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Forming Shapes by Tiling with Triangles (P3)

Forming Shapes by Tiling with Triangles (P3)
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2021-03-26 17:12
TrianglePolygonTransformationGeometric Construction
In collections P3 3S2 Making Triangles
This example demonstrates how to build more complex planar shapes by tiling with triangles and applying mirror symmetry step by step. The key mathematical ideas include geometric construction, triangle and polygon composition, and reflection (mirror) transformations; it also connects to the broader use of vectors/parametric thinking in organizing geometric structure. Construction highlights from the given GeoGebra commands: Point is used to define key vertices, Polygon builds triangle/polygon regions from those vertices, and Mirror generates reflected copies across selected lines. IndexOf and If support conditional selection/updating of point sets during construction. The involved objects include grid controls (ShGrid, buttonShowGridC/buttonShowGridE), instruction texts (Text2InstructionC/Text3InstructionE), reflected point sets (ReflectA1–ReflectA4 and ReflectB1–ReflectB4), and the resulting polygonal assemblies (A, A1, A2, plus grouped vertices such as A1P1/A1P2/A1P3). Suitable for middle/high school geometry, interactive exploration, and GeoGebra lessons on transformations and constructive tiling.

GeoGebra Commands & Parameters

  1. Point(Circle[A1P1, 3]) → A1P2
  2. Polygon(A1P1, A1P2, A1P3) → A1, c, a, b
  3. Point(Circle[A2P1, 3]) → A2P2
  4. Polygon(A2P1, A2P2, A2P3) → A2, c_1, a_1, b_1
  5. Mirror(A1P1, c) → A1P1'
  6. Mirror(A1P2, c) → A1P2'
  7. Mirror(A1P3, c) → A1P3'
  8. Polygon(A1P1', A1P2', A1P3') → A1', a1p3', a1p1', a1p2'
  9. IndexOf(SelectedElement[ListFig], ListFig) → ReflectFig
  10. If(ShowChi, "選擇要反轉的圖形", "Select a figure to reflect") → TextInstSelect
  11. Mirror(A2P1, c_1) → A2P1'
  12. Mirror(A2P2, c_1) → A2P2'
  13. Mirror(A2P3, c_1) → A2P3'
  14. Polygon(A2P1', A2P2', A2P3') → A2', a2p3', a2p1', a2p2'
  15. Point(Circle[A3P1, 3]) → A3P2
  16. Polygon(A3P1, A3P2, A3P3) → A3, c_2, a_2, b_2
  17. Mirror(A3P1, c_2) → A3P1'
  18. Mirror(A3P2, c_2) → A3P2'
  19. Mirror(A3P3, c_2) → A3P3'
  20. Polygon(A3P1', A3P2', A3P3') → A3', a3p3', a3p1', a3p2'
  21. Point(Circle[A4P1, 3]) → A4P2
  22. Polygon(A4P1, A4P2, A4P3) → A4, c_3, a_3, b_3
  23. Mirror(A4P1, c_3) → A4P1'
  24. Mirror(A4P2, c_3) → A4P2'
  25. Mirror(A4P3, c_3) → A4P3'
  26. Polygon(A4P1', A4P2', A4P3') → A4', c', a4p1', a4p2'
  27. Point(Circle[B1P1, 3]) → B1P2
  28. Polygon(B1P1, B1P2, B1P3) → B1, d, a_4, b_4
  29. Mirror(B1P1, d) → B1P1'
  30. Mirror(B1P2, d) → B1P2'
  31. Mirror(B1P3, d) → B1P3'
  32. Polygon(B1P1', B1P2', B1P3') → B1', b1p3', b1p1', b1p2'
  33. Point(Circle[B2P1, 3]) → B2P2
  34. Polygon(B2P1, B2P2, B2P3) → B2, c_4, a_5, b_5
  35. Mirror(B2P1, c_4) → B2P1'
  36. Mirror(B2P2, c_4) → B2P2'
  37. Mirror(B2P3, c_4) → B2P3'
  38. Polygon(B2P1', B2P2', B2P3') → B2', b2p3', a', b2p2'
  39. Point(Circle[B3P1, 3]) → B3P2
  40. Polygon(B3P1, B3P2, B3P3) → B3, c_5, a_6, b_6
  41. Mirror(B3P1, c_5) → B3P1'
  42. Mirror(B3P2, c_5) → B3P2'
  43. Mirror(B3P3, c_5) → B3P3'
  44. Polygon(B3P1', B3P2', B3P3') → B3', f, b3p1', b3p2'
  45. Point(Circle[B4P1, 3]) → B4P2
  46. Polygon(B4P1, B4P2, B4P3) → B4, d_1, a_7, b_7
  47. Mirror(B4P1, d_1) → B4P1'
  48. Mirror(B4P2, d_1) → B4P2'
  49. Mirror(B4P3, d_1) → B4P3'
  50. Polygon(B4P1', B4P2', B4P3') → B4', b4p3', b4p1', b4p2'
