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3.4.4(b) Museo de Onda: Azulejo 4 – 3D Vector and Geometric Transformations

3.4.4(b) Museo de Onda: Azulejo 4 – 3D Vector and Geometric Transformations
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2023-03-22 21:42
VectorCoordinate GeometrySolid GeometryTransformationGeometric Construction
In collections geometriadinamica.es
This GeoGebra example demonstrates the construction of a complex tile pattern in 3D space using a series of geometric transformations including rotation, reflection, and translation, while highlighting relationships among vectors, midpoints, perpendicular lines, and intersections. Core mathematical concepts include 3D vector operations, geometric transformations (rotation, reflection), orthogonal lines, midpoints, and intersections, along with correlation and probability as noted in the original tags. The construction employs GeoGebra commands such as Line, Point, OrthogonalLine, Intersect, Midpoint, Mirror, Min, Rotate, and Vector, involving numerous points (A, B, G, E, J, H, F, I, etc.) and vectors (vp, vg, dif, vs, vr, vt, v, vd) to generate a symmetric azulejo (tile) pattern. This resource is suitable for advanced high school or university students studying 3D geometry, vector transformations, and geometric construction, and can serve as a cross-disciplinary example in mathematical art or cultural mathematics.

GeoGebra Commands & Parameters

  1. Line(A, B) → b
  2. Line(G, b) → e
  3. Point(e) → J
  4. OrthogonalLine(J, e) → h
  5. OrthogonalLine(G, e) → f
  6. Point(f) → H
  7. OrthogonalLine(H, f) → g
  8. Intersect(g, h) → E
  9. Midpoint(E, H) → I
  10. Midpoint(I, E) → B_1
  11. Midpoint(G, J) → D
  12. Midpoint(J, E) → F
  13. Midpoint(H, G) → K
  14. Midpoint(D, I) → L
  15. Midpoint(K, L) → M
  16. Midpoint(L, F) → N
  17. Midpoint(F, J) → O
  18. Midpoint(J, D) → P
  19. Midpoint(G, D) → Q
  20. Midpoint(K, G) → R
  21. Midpoint(Q, M) → S
  22. Midpoint(L, D) → T
  23. Midpoint(P, N) → U
  24. Midpoint(F, E) → V
  25. Midpoint(K, H) → W
  26. Midpoint(W, V) → Z
  27. Midpoint(H, I) → A_1
  28. Midpoint(N, B_1) → C_1
  29. Midpoint(Z, W) → D_1
  30. Midpoint(D, U) → E_1
  31. Midpoint(U, J) → F_1
  32. Midpoint(T, Q) → G_1
  33. Midpoint(S, G) → H_1
  34. Midpoint(S, K) → I_1
  35. Midpoint(M, T) → J_1
  36. Midpoint(T, N) → K_1
  37. Midpoint(N, O) → L_1
  38. Midpoint(F, C_1) → M_1
  39. Midpoint(C_1, L) → N_1
  40. Midpoint(L, D_1) → O_1
  41. Midpoint(D_1, K) → P_1
  42. Midpoint(H, D_1) → Q_1
  43. Midpoint(D_1, I) → R_1
  44. Midpoint(I, C_1) → S_1
  45. Midpoint(C_1, E) → T_1
