利用45-45-90三角形反射生成的镶嵌

利用45-45-90三角形反射生成的镶嵌
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2015-02-10 05:02
通过拖动45-45-90三角形内部的点,可以生成不同的镶嵌图案。本范例展示了反射变换在几何作图中的应用,涉及的核心数学概念包括反射变换和几何作图。使用了Polygon, Segment, PointIn, Mirror等GeoGebra命令,适合初中及以上学生学习几何变换和镶嵌图案。

GeoGebra 命令及参数

  1. Polygon(A, B, 4) → poly1, a_2, b_2, c_2, d, C, D
  2. Segment(A, C) → e
  3. Polygon(A, B, C) → poly2, c, a, b
  4. PointIn(poly2) → E
  5. Mirror(E, a) → E'
  6. Mirror(E, b) → E'_1
  7. Mirror(E, c) → E'_2
  8. Segment(E, E'_1) → f
  9. Segment(E, E') → g
  10. Segment(E, E'_2) → h
  11. Mirror(E'_1, a) → F
  12. Mirror(E'_1, c) → G
  13. Mirror(E', b) → E''
  14. Mirror(E', c) → E''_1
  15. Mirror(E'_2, b) → H
  16. Segment(E'_1, E'') → i
  17. Segment(E'_1, H) → j
  18. Segment(E'_2, G) → k
  19. Segment(E''_1, E') → l
  20. Segment(E'_2, E''_1) → m
  21. Segment(E', F) → p
  22. Mirror(E'', a) → E'''
  23. Mirror(E'', c) → E'''_1
  24. Mirror(F, b) → F'
  25. Mirror(F, c) → F'_1
  26. Mirror(E''_1, b) → I
  27. Mirror(G, b) → G'
  28. Mirror(H, a) → H'
  29. Mirror(H, c) → H'_1
  30. Segment(E'', F') → q
  31. Segment(E'', I) → r
  32. Segment(I, H) → s
  33. Segment(H, G') → t
  34. Segment(G, H'_1) → a_1
  35. Segment(G, E'''_1) → b_1
  36. Segment(E''_1, F'_1) → c_1
  37. Segment(F, H') → d_1
  38. Segment(F, E''') → e_1
  39. Mirror(F', a) → F''
  40. Mirror(F', c) → F''_1
  41. Mirror(I, a) → I'
  42. Mirror(I, c) → I'_1
  43. Mirror(G', a) → G''
  44. Mirror(G', c) → G''_1
  45. Mirror(H'_1, a) → J
  46. Mirror(E'''_1, a) → K
  47. Mirror(E'''_1, b) → L
  48. Mirror(F'_1, b) → M
  49. Mirror(H', b) → H''
  50. Segment(F', H'') → f_1
  51. Segment(I, M) → g_1
  52. Segment(G', L) → h_1
  53. Segment(G''_1, G') → i_1
  54. Segment(G''_1, H'_1) → j_1
  55. Segment(H'_1, I'_1) → k_1
  56. Segment(I'_1, E'''_1) → l_1
  57. Segment(E'''_1, F''_1) → m_1
  58. Segment(K, F'_1) → n_1
  59. Segment(F'_1, J) → p_1
  60. Segment(H', G'') → q_1
  61. Segment(H', I') → r_1
  62. Segment(I', E''') → s_1
  63. Segment(E''', F'') → t_1
  64. Segment(F'', F') → d_2
  65. Mirror(H'', a) → H'''
  66. Mirror(H'', c) → H'''_1
  67. Mirror(M, a) → M'
  68. Mirror(M, c) → M'_1
  69. Mirror(L, a) → L'
  70. Mirror(L, c) → L'_1
  71. Mirror(F''_1, a) → N
  72. Mirror(F''_1, b) → O
  73. Mirror(K, b) → K'
  74. Mirror(J, b) → J'
  75. Mirror(G'', b) → G'''
  76. Mirror(G'', c) → G'''_1
  77. Mirror(I', c) → I''
  78. Segment(H'', G''') → e_2
  79. Segment(M, J') → f_2
  80. Segment(M, K') → g_2
  81. Segment(L, O) → h_2
  82. Segment(L, L'_1) → i_2
  83. Segment(I'_1, M'_1) → j_2
  84. Segment(F''_1, H'''_1) → k_2
  85. Segment(F''_1, N) → l_2
  86. Segment(N, K) → m_2
  87. Segment(K, I'') → n_2
  88. Segment(I'', J) → p_2
  89. Segment(J, G'''_1) → q_2
  90. Segment(G'''_1, G'') → r_2
  91. Segment(I', M') → s_2
  92. Segment(F'', H''') → t_2
  93. Segment(H''', H'') → a_3
  94. Segment(G'', L') → b_3
  95. Mirror(G''', a) → P
  96. Mirror(G''', c) → Q
  97. Mirror(J', a) → J''
  98. Mirror(J', c) → J''_1
  99. Mirror(K', a) → K''
  100. Mirror(K', c) → K''_1
  101. Mirror(O, a) → O'
  102. Mirror(O, c) → O'_1
  103. Mirror(L'_1, a) → R
  104. Mirror(M'_1, a) → S
  105. Mirror(H'''_1, a) → T
  106. Mirror(H'''_1, b) → U
  107. Mirror(I'', b) → I'''
  108. Mirror(G'''_1, b) → V
  109. Mirror(L', b) → L''
  110. Segment(L'', G''') → c_3
  111. Segment(G''', V) → d_3
  112. Segment(V, J') → e_3
  113. Segment(J', I''') → f_3
  114. Segment(I''', K') → g_3
  115. Segment(O, U) → h_3
  116. Segment(L'_1, O'_1) → i_3
  117. Segment(K''_1, M'_1) → j_3
  118. Segment(J''_1, M'_1) → k_3
  119. Segment(Q, H'''_1) → l_3
  120. Segment(H'''_1, T) → m_3
  121. Segment(T, N) → n_3
  122. Segment(S, I'') → p_3
  123. Segment(R, G'''_1) → q_3
  124. Segment(R, L') → r_3
  125. Segment(L', O') → s_3
  126. Segment(K'', M') → t_3
  127. Segment(M', J'') → a_4
  128. Segment(P, H''') → b_4
  129. Segment(G''_1, L'_1) → c_4

AI 绘图提示语参考

先画一个正方形,连接其一条对角线,将正方形分成两个等腰直角三角形。取其中一个三角形,在其内部任意放置一个点。以该三角形的三条边为对称轴,分别反射这个点,得到三个镜像点。用线段连接原始点与三个镜像点,形成一个小三角形。接着,以相邻三角形的边为对称轴,继续反射这些点,并连接相应的镜像点,逐步向外扩展,形成由多个等腰直角三角形拼接而成的密铺图案。整个图形由对称反射生成,点在不同三角形间重复出现,构成复杂的网格状镶嵌结构。
以上提示语为根据原始绘图命令整理而来,仅供参考。