  51. Point(Circle[C1P1, 2]) → C1P2
  52. Polygon(C1P1, C1P2, C1P3) → C1, c_6, a_8, b_8
  53. Polygon(C1P1, C1P2, C1P4) → C1a, c1p4, c1p1, c1p2
  54. Mirror(C1P1, c_6) → C1P1'
  55. Mirror(C1P2, c_6) → C1P2'
  56. Mirror(C1P3, c_6) → C1P3'
  57. Polygon(C1P1', C1P2', C1P3') → C1', c1p3', c1p1', c1p2'
  58. Mirror(C1P1, c_6) → C1P1'_1
  59. Mirror(C1P2, c_6) → C1P2'_1
  60. Mirror(C1P4, c_6) → C1P4'
  61. Polygon(C1P1'_1, C1P2'_1, C1P4') → C1a', c1p4', c1p1'_1, c1p2'_1
  62. Point(Circle[C2P1, 2]) → C2P2
  63. Polygon(C2P1, C2P2, C2P3) → C2, c_7, a_9, b_9
  64. Polygon(C2P1, C2P2, C2P4) → C2a, c2p4, c2p1, c2p2
  65. Mirror(C2P1, c2p4) → C2P1'
  66. Mirror(C2P2, c2p4) → C2P2'
  67. Mirror(C2P4, c2p4) → C2P4'
  68. Polygon(C2P1', C2P2', C2P4') → C2a', c2p4', c2p1', c2p2'
  69. Mirror(C2P1, c2p4) → C2P1'_1
  70. Mirror(C2P2, c2p4) → C2P2'_1
  71. Mirror(C2P3, c2p4) → C2P3'
  72. Polygon(C2P1'_1, C2P2'_1, C2P3') → C2', c2p3', c2p1'_1, c2p2'_1
  73. Point(Circle[C3P1, 2]) → C3P2
  74. Polygon(C3P1, C3P2, C3P3) → C3, c_8, a_{10}, b_{10}
  75. Polygon(C3P1, C3P2, C3P4) → C3a, c3p4, c3p1, c3p2
  76. Mirror(C3P1, c_8) → C3P1'
  77. Mirror(C3P2, c_8) → C3P2'
  78. Mirror(C3P4, c_8) → C3P4'
  79. Polygon(C3P1', C3P2', C3P4') → C3a', c3p4', c3p1', c3p2'
  80. Mirror(C3P1, c_8) → C3P1'_1
  81. Mirror(C3P2, c_8) → C3P2'_1
  82. Mirror(C3P3, c_8) → C3P3'
  83. Polygon(C3P1'_1, C3P2'_1, C3P3') → C3', c3p3', c3p1'_1, c3p2'_1
  84. Point(Circle[C4P1, 2]) → C4P2
  85. Polygon(C4P1, C4P2, C4P3) → C4, c_9, a_{11}, b_{11}
  86. Polygon(C4P1, C4P2, C4P4) → C4a, c4p4, c4p1, c4p2
  87. Mirror(C4P1, c_9) → C4P1'
  88. Mirror(C4P2, c_9) → C4P2'
  89. Mirror(C4P4, c_9) → C4P4'
  90. Polygon(C4P1', C4P2', C4P4') → C4a', c4p4', c4p1', c4p2'
  91. Mirror(C4P1, c_9) → C4P1'_1
  92. Mirror(C4P2, c_9) → C4P2'_1
  93. Mirror(C4P3, c_9) → C4P3'
  94. Polygon(C4P1'_1, C4P2'_1, C4P3') → C4', c4p3', c4p1'_1, c4p2'_1
  95. Point(Circle[D1P1, 4.123105625617661]) → D1P2
  96. Polygon(D1P1, D1P2, D1P3) → D1, c_{10}, a_{12}, b_{12}
  97. Point(Circle[D2P1, 4.123105625617661]) → D2P2
  98. Polygon(D2P1, D2P2, D2P3) → D2, c_{11}, a_{13}, b_{13}
  99. Point(Circle[D3P1, 4.123105625617661]) → D3P2
  100. Polygon(D3P1, D3P2, D3P3) → D3, c_{12}, a_{14}, b_{14}
  101. Point(Circle[D4P1, 4.123105625617661]) → D4P2
  102. Polygon(D4P1, D4P2, D4P3) → D4, c_{13}, a_{15}, b_{15}
  103. Point(Circle[E1P1, 3]) → E1P2
  104. Polygon(E1P1, E1P2, E1P3) → E1, c_{14}, a_{16}, b_{16}
  105. Point(Circle[E2P1, 3]) → E2P2
  106. Polygon(E2P1, E2P2, E2P3) → E2, c_{15}, a_{17}, b_{17}
  107. Point(Circle[E3P1, 3]) → E3P2
  108. Polygon(E3P1, E3P2, E3P3) → E3, c_{16}, a_{18}, b_{18}
  109. Point(Circle[E4P1, 3]) → E4P2
  110. Polygon(E4P1, E4P2, E4P3) → E4, c_{17}, a_{19}, b_{19}
  111. Point(Circle[E5P1, 3]) → E5P2
  112. Polygon(E5P1, E5P2, E5P3) → E5, c_{18}, a_{20}, b_{20}
  113. Point(Circle[E6P1, 3]) → E6P2
  114. Polygon(E6P1, E6P2, E6P3) → E6, c_{19}, a_{21}, b_{21}

AI Drawing Prompt Reference

Draw a complex polygon composed of multiple triangles arranged by mirror symmetry. Start by placing several basic triangles on a grid. Then generate reflected copies of each triangle using mirror transformations. Finally, combine these reflected copies with the original triangles to form larger polygon structures. The construction should proceed step by step to ensure clear symmetry relationships, suitable for geometry teaching and exploration.
The prompt above was compiled from the original drawing commands and is for reference only.