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Octotessellation

Octotessellation
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2020-04-30 04:54
Solid GeometryTransformationGeometric Construction
In collections Tessellation with Pentagrins
This example demonstrates the tessellation structure of an octahedron and its geometric properties. Core math concepts include polygon construction, rotational transformations, spatial geometry, and symmetry. Construction highlights involve commands like Polygon, Point, Intersect, Segment, and Rotate in GeoGebra. Suitable for high school and above students learning solid geometry, spatial visualization, and geometric construction techniques.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 8) → polígono1, f, D, E, F, G, H, g, h, i, j, k, l, m, C
  2. Point(Circle[I, 2.0259275181383116]) → J
  3. Polygon(I, J, K, L, M, N, O, P) → polígono2, i_1, j_1, k_1, l_1, m_1, n, o, p
  4. Intersect(j_1, k_1) → Q
  5. Segment(J, Q) → q
  6. Intersect(k_1, l_1) → R
  7. Segment(Q, R) → r
  8. Intersect(f, g) → S
  9. Segment(R, S) → s
  10. Intersect(j, k) → T
  11. Segment(J, T) → t
  12. Intersect(k, l) → U
  13. Segment(T, U) → a
  14. Intersect(l, m) → V
  15. Segment(U, V) → b
  16. Intersect(f, m) → W
  17. Segment(V, W) → c
  18. Segment(W, S) → d
  19. Polygon(J, T, U, V, W, S, R, Q) → polígono3, j_2, t_1, u, v, w, s_1, r_1, q_1
  20. Rotate(W, 45°, S) → W'
  21. Rotate(W', 45°, S) → W''
  22. Rotate(W'', 45°, S) → W'''
  23. Rotate(W''', 45°, S) → Z
  24. Rotate(Z, 45°, S) → Z'
  25. Rotate(Z', 45°, S) → Z''
  26. Segment(S, Z'') → e
  27. Segment(S, Z') → f_1
  28. Segment(S, Z) → g_1
  29. Segment(S, W''') → h_1
  30. Segment(S, W'') → n_1
  31. Segment(S, W') → p_1
  32. Rotate(S, 135°, W') → S'
  33. Rotate(W', 135°, S') → W''_1
  34. Rotate(S', 135°, W''_1) → S''
  35. Segment(W', S') → a_1
  36. Segment(S', W''_1) → b_1
  37. Segment(W''_1, S'') → c_1
  38. Segment(S'', U) → d_1
  39. Rotate(S, 135°, W'') → S'_1
  40. Rotate(W'', 135°, S'_1) → W'''_1
  41. Rotate(S'_1, 135°, W'''_1) → A_1
  42. Segment(W'', S'_1) → e_1
  43. Segment(S'_1, W'''_1) → f_2
  44. Segment(W'''_1, A_1) → g_2
  45. Segment(A_1, W''_1) → h_2
  46. Polygon(S, W', S', W''_1, S'', U, V, W) → polígono4, s_2, w', s', w''_1, s'', u_1, v_1, w_1
  47. Polygon(S, W'', S'_1, W'''_1, A_1, W''_1, S', W') → polígono5, s_3, w'', s'_1, w'''_1, i_2, k_2, l_2, m_2
  48. Rotate(S, 135°, W''') → S'_2
  49. Rotate(W''', 135°, S'_2) → B_1
  50. Rotate(S'_2, 135°, B_1) → C_1
  51. Segment(W''', S'_2) → n_2
  52. Segment(S'_2, B_1) → p_2
  53. Segment(B_1, C_1) → q_2
  54. Segment(C_1, W'''_1) → r_2
  55. Rotate(S, 135°, Z) → S'_3
  56. Rotate(Z, 135°, S'_3) → Z'_1
  57. Rotate(S'_3, 135°, Z'_1) → D_1
  58. Rotate(S, 135°, Z') → S'_4
  59. Rotate(Z', 135°, S'_4) → Z''_1
  60. Rotate(S'_4, 135°, Z''_1) → E_1
  61. Rotate(S, 135°, Z'') → S'_5
  62. Rotate(Z'', 135°, S'_5) → Z'''
  63. Rotate(S'_5, 135°, Z''') → F_1
  64. Segment(Z'', S'_5) → t_2
  65. Segment(S'_5, Z''') → a_2
  66. Rotate(Q, 135°, J) → Q'
  67. Segment(J, Q') → b_2
  68. Segment(Q', Z''') → c_2
  69. Segment(Z', S'_4) → d_2
  70. Segment(Z''', F_1) → e_2
  71. Segment(F_1, Z''_1) → f_3
  72. Segment(Z''_1, S'_4) → g_3
  73. Segment(Z, S'_3) → h_3
  74. Segment(S'_3, Z'_1) → i_3
  75. Segment(E_1, Z'_1) → j_3
  76. Segment(E_1, Z''_1) → k_3
  77. Segment(Z'_1, D_1) → l_3
  78. Segment(D_1, B_1) → m_3
  79. Polygon(S, R, Q, J, Q', Z''', S'_5, Z'') → polígono6, s_4, r_3, q_3, j_4, q', z''', s'_5, z''
  80. Polygon(S, Z', S'_4, Z''_1, F_1, Z''', S'_5, Z'') → polígono7, s_5, z', s'_4, z''_1, n_3, p_3, t_3, a_3
  81. Polygon(S, Z, S'_3, Z'_1, E_1, Z''_1, S'_4, Z') → polígono8, s_6, z_1, s'_3, z'_1, b_3, c_3, d_3, e_3
  82. Polygon(S, Z, S'_3, Z'_1, D_1, B_1, S'_2, W''') → polígono9, s_7, z_2, f_4, g_4, h_4, i_4, s'_2, w'''
  83. Polygon(S, W''', S'_2, B_1, C_1, W'''_1, S'_1, W'') → polígono10, s_8, k_4, l_4, m_4, n_4, p_4, q_4, r_4
  84. Segment(Q', T) → t_4
  85. Segment(T, S'') → a_4
  86. Segment(S'', A_1) → b_4
  87. Segment(A_1, C_1) → c_4
  88. Segment(C_1, D_1) → d_4
  89. Segment(D_1, E_1) → e_4
  90. Segment(E_1, F_1) → f_5
  91. Segment(F_1, Q') → g_5
  92. Segment(S, Z'') → h_5
  93. Segment(Z'', S'_5) → i_5
  94. Segment(S'_5, Z''') → j_5
  95. Segment(S, R) → k_5
  96. Segment(R, Q) → l_5
  97. Segment(Q, J) → m_5
  98. Segment(S, W) → n_5
  99. Segment(W, V) → p_5
  100. Segment(V, U) → q_5
  101. Segment(S, W') → r_5
  102. Segment(W', S') → t_5
  103. Segment(S', W''_1) → a_5
  104. Segment(S, W'') → b_5
  105. Segment(W'', S'_1) → c_5
  106. Segment(S'_1, W'''_1) → d_5
  107. Segment(S, W''') → e_5
  108. Segment(W''', S'_2) → f_6
  109. Segment(S'_2, B_1) → g_6
  110. Segment(S, Z) → h_6
  111. Segment(Z, S'_3) → i_6
  112. Segment(S'_3, Z'_1) → j_6
  113. Segment(S, Z') → k_6
  114. Segment(Z', S'_4) → l_6
  115. Segment(S'_4, Z''_1) → m_6