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Do Triangles Tessellate? Exploring Midpoint Rotation

Do Triangles Tessellate? Exploring Midpoint Rotation
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2014-04-09 18:37
TrianglePolygonAngleTransformationGeometric Construction
In collections Tessellations of Common Figures
① This animation demonstrates whether any triangle can tessellate the plane using the midpoint rotation rule (one of the 17 wallpaper groups), visually showing how triangles cover the plane without gaps or overlaps. ② Core mathematical concepts include plane tessellation, rotational symmetry, midpoint rotation, wallpaper groups, and the fact that the sum of triangle angles is 180°. ③ Construction highlights: using GeoGebra commands Polygon to create triangle poly1, Midpoint to find side midpoints, and Rotate to rotate the triangle about these midpoints by specified angles, generating multiple rotated copies (e.g., A', B', C') and repeating to form a tessellation pattern; objects include points A, B, C, sides a, b, c, and successive rotated triangles. ④ Suitable for middle/high school geometry or enrichment classes, helping students understand tessellation conditions and transformation geometry, and useful for exploratory learning or teacher demonstrations.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → poly1, c, a, b
  2. Midpoint(B, C) → D
  3. Midpoint(C, A) → E
  4. Midpoint(A, B) → F
  5. Rotate(A, rot, F) → A'
  6. Rotate(B, rot, F) → B'
  7. Rotate(C, rot, F) → C'
  8. Polygon(A', B', C') → poly1', c', a', b'
  9. Rotate(A, rot, D) → A'_1
  10. Rotate(B, rot, D) → B'_1
  11. Rotate(C, rot, D) → C'_1
  12. Polygon(A'_1, B'_1, C'_1) → poly1'_1, c'_1, a'_1, b'_1
  13. Rotate(A, rot, E) → A'_2
  14. Rotate(B, rot, E) → B'_2
  15. Rotate(C, rot, E) → C'_2
  16. Polygon(A'_2, B'_2, C'_2) → poly1'_2, c'_2, a'_2, b'_2
  17. Rotate(A, 180*°, F) → A'_3
  18. Rotate(B, 180*°, F) → B'_3
  19. Rotate(C, 180*°, F) → C'_3
  20. Polygon(A'_3, B'_3, C'_3) → poly1'_3, c'_3, a'_3, b'_3
  21. Rotate(A, 180*°, D) → A'_4
  22. Rotate(B, 180*°, D) → B'_4
  23. Rotate(C, 180*°, D) → C'_4
  24. Polygon(A'_4, B'_4, C'_4) → poly1'_4, c'_4, a'_4, b'_4
  25. Rotate(A, 180*°, E) → A'_5
  26. Rotate(B, 180*°, E) → B'_5
  27. Rotate(C, 180*°, E) → C'_5
  28. Polygon(A'_5, B'_5, C'_5) → poly1'_5, c'_5, a'_5, b'_5
  29. Midpoint(C'_3, B) → G
  30. Midpoint(B, A'_4) → H
  31. Midpoint(A'_4, C) → I
  32. Midpoint(C, B'_5) → J
  33. Midpoint(B'_5, A) → K
  34. Midpoint(C'_3, A) → L
  35. Rotate(A'_3, rot, G) → M
  36. Rotate(B'_3, rot, G) → N
  37. Rotate(C'_3, rot, G) → O
  38. Polygon(M, N, O) → poly2, o, m, n
  39. Rotate(A'_3, rot, L) → P
  40. Rotate(B'_3, rot, L) → Q
  41. Rotate(C'_3, rot, L) → R
  42. Polygon(P, Q, R) → poly3, r, p, q
  43. Rotate(A'_4, rot, H) → S
