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Forming Shapes with Triangles: Mirror Reflection and Polygon Construction

Forming Shapes with Triangles: Mirror Reflection and Polygon Construction
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2021-03-26 17:01
TrianglePolygonAngleTransformationGeometric Construction
In collections P3 3S2 製作三角形
This example demonstrates how to build more complex planar figures by assembling triangles and applying mirror (reflection) transformations. The key mathematical ideas include triangle/polygon geometry, symmetry via reflection, and systematic organization of vertices and construction steps. In GeoGebra, the figure is constructed using Point to define key vertices, Polygon to create triangle and composite polygonal regions, Mirror to reflect selected parts across specified symmetry lines, and If combined with IndexOf for conditional logic that controls stepwise display. The workflow is supported by teaching interface elements such as Text instructions and buttons, alongside collections of reflected points/objects (e.g., ReflectA*/ReflectB*, A*P* sets) that generate the final tessellated structure. Suitable for middle and early high school geometry lessons and for guided visualization of geometric construction and transformation-based reasoning.

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 symmetric pattern composed of multiple triangles. First, create four separate triangles, each mirrored across one of its sides to form a reflected copy, resulting in four pairs of symmetric triangles. Then, create four more triangles, each also mirrored across one of its sides to form reflected copies, yielding four additional pairs of symmetric triangles. All triangles are of equal size and arranged in a regular grid, creating an overall symmetric and repetitive geometric design.
The prompt above was compiled from the original drawing commands and is for reference only.