  46. Mirror(F_1, h) → U_1
  47. Mirror(L_1, h) → V_1
  48. Mirror(M_1, h) → W_1
  49. Mirror(T_1, h) → Z_1
  50. Mirror(B_1, h) → A_2
  51. Mirror(L_1, h) → C_2
  52. Line(Q, A_2) → d
  53. Line(F, d) → i
  54. Line(C_1, d) → k
  55. Line(S_1, k) → l
  56. Line(R_1, L_1) → j
  57. Line(D_1, j) → m
  58. Line(M, j) → n
  59. Line(I, j) → p
  60. Line(E_1, N_1) → c_1
  61. Line(F_1, M_1) → d_1
  62. Line(G_1, R_1) → e_1
  63. Line(W_1, P_1) → g_1
  64. Line(Q_1, Z_1) → h_1
  65. Mirror(T, m) → T'
  66. Mirror(K_1, m) → I_2
  67. Mirror(T, m) → T'_1
  68. Mirror(L, m) → L'
  69. Mirror(T, m) → T'_2
  70. Mirror(U, m) → U'_1
  71. Mirror(T, m) → T'_3
  72. Point(b) → A1
  73. Min(90, ((a - 1) * 90)) → r1
  74. Rotate(T, ((-r1))°, T) → T'_7
  75. Rotate(U'_1, ((-r1))°, T) → D_{12}
  76. Rotate(T'_3, ((-r1))°, T) → E_{12}
  77. Rotate(U, ((-r1))°, T) → U'_3
  78. Rotate(T, ((-r1))°, T) → T'_8
  79. Rotate(T, ((-r1))°, T'_8) → T'_9
  80. Rotate(T'_2, ((-r1))°, T) → I_{12}
  81. Rotate(T'_1, ((-r1))°, T) → T_{12}
  82. Rotate(L', ((-r1))°, T) → L''_2
  83. Rotate(L, ((-r1))°, T) → L'_2
  84. Midpoint(C_1, E) → K_{20}
  85. Rotate(U'_1, 270°, L') → D_{21}
  86. Rotate(D_{21}, 270°, L') → G_{21}
  87. Rotate(G_{21}, 270°, L') → M_{21}
  88. Line(U'_1, D_{21}) → n_1
  89. Mirror(U'_1, n_1) → V_{21}
  90. Mirror(M_{21}, n_1) → D_{22}
  91. Mirror(D_{21}, n_1) → M_{24}
  92. Mirror(G_{21}, n_1) → P_{24}
  93. Line(G_1, O_1) → l_1
  94. OrthogonalLine(G_1, l_1) → m_1
  95. OrthogonalLine(J_1, l_1) → q_1
  96. OrthogonalLine(O_1, l_1) → r_1
  97. OrthogonalLine(R_1, l_1) → s_1
  98. OrthogonalLine(E_1, m_1) → t_1
  99. OrthogonalLine(F_1, m_1) → a_2
  100. Min(180, ((a - 1) * 180)) → r2
  101. Min(270, ((a - 1) * 270)) → r3
  102. Rotate(T, r1°, T) → T'_4
  103. Rotate(T, r2°, T) → T'_5
  104. Rotate(T, r3°, T) → T'_6
  105. Rotate(L, r1°, T) → L'_1
  106. Rotate(K_1, r1°, T) → Q_3
  107. Rotate(L, r2°, T) → L'_3
  108. Rotate(K_1, r2°, T) → Z_3
  109. Rotate(L, r3°, T) → L'_4
  110. Rotate(K_1, r3°, T) → H_4
  111. Min(180, ((a - 5.5) * 180)) → r4
  112. Vector(S, D) → u
  113. Vector(S, L) → w
  114. Line(S, O) → q
  115. Line(F, M) → r
  116. Line(D_1, V) → s
  117. Mirror(pic1, s) → pic1'_1
  118. Mirror(pic1'_1, r) → pic4
  119. Mirror(pic4, q) → pic4'
  120. Mirror(pic4', e) → pic4''
  121. Midpoint(Q_1, R_1) → B_2
  122. Midpoint(R_1, S_1) → P_2
  123. Midpoint(S_1, T_1) → R_2
  124. Midpoint(T_1, Z_1) → S_2
  125. Mirror(pic1, B_2) → pic1'
  126. Mirror(pic1', P_2) → pic1''
  127. Mirror(pic1'', R_2) → pic1'''
  128. Mirror(pic1''', S_2) → pic2
  129. Mirror(pic1', s) → pic1''_1