  44. Rotate(B'_4, rot, H) → T
  45. Rotate(C'_4, rot, H) → U
  46. Polygon(S, T, U) → poly4, u, s, t
  47. Rotate(A'_4, rot, I) → V
  48. Rotate(B'_4, rot, I) → W
  49. Rotate(C'_4, rot, I) → Z
  50. Polygon(V, W, Z) → poly5, z, v, w
  51. Rotate(A'_5, rot, K) → A_1
  52. Rotate(B'_5, rot, K) → B_1
  53. Rotate(C'_5, rot, K) → C_1
  54. Polygon(A_1, B_1, C_1) → poly6, c_1, a_1, b_1
  55. Rotate(A'_5, rot, J) → D_1
  56. Rotate(B'_5, rot, J) → E_1
  57. Rotate(C'_5, rot, J) → F_1
  58. Polygon(D_1, E_1, F_1) → poly7, f_1, d_1, e_1
  59. Angle(C, A, B) → α
  60. Angle(A, B, C) → β
  61. Angle(B, C, A) → γ
  62. Midpoint(a'_3) → G_1
  63. Midpoint(b'_3) → H_1
  64. Midpoint(b'_4) → I_1
  65. Midpoint(c'_4) → J_1
  66. Midpoint(c'_5) → K_1
  67. Midpoint(a'_5) → L_1
  68. Rotate(A'_3, 180*°, H_1) → M_1
  69. Rotate(B'_3, 180*°, H_1) → N_1
  70. Rotate(C'_3, 180*°, H_1) → O_1
  71. Polygon(M_1, N_1, O_1) → poly8, o_1, m_1, n_1
  72. Rotate(A'_4, 180*°, I_1) → P_1
  73. Rotate(B'_4, 180*°, I_1) → Q_1
  74. Rotate(C'_4, 180*°, I_1) → R_1
  75. Polygon(P_1, Q_1, R_1) → poly9, r_1, p_1, q_1
  76. Rotate(A'_4, 180*°, J_1) → S_1
  77. Rotate(B'_4, 180*°, J_1) → T_1
  78. Rotate(C'_4, 180*°, J_1) → U_1
  79. Polygon(S_1, T_1, U_1) → poly10, u_1, s_1, t_1
  80. Rotate(A'_5, 180*°, K_1) → V_1
  81. Rotate(B'_5, 180*°, K_1) → W_1
  82. Rotate(C'_5, 180*°, K_1) → Z_1
  83. Polygon(V_1, W_1, Z_1) → poly11, z_1, v_1, w_1
  84. Rotate(A'_5, 180*°, L_1) → A_2
  85. Rotate(B'_5, 180*°, L_1) → B_2
  86. Rotate(C'_5, 180*°, L_1) → C_2
  87. Polygon(A_2, B_2, C_2) → poly12, c_2, a_2, b_2
  88. Rotate(A'_3, 180*°, G_1) → D_2
  89. Rotate(B'_3, 180*°, G_1) → E_2
  90. Rotate(C'_3, 180*°, G_1) → F_2
  91. Polygon(D_2, E_2, F_2) → poly13, f_2, d_2, e_2
  92. Midpoint(t_1) → G_2
  93. Midpoint(r_1) → H_2
  94. Rotate(P_1, 180*°, H_2) → I_2
  95. Rotate(Q_1, 180*°, H_2) → J_2
  96. Rotate(R_1, 180*°, H_2) → K_2
  97. Polygon(I_2, J_2, K_2) → poly9', k_2, i_2, j_2
  98. Rotate(S_1, 180*°, G_2) → L_2
  99. Rotate(T_1, 180*°, G_2) → M_2
  100. Rotate(U_1, 180*°, G_2) → N_2
  101. Polygon(L_2, M_2, N_2) → poly10', n_2, l_2, m_2
  102. Midpoint(a) → O_2
  103. Midpoint(m_1) → P_2
  104. Midpoint(c) → Q_2
  105. Angle(I_1, B, H_2) → δ
  106. Angle(H_1, B, Q_2) → ε
  107. Angle(H_2, B, P_2) → ζ
  108. Angle(P_2, B, H_1) → η
  109. Angle(O_2, B, I_1) → θ

AI Drawing Prompt Reference

Draw a triangle and find the midpoint of each side. Rotate the triangle around these midpoints to show how the rotated triangles cover the plane without gaps, demonstrating rotational symmetry and tessellation. Repeat the rotation and arrangement multiple times to create a seamless pattern, illustrating the concepts of rotational symmetry, tessellation, and wallpaper groups.
The prompt above was compiled from the original drawing commands and is for reference only.