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GeoGebra Fractal Trees + Snow: recursive rules that grow a winter scene

GeoGebra Fractal Trees + Snow: recursive rules that grow a winter scene
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2023-12-11 15:23
SequenceTransformationGeometric ConstructionSlider Animation
This example combines a GeoGebra “Snow” construction with a fractal tree to demonstrate how recursive, rule-based geometry can generate highly detailed, snow-like patterns. The key mathematical ideas include fractal geometry via iteration, piecewise/rule-driven construction, and geometric transformations that position and scale growing branches. It uses RandomUniform and Random to create randomized or perturbed parameters, and employs Translate (and related transformation logic) to place each newly generated segment in the correct location. Parameters such as initialAngle, time, angle, and radius, together with coordinate/size variables (e.g., posX/posY and size sets like size1–size4), manage direction, step length, and scaling across iterations. Objects like A, B, pic1, and buttons (button1/button2) support interactive control, while collections of posX1–posX10 and posY1–posY10 reflect the layered construction of each iteration level. Suitable for secondary school enrichment, math visualization projects, and learners using GeoGebra to connect fractals with transformations and parametric iteration.

GeoGebra Commands & Parameters

  1. RandomUniform(2, 5) → size
  2. Random(0, (2 * pi)) → initialAngle
  3. Translate(Rotate[pic1, angle], Vector[(0, 0), (posX, posY)]) → S1
  4. RandomUniform(0.3, 1.2) → size1
  5. RandomUniform(0.3, 1.2) → size2
  6. RandomUniform(0.3, 1.2) → size3
  7. RandomUniform(0.3, 1.2) → size4
  8. RandomUniform(0.3, 1.2) → size5
  9. RandomUniform(0.3, 1.2) → size6
  10. RandomUniform(0.3, 1.2) → size7
  11. RandomUniform(0.3, 1.2) → size8
  12. RandomUniform(0.3, 1.2) → size9
  13. RandomUniform(0.3, 1.2) → size10
  14. RandomUniform(0, (2 * pi)) → initialAngle1
  15. Random(1, 3) → w1
  16. Random(1, 3) → w2
  17. Random(1, 3) → w3
  18. Random(1, 3) → w4
  19. Random(1, 3) → w5
  20. Random(1, 3) → w6
  21. Random(1, 3) → w7
  22. Random(1, 3) → w8
  23. Random(1, 3) → w9
  24. Random(1, 3) → w10
  25. RandomUniform(0, (2 * pi)) → initialAngle2
  26. RandomUniform(0, (2 * pi)) → initialAngle3
  27. RandomUniform(0, (2 * pi)) → initialAngle4
  28. RandomUniform(0, (2 * pi)) → initialAngle5
  29. RandomUniform(0, (2 * pi)) → initialAngle6
  30. RandomUniform(0, (2 * pi)) → initialAngle7
  31. RandomUniform(0, (2 * pi)) → initialAngle8
  32. RandomUniform(0, (2 * pi)) → initialAngle9
  33. RandomUniform(0, (2 * pi)) → initialAngle10
  34. RandomUniform(0.3, 1.2) → size11
  35. RandomUniform(0.3, 1.2) → size12
  36. RandomUniform(0.3, 1.2) → size13
  37. RandomUniform(0.3, 1.2) → size14
  38. RandomUniform(0.3, 1.2) → size15
  39. RandomUniform(0.3, 1.2) → size16
  40. RandomUniform(0.3, 1.2) → size17
  41. RandomUniform(0.3, 1.2) → size18
  42. RandomUniform(0.3, 1.2) → size19
