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Kaleidoscope Transformations: Rotations and Reflections

Kaleidoscope Transformations: Rotations and Reflections
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2016-10-31 03:54
PolygonTransformationGeometric Construction
This example demonstrates the creation of kaleidoscope patterns through rotations and reflections. It involves core mathematical concepts such as rotation, reflection, and polygon construction. Key GeoGebra commands used include Circle, Line, OrthogonalLine, Intersect, Point, Segment, Mirror, Polygon, and Rotate. Suitable for middle school students and above to learn geometric transformations and symmetry, or for math clubs.

GeoGebra Commands & Parameters

  1. Circle(A, b) → p
  2. Circle(A, B) → c
  3. Circle(A, C) → d
  4. Circle(A, D) → e
  5. Circle(A, E) → f
  6. Line(F, A) → g
  7. OrthogonalLine(A, g) → h
  8. Intersect(f, g) → G, H
  9. Point(e) → I_1
  10. Segment(G, I_1) → i
  11. Point(g) → J
  12. Segment(J, I_1) → j
  13. Mirror(G, g) → G'
  14. Mirror(I_1, g) → I'
  15. Segment(G', I') → i'
  16. Mirror(J, g) → J'
  17. Mirror(I_1, g) → I'_1
  18. Segment(J', I'_1) → j'
  19. Polygon(G, I_1, J, I') → PreImageKite, g_1, i_1, j_1, i'_1
  20. Mirror(G, h) → G'_1
  21. Mirror(I_1, h) → I'_2
  22. Mirror(J, h) → J'_1
  23. Mirror(I', h) → I''
  24. Polygon(G'_1, I'_2, J'_1, I'') → ReflectedKite, g'_1, i'_2, j'_1, i''
  25. Rotate(G, (-(90°)), A) → G'_2
  26. Rotate(I_1, (-(90°)), A) → I'_3
  27. Rotate(J, (-(90°)), A) → J'_2
  28. Rotate(I', (-(90°)), A) → I''_1
  29. Polygon(G'_2, I'_3, J'_2, I''_1) → RotatedKite90, g'_2, i'_3, j'_2, i''_1
  30. Rotate(G, (-(270°)), A) → G'_3
  31. Rotate(I_1, (-(270°)), A) → I'_4
  32. Rotate(J, (-(270°)), A) → J'_3
  33. Rotate(I', (-(270°)), A) → I''_2
  34. Polygon(G'_3, I'_4, J'_3, I''_2) → RotatedKite270, g'_3, i'_4, j'_3, i''_2
  35. Rotate(G, (-(60°)), A) → G'_4
  36. Rotate(I_1, (-(60°)), A) → I
  37. Rotate(J, (-(60°)), A) → J'_4
  38. Rotate(I', (-(60°)), A) → I''_3
  39. Polygon(G'_4, I, J'_4, I''_3) → RotatedKite60, g'_4, i_2, j'_4, i''_3
  40. Rotate(G, (-(120°)), A) → G'_5
  41. Rotate(I_1, (-(120°)), A) → K
  42. Rotate(J, (-(120°)), A) → J'_5
  43. Rotate(I', (-(120°)), A) → I''_4
  44. Polygon(G'_5, K, J'_5, I''_4) → RotatedKite120, g'_5, k, j'_5, i''_4
  45. Rotate(G, (-(2400°)), A) → G'_6
  46. Rotate(I_1, (-(2400°)), A) → L
  47. Rotate(J, (-(2400°)), A) → J'_6
  48. Rotate(I', (-(2400°)), A) → I''_5
  49. Polygon(G'_6, L, J'_6, I''_5) → RotatedKite240, g'_6, l, j'_6, i''_5
  50. Intersect(h, j_1) → M
  51. Rotate(M, 90°, A) → M'
  52. Point(p) → N
  53. Polygon(A, M', N, M) → Quad2, a, m', n, m
  54. Mirror(A, A) → A'
  55. Mirror(M', A) → M''
  56. Mirror(N, A) → N'
  57. Mirror(M, A) → M'_1
  58. Polygon(A', M'', N', M'_1) → Quad2Reflected, a', m'', n', m'_1
  59. Rotate(A, 90°, A) → A'_1
  60. Rotate(M', 90°, A) → M''_1
  61. Rotate(N, 90°, A) → N'_1
  62. Rotate(M, 90°, A) → M'_2
  63. Polygon(A'_1, M''_1, N'_1, M'_2) → Quad2Rotated90, a'_1, m''_1, n'_1, m'_2
  64. Rotate(A, 270°, A) → A'_2
  65. Rotate(M', 270°, A) → M''_2
  66. Rotate(N, 270°, A) → N'_2
  67. Rotate(M, 270°, A) → M'_3
  68. Polygon(A'_2, M''_2, N'_2, M'_3) → Quad2Rotated270, a'_2, m''_2, n'_2, m'_3
  69. Rotate(G, (-(300°)), A) → G'_7
  70. Rotate(I_1, (-(300°)), A) → O
  71. Rotate(J, (-(300°)), A) → J'_7
  72. Rotate(I', (-(300°)), A) → I''_6
  73. Polygon(G'_7, O, J'_7, I''_6) → RotatedKite300, g'_7, o, j'_7, i''_6
  74. Rotate(G, (-(30°)), A) → G'_8
  75. Rotate(I_1, (-(30°)), A) → P
  76. Rotate(J, (-(30°)), A) → J'_8
  77. Rotate(I', (-(30°)), A) → I''_7
  78. Polygon(G'_8, P, J'_8, I''_7) → RotatedKite30, g'_8, p_1, j'_8, i''_7
  79. Rotate(G, (-(150°)), A) → G'_9
  80. Rotate(I_1, (-(150°)), A) → Q
  81. Rotate(J, (-(150°)), A) → J'_9
  82. Rotate(I', (-(150°)), A) → I''_8
  83. Polygon(G'_9, Q, J'_9, I''_8) → RotatedKite150, g'_9, q, j'_9, i''_8
  84. Rotate(G, (-(210°)), A) → G'_{10}
  85. Rotate(I_1, (-(210°)), A) → R
  86. Rotate(J, (-(210°)), A) → J'_{10}
  87. Rotate(I', (-(210°)), A) → I''_9
  88. Polygon(G'_{10}, R, J'_{10}, I''_9) → RotatedKite210, g'_{10}, r, j'_{10}, i''_9
  89. Rotate(G, (-(330°)), A) → G'_{11}
  90. Rotate(I_1, (-(330°)), A) → S
  91. Rotate(J, (-(330°)), A) → J'_{11}
  92. Rotate(I', (-(330°)), A) → I''_{10}
  93. Polygon(G'_{11}, S, J'_{11}, I''_{10}) → RotatedKite330, g'_{11}, s, j'_{11}, i''_{10}
  94. Rotate(A, (-(30°)), A) → A'_3
  95. Rotate(M', (-(30°)), A) → M''_3
  96. Rotate(N, (-(30°)), A) → N'_3
  97. Rotate(M, (-(30°)), A) → M'_4
  98. Polygon(A'_3, M''_3, N'_3, M'_4) → QuadRotated30, a'_3, m''_3, n'_3, m'_4
  99. Rotate(A, (-(120°)), A) → A'_4
  100. Rotate(M', (-(120°)), A) → M''_4
  101. Rotate(N, (-(120°)), A) → N'_4
  102. Rotate(M, (-(120°)), A) → M'_5
  103. Polygon(A'_4, M''_4, N'_4, M'_5) → QuadRotated120, a'_4, m''_4, n'_4, m'_5