, U_2) → p_3
  118. Midpoint(e_2) → W_2
  119. Rotate(K_2, 54°, W_2) → Z_2
  120. Line(W_2, Z_2) → q_3
  121. Intersect(p_3, q_3) → A_3
  122. Line(T_2, S_2) → r_3
  123. Intersect(q_3, r_3) → B_3
  124. Midpoint(g_3) → C_3
  125. Rotate(L_2, (-(54°)), C_3) → D_3
  126. Line(C_3, D_3) → s_3
  127. Intersect(n_3, s_3) → E_3
  128. PolyLine(T_2, E_3, B_3, A_3, T_2) → t_3
  129. Sequence(PolyLine[Rotate[T_2, (nn * 36°), S_2], Rotate[E_3, (nn * 36°), S_2], Rotate[B_3, (nn * 36°), S_2], Rotate[A_3, (nn * 36°), S_2], Rotate[T_2, (nn * 36°), S_2]], nn, 1, 9) → l1
  130. Point(Circle[F_3, 2.902117938185601]) → G_3
  131. Point(Circle[E_4, 4]) → F_4
  132. Rotate(F_4, 72°, E_4) → FF_1
  133. Rotate(E_4, (-(144°)), F_4) → K_3
  134. Rotate(F_4, (-(144°)), K_3) → F'_1
  135. Rotate(E_4, 144°, FF_1) → C_4
  136. Polygon(E_4, F_4, K_3, F'_1, C_4, FF_1) → veelhoek2_1, e_3, f_4, k_4, f'_1, c_4, ff_1
  137. Midpoint(ff_1) → D_4
  138. Midpoint(c_4) → E_5
  139. Midpoint(f'_1) → G_4
  140. Midpoint(k_4) → F_5
  141. Midpoint(f_4) → G_5
  142. Midpoint(e_3) → H_4
  143. Segment(D_4, H_4) → g_4
  144. Segment(G_4, F_5) → l_4
  145. Rotate(D_4, 72°, D_4) → D'_1
  146. Rotate(H_4, 72°, D_4) → I_3
  147. Segment(D'_1, I_3) → m_4
  148. Segment(I_3, E_5) → n_4
  149. Rotate(I_3, 72°, E_5) → J_3
  150. Rotate(E_5, 72°, E_5) → E'_1
  151. Segment(J_3, E'_1) → p_4
  152. Segment(J_3, G_4) → q_4
  153. Segment(E_4, F'_1) → r_4
  154. Mirror(J_3, r_4) → L_3
  155. Mirror(I_3, r_4) → M_3
  156. Segment(H_4, M_3) → s_4
  157. Segment(M_3, G_5) → t_4
  158. Segment(G_5, L_3) → a_4
  159. Segment(L_3, F_5) → b_4
  160. Point(Circle[O_3, 4]) → N_3
  161. Rotate(N_3, 72°, O_3) → P_3
  162. Rotate(O_3, 72°, P_3) → Q_3
  163. Rotate(P_3, (-(144°)), Q_3) → P'_1
  164. Rotate(O_3, 144°, N_3) → O'_1
  165. Polygon(P_3, O_3, N_3, O'_1, P'_1, Q_3) → veelhoek1_1, p_5, o_1, n_5, o'_1, p'_1, q_5
  166. Midpoint(o_1) → R_3
  167. Midpoint(p_5) → S_3
  168. Midpoint(q_5) → T_3
  169. Midpoint(n_5) → U_3
  170. Midpoint(o'_1) → V_3
  171. Midpoint(p'_1) → W_3
  172. Segment(S_3, R_3) → f_5
  173. Segment(S_3, T_3) → g_5
  174. Segment(W_3, V_3) → h_4
  175. Segment(V_3, U_3) → i_4
  176. Rotate(V_3, 72°, U_3) → V'_2
  177. Rotate(U_3, 72°, U_3) → U'_1
  178. Segment(V'_2, U'_1) → i'_1
  179. Rotate(W_3, (-(72°)), W_3) → W'_1
  180. Rotate(V_3, (-(72°)), W_3) → V'_3
  181. Segment(W'_1, V'_3) → h'_1
  182. Intersect(i'_1, h'_1) → Z_3
  183. Segment(Q_3, N_3) → j_4
  184. Mirror(Z_3, j_4) → Z'_1
  185. Segment(T_3, Z'_1) → k_5
  186. Segment(Z'_1, R_3) → l_5
  187. Segment(W_3, Z_3) → m_5
  188. Segment(Z_3, U_3) → r_5
  189. Point(Circle[I_4, 4]) → J_4
  190. Rotate(J_4, 72°, I_4) → J'_1
  191. Rotate(I_4, 108°, J'_1) → H_5
  192. Polygon(I_4, J_4, H_5, J'_1) → vierhoek1_1, i_5, j_5, h_5, j'_1
  193. Midpoint(h_5) → K_4
  194. Midpoint(j'_1) → L_4
  195. Midpoint(i_5) → M_4
  196. Midpoint(j_5) → A_4
  197. Segment(K_4, A_4) → s_5
  198. Segment(M_4, L_4) → t_5
  199. Rotate(M_4, 72°, L_4) → M'_1
  200. Rotate(L_4, 72°, L_4) → L'_1
  201. Segment(M'_1, L'_1) → t'_1
  202. Rotate(K_4, (-(72°)), K_4) → K'_1
  203. Rotate(A_4, (-(72°)), K_4) → B_4
  204. Segment(K'_1, B_4) → s'_1
  205. Intersect(t'_1, s'_1) → C_5
  206. Segment(L_4, C_5) → a_5
  207. Segment(C_5, K_4) → b_5
  208. Segment(I_4, H_5) → c_5
  209. Mirror(C_5, c_5) → D_5
  210. Segment(M_4, D_5) → d_3
  211. Segment(D_5, A_4) → e_4
  212. Point(Circle[N_4, 4]) → O_4
  213. Polygon(N_4, O_4, 5) → veelhoek3_1, d_4, f_6, g_6, h_6, i_6, P_4, Q_4, R_4
  214. Centroid(veelhoek3_1) → S_4
  215. Midpoint(g_6) → T_4
  216. Midpoint(d_4) → U_4
  217. Midpoint(f_6) → V_4
  218. Rotate(O_4, 54°, U_4) → W_4
  219. Segment(U_4, W_4) → j_6
  220. Rotate(O_4, (-(54°)), V_4) → Z_4
  221. Segment(V_4, Z_4) → k_6
  222. Intersect(j_6, k_6) → A_5
  223. Segment(V_4, A_5) → l_6
  224. Segment(A_5, U_4) → m_6
  225. Midpoint(i_6) → C_6
  226. Midpoint(h_6) → D_6
  227. Rotate(V_4, 72°, S_4) → E_6
  228. Rotate(A_5, 72°, S_4) → F_6
  229. Segment(E_6, F_6) → n_6
  230. Segment(V_4, F_6) → p_6
  231. Segment(T_4, N_4) → q_6
  232. Rotate(A_5, (-(72°)), S_4) → G_6
  233. Segment(U_4, G_6) → r_6
  234. Segment(G_6, C_6) → s_6
  235. Rotate(A_5, (-(144°)), S_4) → B_5
  236. Segment(C_6, B_5) → t_6
  237. Segment(B_5, D_6) → a_6
  238. Rotate(A_5, 144°, S_4) → H_6
  239. Segment(T_4, H_6) → b_6
  240. Segment(H_6, D_6) → c_6