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新发现的无周期单形!

新发现的无周期单形!
几何画板及绘图内容加载中，请稍等……
2023-03-30 23:48
多边形变换几何作图
这是一个展示新发现的无周期单形的几何作图。无周期单形是一种特殊的几何形状，它不能通过平移或旋转来重复自身。本例通过使用旋转、反射等变换命令，展示了如何构造这种独特的几何形状。适合高中及以上学生或数学爱好者学习和探索。

GeoGebra 命令及参数

  1. Point(Circle[A, 2]) → B
  2. Segment(A, B) → f
  3. Circle(B, 1) → c
  4. Rotate(A, (2 * pi / 3), B) → A'
  5. Ray(B, A') → g
  6. Intersect(c, g, 1) → C
  7. Segment(B, C) → h
  8. OrthogonalLine(C, h) → i
  9. Circle(C, sqrt(3)) → d
  10. Intersect(d, i, 1) → D
  11. OrthogonalLine(D, f) → j
  12. Circle(D, sqrt(3)) → e
  13. Intersect(e, j, 1) → E
  14. OrthogonalLine(E, j) → k
  15. Circle(E, 1) → p
  16. Intersect(p, k, 2) → F
  17. Rotate(E, (-2 * pi / 3), F) → E'
  18. Segment(D, E) → l
  19. Segment(E, F) → m
  20. Segment(F, E') → n
  21. OrthogonalLine(E', n) → q
  22. Circle(E', sqrt(3)) → r
  23. Intersect(r, q, 1) → G
  24. Rotate(E', (2 * pi / 3), G) → E''
  25. Segment(G, E'') → s
  26. OrthogonalLine(E'', s) → t
  27. Circle(E'', 1) → c_1
  28. Intersect(c_1, t, 2) → H
  29. Rotate(E'', (2 * pi / 3), H) → E'''
  30. Segment(E'', H) → a
  31. Segment(H, E''') → b
  32. OrthogonalLine(E''', b) → f_1
  33. Circle(E''', sqrt(3)) → d_1
  34. Intersect(d_1, f_1, 1) → I
  35. Rotate(E''', (2 * pi / 3), I) → J
  36. Polygon(A, B, E''', I, J, C, D, E, F, E', G, E'', H) → poly1, a_1, h_1, e''', i_1, j_1, b_1, c_2, d_2, e_1, f_2, e', g_1, e''
  37. Circle(K, 2) → k_1
  38. Point(k_1) → L
  39. Rotate(K, (-2 * pi / 3), L) → K'
  40. Ray(L, K') → m_1
  41. Circle(L, 1) → p_1
  42. Intersect(p_1, m_1, 1) → P
  43. Segment(K, L) → n_1
  44. Segment(L, P) → q_1
  45. OrthogonalLine(P, q_1) → r_1
  46. Circle(P, sqrt(3)) → s_1
  47. Intersect(s_1, r_1, 2) → Q
  48. Rotate(P, (2 * pi / 3), Q) → P'
  49. Segment(P, Q) → t_1
  50. Segment(Q, P') → g_2
  51. OrthogonalLine(P', g_2) → h_2
  52. Circle(P', 1) → e_2
  53. Intersect(e_2, h_2, 2) → R
  54. Rotate(P', (2 * pi / 3), R) → P''
  55. Segment(P', R) → i_2
  56. Segment(R, P'') → j_2
  57. OrthogonalLine(P'', j_2) → k_2
  58. Circle(P'', sqrt(3)) → p_2
  59. Intersect(p_2, k_2, 2) → S
  60. Rotate(P'', (-2 * pi / 3), S) → P'''
  61. Segment(P''', S) → l_2
  62. Segment(P'', S) → m_2
  63. OrthogonalLine(P''', l_2) → n_2
  64. Circle(P''', 1) → q_2
  65. Intersect(q_2, n_2, 2) → T
  66. Rotate(P''', (-2 * pi / 3), T) → U
  67. Segment(T, U) → r_2
  68. OrthogonalLine(U, r_2) → s_2
  69. Circle(U, sqrt(3)) → t_2
  70. Intersect(t_2, s_2, 2) → V
  71. Rotate(U, (-2 * pi / 3), V) → U'
  72. Polygon(K, L, U, V, U', P, Q, P', R, P'', S, P''', T) → poly2, k_3, t_3, u, v, u', l_3, p_3, q_3, p', r_3, p'', s_3, p'''
  73. Segment(B, A) → l_1
  74. Segment(K, L) → a_2
  75. Segment(D, E) → b_2
  76. Segment(D, C) → f_3
  77. Segment(C, B) → g_3
  78. Segment(E, F) → h_3
  79. Segment(F, E') → i_3
  80. Segment(E', G) → j_3
  81. Segment(G, E'') → m_3
  82. Segment(E'', H) → n_3
  83. Segment(H, E''') → a_3
  84. Segment(E''', I) → b_3
  85. Segment(I, J) → c_3
  86. Segment(J, A) → d_3
  87. Segment(K, U') → e_3
  88. Segment(U', V) → f_4
  89. Segment(V, U) → g_4
  90. Segment(U, T) → h_4
  91. Segment(T, P''') → i_4
  92. Segment(P''', S) → j_4
  93. Segment(S, P'') → k_4
  94. Segment(P'', R) → l_4
  95. Segment(R, P') → m_4
  96. Segment(P', Q) → n_4
  97. Segment(Q, P) → p_4
  98. Segment(P, L) → q_4
  99. AperiodicMonotile(M) → poly3, N, f_5, g_5, h_5, i_5, j_5, r_4, s_4, t_4, a_4, b_4, c_4, d_4, e_4
  100. ReflectAperiodicMonotile(O) → W, poly4, t_5, a_5, b_5, c_5, d_5, k_5, l_5, m_5, n_5, p_5, q_5, r_5, s_5

AI 绘图提示语参考

这是一个新的非周期多边形，不能通过平移或旋转重复。图形由多个点、线段和圆组成，通过旋转和反射构造。关键对象包括点A、B、E'''，I、J、C、D、E、F、E'、G、E''、H等，它们通过线段和圆连接。构造顺序从点A开始，依次添加点B、E'''，I、J等，然后通过圆和线段连接这些点，形成复杂的几何形状。图形适合数学爱好者或高年级学生研究。
以上提示语为根据原始绘图命令整理而来，仅供参考。