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3.2 Tiles: Exploring Vector Translation and Symmetry Transformations

3.2 Tiles: Exploring Vector Translation and Symmetry Transformations
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2023-03-22 20:45
TrianglePolygonVectorTransformationGeometric Construction
In collections geometriadinamica.es
① This example demonstrates how to construct a tiling-like pattern using vector translations, reflections, and rotations of basic geometric shapes, showing the combined effects of transformations in the 2D plane. ② Core concepts include 2D vectors, translation, axial symmetry, rotation, midpoints, perpendicular lines, and properties of triangles and rectangles, along with angle and trigonometric relationships. ③ Construction uses GeoGebra commands such as Line, Circle, Intersect, Polygon, Point, Vector, Translate, Midpoint, Segment, Mirror, OrthogonalLine, Rotate, and Angle, with multiple objects (A, B, C, poly1, u, A', C', D', E', etc.) to perform successive transformations. ④ Suitable for middle or high school geometry or vector lessons, helping students understand composition of transformations, symmetry, and how to create patterns through construction commands, also useful for math-art integration activities.

GeoGebra Commands & Parameters

  1. Line(A, B) → b
  2. Circle(A, a) → c
  3. Intersect(c, b, 2) → C
  4. Polygon(A, C, 4) → poly1, d, e, f, g, D, E
  5. Point(b) → F
  6. Vector(A, F) → u
  7. Translate(A, u) → A'
  8. Translate(C, u) → C'
  9. Translate(D, u) → D'
  10. Translate(E, u) → E'
  11. Polygon(A', C', D', E') → poly1', a', c', d', e'
  12. Translate(A', u) → A''
  13. Translate(C', u) → C''
  14. Translate(D', u) → D''
  15. Translate(E', u) → E''
  16. Polygon(A'', C'', D'', E'') → poly1'', a'', c'', d'', e''
  17. Translate(A'', u) → A'''
  18. Translate(C'', u) → C'''
  19. Translate(D'', u) → D'''
  20. Translate(E'', u) → E'''
  21. Translate(A''', u) → G
  22. Translate(C''', u) → H
  23. Translate(D''', u) → I
  24. Translate(E''', u) → J
  25. Translate(G, u) → G'
  26. Translate(H, u) → H'
  27. Translate(I, u) → I'
  28. Translate(J, u) → J'
  29. Point(e') → S
  30. Point(d'') → T
  31. Point(e'') → U
  32. Vector(A, L_1) → v_1
  33. Translate(A, v_1) → A'_1
  34. Translate(C, v_1) → C'_1
  35. Translate(D, v_1) → D'_1
  36. Translate(E, v_1) → E'_1
  37. Polygon(A'_1, C'_1, D'_1, E'_1) → poly1'_1, a'_1, c'_1, d'_1, e'_1
  38. Translate(A', v_1) → A''_1
  39. Translate(C', v_1) → C''_1
  40. Translate(D', v_1) → D''_1
  41. Translate(E', v_1) → E''_1
  42. Polygon(A''_1, C''_1, D''_1, E''_1) → poly1''_1, a''_1, c''_1, d''_1, e''_1
  43. Midpoint(e'_1) → M_1
  44. Midpoint(c'_1) → N_1
  45. Midpoint(d'_1) → O_1
  46. Midpoint(a'_1) → P_1
  47. Midpoint(O_1, P_1) → Q_1
  48. Segment(M_1, Q_1) → d_1
  49. Point(d_1) → R_1
  50. Midpoint(e''_1) → S_1
  51. Midpoint(d''_1) → T_1
  52. Midpoint(a''_1) → U_1
  53. Midpoint(T_1, U_1) → V_1
  54. Mirror(R_1, Q_1) → Z_1