  130. Mirror(pic1''_1, r) → pic3
  131. Mirror(pic1'', s) → pic1'''_1
  132. Mirror(pic1''', s) → pic5
  133. Mirror(pic2, s) → pic2'
  134. Mirror(pic2', r) → pic2''
  135. Mirror(pic5, r) → pic5'
  136. Mirror(pic1'''_1, r) → pic6
  137. Mirror(pic3, q) → pic3'
  138. Mirror(pic3', e) → pic3''
  139. Mirror(pic6, q) → pic6'
  140. Mirror(pic6', e) → pic6''
  141. Mirror(pic5', q) → pic5''
  142. Mirror(pic5'', e) → pic5'''
  143. Mirror(pic2'', q) → pic2'''
  144. Mirror(pic2''', e) → pic7
  145. Midpoint(Q_1, P_1) → T_2
  146. Midpoint(P_1, I_1) → W_2
  147. Midpoint(I_1, H_1) → A_3
  148. Midpoint(H_1, G_1) → B_3
  149. Midpoint(G_1, E_1) → C_3
  150. Midpoint(E_1, F_1) → E_3
  151. Midpoint(F_1, U_1) → G_3
  152. Midpoint(U_1, V_1) → I_3
  153. Midpoint(V_1, W_1) → J_3
  154. Midpoint(W_1, Z_1) → K_3
  155. Midpoint(W_1, M_1) → L_3
  156. Midpoint(M_1, N_1) → N_3
  157. Midpoint(N_1, O_1) → P_3
  158. Midpoint(O_1, P_1) → R_3
  159. Midpoint(O_1, J_1) → S_3
  160. Midpoint(J_1, G_1) → T_3
  161. Midpoint(E_1, K_1) → U_3
  162. Midpoint(L_1, F_1) → W_3
  163. Mirror(A_3, H_1) → A_4
  164. Mirror(T_3, G_1) → D_4
  165. Mirror(U_3, E_1) → E_4
  166. Mirror(W_3, F_1) → F_4
  167. Midpoint(B_3, R_3) → G_4
  168. Midpoint(C_3, P_3) → I_4
  169. Midpoint(E_3, N_3) → K_4
  170. Midpoint(G_3, L_3) → L_4
  171. Mirror(J_3, I_3) → M_4
  172. Mirror(F_4, W_3) → N_4
  173. Mirror(W_3, N_4) → O_4
  174. Mirror(E_4, U_3) → P_4
  175. Mirror(U_3, P_4) → Q_4
  176. Mirror(T_3, S_3) → R_4
  177. Line(F_4, O_4) → t
  178. Line(Q_4, E_4) → a_1
  179. Line(D_4, R_4) → b_1
  180. Line(K_3, M_4) → f_1
  181. Line(A_4, T_2) → i_1
  182. Line(L_4, G_4) → j_1
  183. Line(B_3, G_3) → k_1
  184. Line(G_3, S_2) → p_1
  185. Line(C_3, P_2) → b_2
  186. Line(B_3, B_2) → c_2
  187. Vector(A_3, D_4) → z
  188. Vector(A_3, S_3) → u_1

AI Drawing Prompt Reference

Draw a complex 3D tile pattern with multiple points, line segments, and symmetric structures. The pattern includes points A, B, G, E, J, H, F, I, and others, generated using geometric transformations like rotation, reflection, and translation. Key objects include line segments b, e, h, f, g, e, I, E, and points J, H, E, I, F, K, L, M, N, O, P, Q, R, S, T, U, V, W, A_1, C_1, D_1, E_1, F_1, G_1, H_1, I_1, J_1, K_1, L_1, M_1, N_1, O_1, P_1, Q_1, R_1, S_1, T_1. The construction order starts from point A, using midpoints, perpendicular lines, and intersection points to generate the pattern, followed by reflection operations to create symmetry.
The prompt above was compiled from the original drawing commands and is for reference only.