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Tessellation: Constructing a Geometric Tiling via Rotation and Translation

Tessellation: Constructing a Geometric Tiling via Rotation and Translation
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2017-12-25 11:10
PolygonAngleVectorTransformationGeometric Construction
This GeoGebra example demonstrates how to create a seamless tessellation (tiling) pattern by repeatedly applying rotation and translation to a basic polygon composed of semicircles and line segments. The core mathematical concepts involve geometric transformations (rotation, translation), symmetry, and the fundamental principle of tessellation—covering the plane with congruent shapes without gaps or overlaps. The construction uses GeoGebra commands such as Semicircle, Point, Polygon, LineBisector, Intersect, Rotate, Vector, and Translate to generate the base shape and its multiple copies (e.g., poly1', poly1'', etc.), arranged in a periodic grid. This is suitable for middle or high school students learning about transformations, symmetry, and tessellation, and can be used in classroom demonstrations or exploratory activities connecting mathematics with visual art.

GeoGebra Commands & Parameters

  1. Semicircle(A, B) → c
  2. Semicircle(B, A) → d
  3. Point(c) → C
  4. Point(d) → D
  5. Polygon(A, C, B, D) → poly1, a, c_1, b, d_1
  6. LineBisector(A, B) → h
  7. Intersect(d, h, 1) → E
  8. Rotate(A, (-(90°)), E) → A'
  9. Rotate(C, (-(90°)), E) → C'
  10. Rotate(B, (-(90°)), E) → B'
  11. Rotate(D, (-(90°)), E) → D'
  12. Polygon(A', C', B', D') → poly1', a', c', b', d'
  13. Rotate(A, 180°, E) → A'_1
  14. Rotate(C, 180°, E) → C'_1
  15. Rotate(B, 180°, E) → B'_1
  16. Rotate(D, 180°, E) → D'_1
  17. Polygon(A'_1, C'_1, B'_1, D'_1) → poly1'_1, a'_1, c'_1, b'_1, d'_1
  18. Rotate(A', 180°, E) → A''
  19. Rotate(C', 180°, E) → C''
  20. Rotate(B', 180°, E) → B''
  21. Rotate(D', 180°, E) → D''
  22. Polygon(A'', C'', B'', D'') → poly1'', a'', c'', b'', d''
  23. Intersect(c, h, 1) → F
  24. Rotate(A, 180°, F) → A'_2
  25. Rotate(C, 180°, F) → C'_2
  26. Rotate(B, 180°, F) → B'_2
  27. Rotate(D, 180°, F) → D'_2
  28. Polygon(A'_2, C'_2, B'_2, D'_2) → poly1'_2, a'_2, c'_2, b'_2, d'_2
  29. Rotate(A'', 180°, A) → A'''
  30. Rotate(C'', 180°, A) → C'''
  31. Rotate(B'', 180°, A) → B'''
  32. Rotate(D'', 180°, A) → D'''
  33. Polygon(A''', C''', B''', D''') → poly1''', a''', c''', b''', d'''
  34. Rotate(A', 180°, B) → A''_1
  35. Rotate(C', 180°, B) → C''_1
  36. Rotate(B', 180°, B) → B''_1
  37. Rotate(D', 180°, B) → D''_1
  38. Polygon(A''_1, C''_1, B''_1, D''_1) → poly1''_1, a''_1, c''_1, b''_1, d''_1
  39. Rotate(A', 180°, B') → A''_2
  40. Rotate(C', 180°, B') → C''_2
  41. Rotate(B', 180°, B') → B''_2
  42. Rotate(D', 180°, B') → D''_2
  43. Polygon(A''_2, C''_2, B''_2, D''_2) → poly1''_2, a''_2, c''_2, b''_2, d''_2
  44. Rotate(A'', 180°, B'_1) → A'''_1
  45. Rotate(C'', 180°, B'_1) → C'''_1
  46. Rotate(B'', 180°, B'_1) → B'''_1
  47. Rotate(D'', 180°, B'_1) → D'''_1
  48. Polygon(A'''_1, C'''_1, B'''_1, D'''_1) → poly1'''_1, a'''_1, c'''_1, b'''_1, d'''_1
  49. Vector(A, B) → u
  50. Translate(E, u) → E'
  51. Translate(F, u) → F'
  52. Rotate(A''_1, 180°, F') → M