  55. Polygon(O_1, R_1, P_1, Z_1) → poly10, o_1, r_1, p_1, z_1
  56. Segment(C, E) → q_1
  57. Segment(A, D) → s_1
  58. Segment(A'', D'') → b_2
  59. Segment(P_1, O_1) → c_2
  60. Segment(A''_1, D''_1) → d_2
  61. Segment(E''_1, C''_1) → e_2
  62. Segment(M_1, N_1) → f_2
  63. Midpoint(f) → K
  64. Midpoint(g) → M
  65. Midpoint(d) → N
  66. Midpoint(e) → O
  67. Polygon(K, M, N, O) → poly3, k, m, n, o
  68. Midpoint(d') → P
  69. Midpoint(a') → R
  70. OrthogonalLine(S, e') → p
  71. OrthogonalLine(T, b_2) → q
  72. OrthogonalLine(U, b_2) → r
  73. Intersect(c', p) → Q
  74. Intersect(a'', r) → V
  75. Intersect(q, c'') → W
  76. Polygon(S, P, Q, R) → poly4, s, p_2, q_2, r_2
  77. Polygon(U, T, W, V) → poly5, u_1, t, w, v
  78. Segment(V_1, E''_1) → c_1
  79. Segment(P, R) → e_1
  80. Segment(K, N) → l_1
  81. Segment(O, M) → m_1
  82. Vector(A, B_2) → n_1
  83. Translate(A, n_1) → A'_2
  84. Translate(C, n_1) → C'_2
  85. Translate(D, n_1) → D'_2
  86. Translate(E, n_1) → E'_2
  87. Polygon(A'_2, C'_2, D'_2, E'_2) → poly1'_2, a'_2, c'_2, d'_2, e'_2
  88. Translate(A', n_1) → A''_2
  89. Translate(C', n_1) → C''_2
  90. Translate(D', n_1) → D''_2
  91. Translate(E', n_1) → E''_2
  92. Polygon(A''_2, C''_2, D''_2, E''_2) → poly1''_2, a''_3, c''_2, d''_3, e''_2
  93. Translate(A'', n_1) → A'''_1
  94. Translate(C'', n_1) → C'''_1
  95. Translate(D'', n_1) → D'''_1
  96. Translate(E'', n_1) → E'''_1
  97. Polygon(A'''_1, C'''_1, D'''_1, E'''_1) → poly1'''_1, a'''_1, c'''_1, d'''_1, e'''_1
  98. Point(d'_2) → D_2
  99. Point(e'_2) → H_2
  100. Midpoint(E'_2, C'_2) → I_2
  101. Midpoint(E''_2, C''_2) → J_2
  102. Point(d''_3) → K_2
  103. Point(d'''_1) → L_2
  104. Point(e'''_1) → M_2
  105. Point(a'''_1) → N_2
  106. Point(c'''_1) → P_2
  107. Mirror(D_2, I_2) → O_2
  108. Mirror(H_2, I_2) → Q_2
  109. Rotate(K_2, 90°, J_2) → R_2
  110. Mirror(K_2, J_2) → S_2
  111. Polygon(D_2, H_2, O_2, Q_2) → poly11, d_3, h_2, o_2, q_3
  112. Mirror(R_2, J_2) → T_2
  113. Polygon(K_2, R_2, S_2, T_2) → poly12, k_2, r_3, s_2, t_2
  114. Polygon(L_2, M_2, N_2, P_2) → poly13, l_2, m_2, n_2, p_3
  115. Angle(Q_2, I_2, H_2) → γ
  116. Angle(T_2, J_2, K_2) → δ
  117. Point(d''_1) → L
  118. Mirror(L, d_2) → L'
  119. Mirror(L, e_2) → L'_1
  120. Mirror(L', e_2) → L''
  121. Polygon(L, L', L'', L'_1) → poly2, l, l', l'', l'_1
  122. Angle(D''_1, V_1, A''_1) → α
  123. Intersect(l_1, m_1) → Z
  124. Angle(O, Z, K) → β
  125. Angle(Z_1, Q_1, R_1) → ε

AI Drawing Prompt Reference

Create a pattern composed of triangles and rectangles using transformations such as translation, reflection, and rotation. Start by drawing a triangle and a rectangle, then generate a series of repeated patterns by translating these basic shapes. Use reflection and rotation to further transform these patterns, creating a complex symmetric pattern. Finally, generate multiple repeated patterns through translation and rotation to form a complete set of patterns.
The prompt above was compiled from the original drawing commands and is for reference only.