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Tessellations by Reflecting over 45-45-90 Triangle

Tessellations by Reflecting over 45-45-90 Triangle
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2015-02-10 05:02
TriangleTransformationGeometric Construction
Drag a point inside a 45-45-90 triangle to create different tessellations. This example demonstrates the application of reflection transformations in geometric constructions, involving key concepts such as reflection and geometric construction. It uses commands like Polygon, Segment, PointIn, Mirror, and is suitable for middle school and above students learning about transformations and tessellations.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 4) → poly1, a_2, b_2, c_2, d, C, D
  2. Segment(A, C) → e
  3. Polygon(A, B, C) → poly2, c, a, b
  4. PointIn(poly2) → E
  5. Mirror(E, a) → E'
  6. Mirror(E, b) → E'_1
  7. Mirror(E, c) → E'_2
  8. Segment(E, E'_1) → f
  9. Segment(E, E') → g
  10. Segment(E, E'_2) → h
  11. Mirror(E'_1, a) → F
  12. Mirror(E'_1, c) → G
  13. Mirror(E', b) → E''
  14. Mirror(E', c) → E''_1
  15. Mirror(E'_2, b) → H
  16. Segment(E'_1, E'') → i
  17. Segment(E'_1, H) → j
  18. Segment(E'_2, G) → k
  19. Segment(E''_1, E') → l
  20. Segment(E'_2, E''_1) → m
  21. Segment(E', F) → p
  22. Mirror(E'', a) → E'''
  23. Mirror(E'', c) → E'''_1
  24. Mirror(F, b) → F'
  25. Mirror(F, c) → F'_1
  26. Mirror(E''_1, b) → I
  27. Mirror(G, b) → G'
  28. Mirror(H, a) → H'
  29. Mirror(H, c) → H'_1
  30. Segment(E'', F') → q
  31. Segment(E'', I) → r
  32. Segment(I, H) → s
  33. Segment(H, G') → t
  34. Segment(G, H'_1) → a_1
  35. Segment(G, E'''_1) → b_1
  36. Segment(E''_1, F'_1) → c_1
  37. Segment(F, H') → d_1
  38. Segment(F, E''') → e_1
  39. Mirror(F', a) → F''
  40. Mirror(F', c) → F''_1
  41. Mirror(I, a) → I'
  42. Mirror(I, c) → I'_1
  43. Mirror(G', a) → G''
  44. Mirror(G', c) → G''_1
  45. Mirror(H'_1, a) → J
  46. Mirror(E'''_1, a) → K
  47. Mirror(E'''_1, b) → L
  48. Mirror(F'_1, b) → M
  49. Mirror(H', b) → H''
  50. Segment(F', H'') → f_1
  51. Segment(I, M) → g_1
  52. Segment(G', L) → h_1
  53. Segment(G''_1, G') → i_1
  54. Segment(G''_1, H'_1) → j_1
  55. Segment(H'_1, I'_1) → k_1
  56. Segment(I'_1, E'''_1) → l_1
  57. Segment(E'''_1, F''_1) → m_1
  58. Segment(K, F'_1) → n_1
  59. Segment(F'_1, J) → p_1
  60. Segment(H', G'') → q_1
  61. Segment(H', I') → r_1
  62. Segment(I', E''') → s_1
  63. Segment(E''', F'') → t_1
  64. Segment(F'', F') → d_2
  65. Mirror(H'', a) → H'''
  66. Mirror(H'', c) → H'''_1
  67. Mirror(M, a) → M'
  68. Mirror(M, c) → M'_1
  69. Mirror(L, a) → L'
  70. Mirror(L, c) → L'_1
  71. Mirror(F''_1, a) → N
  72. Mirror(F''_1, b) → O
  73. Mirror(K, b) → K'
  74. Mirror(J, b) → J'
  75. Mirror(G'', b) → G'''
  76. Mirror(G'', c) → G'''_1
  77. Mirror(I', c) → I''
  78. Segment(H'', G''') → e_2
  79. Segment(M, J') → f_2
  80. Segment(M, K') → g_2
  81. Segment(L, O) → h_2
  82. Segment(L, L'_1) → i_2
  83. Segment(I'_1, M'_1) → j_2
  84. Segment(F''_1, H'''_1) → k_2
  85. Segment(F''_1, N) → l_2
  86. Segment(N, K) → m_2
  87. Segment(K, I'') → n_2
  88. Segment(I'', J) → p_2
  89. Segment(J, G'''_1) → q_2
  90. Segment(G'''_1, G'') → r_2
  91. Segment(I', M') → s_2
  92. Segment(F'', H''') → t_2
  93. Segment(H''', H'') → a_3
  94. Segment(G'', L') → b_3
  95. Mirror(G''', a) → P
  96. Mirror(G''', c) → Q
  97. Mirror(J', a) → J''
  98. Mirror(J', c) → J''_1
  99. Mirror(K', a) → K''
  100. Mirror(K', c) → K''_1
  101. Mirror(O, a) → O'
  102. Mirror(O, c) → O'_1
  103. Mirror(L'_1, a) → R
  104. Mirror(M'_1, a) → S
  105. Mirror(H'''_1, a) → T
  106. Mirror(H'''_1, b) → U
  107. Mirror(I'', b) → I'''
  108. Mirror(G'''_1, b) → V
  109. Mirror(L', b) → L''
  110. Segment(L'', G''') → c_3
  111. Segment(G''', V) → d_3
  112. Segment(V, J') → e_3
  113. Segment(J', I''') → f_3
  114. Segment(I''', K') → g_3
  115. Segment(O, U) → h_3
  116. Segment(L'_1, O'_1) → i_3
  117. Segment(K''_1, M'_1) → j_3
  118. Segment(J''_1, M'_1) → k_3
  119. Segment(Q, H'''_1) → l_3
  120. Segment(H'''_1, T) → m_3
  121. Segment(T, N) → n_3
  122. Segment(S, I'') → p_3
  123. Segment(R, G'''_1) → q_3
  124. Segment(R, L') → r_3
  125. Segment(L', O') → s_3
  126. Segment(K'', M') → t_3
  127. Segment(M', J'') → a_4
  128. Segment(P, H''') → b_4
  129. Segment(G''_1, L'_1) → c_4

AI Drawing Prompt Reference

Start with a square and draw one diagonal to split it into two 45-45-90 triangles. Choose one triangle and place a point anywhere inside it. Reflect this point across each of the triangle's three sides to get three mirrored points. Connect the original point to each mirrored point with segments, forming a small triangle. Then continue reflecting these points across adjacent triangle sides and connect corresponding mirrored points, expanding outward step by step. This creates a tessellation of many right isosceles triangles, with the point repeated in each triangle through reflections, forming an intricate grid-like pattern.
The prompt above was compiled from the original drawing commands and is for reference only.