  53. Rotate(C''_1, 180°, F') → N
  54. Rotate(B''_1, 180°, F') → O
  55. Rotate(D''_1, 180°, F') → P
  56. Polygon(M, N, O, P) → poly3, m, n_1, o, p
  57. Rotate(A'_2, 180°, A'_2) → Q
  58. Rotate(C'_2, 180°, A'_2) → R
  59. Rotate(B'_2, 180°, A'_2) → S
  60. Rotate(D'_2, 180°, A'_2) → T
  61. Polygon(Q, R, S, T) → poly4, q, r, s, t
  62. Rotate(A, 180°, B) → A'_3
  63. Rotate(C, 180°, B) → C'_3
  64. Rotate(B, 180°, B) → B'_3
  65. Rotate(D, 180°, B) → D'_3
  66. Polygon(A'_3, C'_3, B'_3, D'_3) → poly1'_3, a'_3, c'_3, b'_3, d'_3
  67. Rotate(A', 180°, E') → A''_3
  68. Rotate(C', 180°, E') → C''_3
  69. Rotate(B', 180°, E') → B''_3
  70. Rotate(D', 180°, E') → D''_3
  71. Polygon(A''_3, C''_3, B''_3, D''_3) → poly1''_3, a''_3, c''_3, b''_3, d''_3
  72. Rotate(A'_1, 180°, B') → A_1
  73. Rotate(C'_1, 180°, B') → B_1
  74. Rotate(B'_1, 180°, B') → C_1
  75. Rotate(D'_1, 180°, B') → D_1
  76. Polygon(A_1, B_1, C_1, D_1) → poly6, a_1, b_1, c_2, d_2
  77. Translate(F, Vector[(-u)]) → F'_1
  78. Translate(E, Vector[(-u)]) → E'_1
  79. Rotate(A''', 180°, F'_1) → E_1
  80. Rotate(C''', 180°, F'_1) → F_1
  81. Rotate(B''', 180°, F'_1) → G_1
  82. Rotate(D''', 180°, F'_1) → H_1
  83. Polygon(E_1, F_1, G_1, H_1) → poly7, e_1, f_1, g_1, h_1
  84. Rotate(A'', 180°, E'_1) → A'''_2
  85. Rotate(C'', 180°, E'_1) → C'''_2
  86. Rotate(B'', 180°, E'_1) → B'''_2
  87. Rotate(D'', 180°, E'_1) → D'''_2
  88. Polygon(A'''_2, C'''_2, B'''_2, D'''_2) → poly1'''_2, a'''_2, c'''_2, b'''_2, d'''_2
  89. Rotate(A'_2, 180°, B'_2) → M_1
  90. Rotate(C'_2, 180°, B'_2) → N_1
  91. Rotate(B'_2, 180°, B'_2) → O_1
  92. Rotate(D'_2, 180°, B'_2) → P_1
  93. Polygon(M_1, N_1, O_1, P_1) → poly9, m_1, n_2, o_1, p_1
  94. Rotate(A, 180°, A) → A'_4
  95. Rotate(C, 180°, A) → C'_4
  96. Rotate(B, 180°, A) → B'_4
  97. Rotate(D, 180°, A) → D'_4
  98. Polygon(A'_4, C'_4, B'_4, D'_4) → poly1'_4, a'_4, c'_4, b'_4, d'_4
  99. Rotate(A'_1, 180°, B'_1) → Q_1
  100. Rotate(C'_1, 180°, B'_1) → R_1
  101. Rotate(B'_1, 180°, B'_1) → S_1
  102. Rotate(D'_1, 180°, B'_1) → T_1
  103. Polygon(Q_1, R_1, S_1, T_1) → poly10, q_1, r_1, s_1, t_1
  104. Rotate(F, 180°, E) → H
  105. Rotate(A'_1, 180°, H) → W_1
  106. Rotate(C'_1, 180°, H) → Z_1
  107. Rotate(B'_1, 180°, H) → A_2
  108. Rotate(D'_1, 180°, H) → B_2
  109. Polygon(W_1, Z_1, A_2, B_2) → poly2, w_1, z_2, a_2, b_2
  110. Rotate(F', 180°, E') → H'
  111. Rotate(F'_1, 180°, E'_1) → H'_1
  112. Rotate(A_1, 180°, H') → G
  113. Rotate(B_1, 180°, H') → I
  114. Rotate(C_1, 180°, H') → J
  115. Rotate(D_1, 180°, H') → K
  116. Polygon(G, I, J, K) → poly6', g, i, j, k
  117. Rotate(Q_1, 180°, H'_1) → L
  118. Rotate(R_1, 180°, H'_1) → U
  119. Rotate(S_1, 180°, H'_1) → V
  120. Rotate(T_1, 180°, H'_1) → W
  121. Polygon(L, U, V, W) → poly10', l, u_1, v, w
  122. Rotate(A'''_2, 180°, B'''_2) → Z
  123. Rotate(C'''_2, 180°, B'''_2) → I_1
  124. Rotate(B'''_2, 180°, B'''_2) → J_1
  125. Rotate(D'''_2, 180°, B'''_2) → K_1
  126. Polygon(Z, I_1, J_1, K_1) → poly5, z_1, i_1, j_1, k_1
  127. Rotate(A''_3, 180°, A''_3) → L_1
  128. Rotate(C''_3, 180°, A''_3) → U_1
  129. Rotate(B''_3, 180°, A''_3) → V_1
  130. Rotate(D''_3, 180°, A''_3) → C_2
  131. Polygon(L_1, U_1, V_1, C_2) → poly8, l_1, u_2, v_1, c_3