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New Aperiodic Monotile!

New Aperiodic Monotile!
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2023-03-30 23:48
PolygonTransformationGeometric Construction
This example demonstrates a newly discovered aperiodic monotile, a special geometric shape that cannot be repeated through translation or rotation. Using transformation commands such as rotation and reflection, the construction of this unique shape is shown. Suitable for high school and above students or math enthusiasts.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

This is a new aperiodic monotile, which cannot be repeated by translation or rotation. The figure consists of points, line segments, and circles, constructed using rotations and reflections. Key objects include points A, B, E''', I, J, C, D, E, F, E', G, E'', H, connected by line segments and circles. The construction order starts from point A, then adds points B, E''', I, J, etc., and connects them through circles and line segments to form a complex geometric shape. This figure is suitable for math enthusiasts or high school students.
The prompt above was compiled from the original drawing commands and is for reference only.