  43. RandomUniform(0.3, 1.2) → size20
  44. Random(1, 3) → w11
  45. Random(1, 3) → w12
  46. Random(1, 3) → w13
  47. Random(1, 3) → w14
  48. Random(1, 3) → w15
  49. Random(1, 3) → w16
  50. Random(1, 3) → w17
  51. Random(1, 3) → w18
  52. Random(1, 3) → w19
  53. Random(1, 3) → w20
  54. RandomUniform(0, (2 * pi)) → initialAngle11
  55. RandomUniform(0, (2 * pi)) → initialAngle12
  56. RandomUniform(0, (2 * pi)) → initialAngle13
  57. RandomUniform(0, (2 * pi)) → initialAngle14
  58. RandomUniform(0, (2 * pi)) → initialAngle15
  59. RandomUniform(0, (2 * pi)) → initialAngle16
  60. RandomUniform(0, (2 * pi)) → initialAngle17
  61. RandomUniform(0, (2 * pi)) → initialAngle18
  62. RandomUniform(0, (2 * pi)) → initialAngle19
  63. RandomUniform(0, (2 * pi)) → initialAngle20
  64. RandomUniform(0, 10^(2)) → radius1
  65. RandomUniform(0, 10^(2)) → radius2
  66. RandomUniform(0, 10^(2)) → radius3
  67. RandomUniform(0, 10^(2)) → radius4
  68. RandomUniform(0, 10^(2)) → radius5
  69. RandomUniform(0, 10^(2)) → radius6
  70. RandomUniform(0, 10^(2)) → radius7
  71. RandomUniform(0, 10^(2)) → radius8
  72. RandomUniform(0, 10^(2)) → radius9
  73. RandomUniform(0, 10^(2)) → radius10
  74. RandomUniform(0, 10^(2)) → radius11
  75. RandomUniform(0, 10^(2)) → radius12
  76. RandomUniform(0, 10^(2)) → radius13
  77. RandomUniform(0, 10^(2)) → radius14
  78. RandomUniform(0, 10^(2)) → radius15
  79. RandomUniform(0, 10^(2)) → radius16
  80. RandomUniform(0, 10^(2)) → radius17
  81. RandomUniform(0, 10^(2)) → radius18
  82. RandomUniform(0, 10^(2)) → radius19
  83. RandomUniform(0, 10^(2)) → radius20
  84. Translate(Rotate[Dilate[pic1, size1], ((2 * pi) * sin(angle1))], Vector[(0, 0), (posX1, posY1)]) → T1
  85. Translate(Rotate[Dilate[pic1, size2], ((2 * pi) * sin(angle2))], Vector[(0, 0), (posX2, posY2)]) → T2
  86. Translate(Rotate[Dilate[pic1, size3], ((2 * pi) * sin(angle3))], Vector[(0, 0), (posX3, posY3)]) → T3
  87. Translate(Rotate[Dilate[pic1, size4], ((2 * pi) * sin(angle4))], Vector[(0, 0), (posX4, posY4)]) → T4
  88. Translate(Rotate[Dilate[pic1, size5], ((2 * pi) * sin(angle5))], Vector[(0, 0), (posX5, posY5)]) → T5
  89. Translate(Rotate[Dilate[pic1, size6], ((2 * pi) * sin(angle6))], Vector[(0, 0), (posX6, posY6)]) → T6
  90. Translate(Rotate[Dilate[pic1, size7], ((2 * pi) * sin(angle7))], Vector[(0, 0), (posX7, posY7)]) → T7
  91. Translate(Rotate[Dilate[pic1, size8], ((2 * pi) * sin(angle8))], Vector[(0, 0), (posX8, posY8)]) → T8
  92. Translate(Rotate[Dilate[pic1, size9], ((2 * pi) * sin(angle9))], Vector[(0, 0), (posX9, posY9)]) → T9
  93. Translate(Rotate[Dilate[pic1, size10], ((2 * pi) * sin(angle10))], Vector[(0, 0), (posX10, posY10)]) → T10
  94. Translate(Rotate[Dilate[pic1, size11], ((2 * pi) * sin(angle11))], Vector[(0, 0), (posX11, posY11)]) → T11
  95. Translate(Rotate[Dilate[pic1, size12], ((2 * pi) * sin(angle12))], Vector[(0, 0), (posX12, posY12)]) → T12
  96. Translate(Rotate[Dilate[pic1, size13], ((2 * pi) * sin(angle13))], Vector[(0, 0), (posX13, posY13)]) → T13
  97. Translate(Rotate[Dilate[pic1, size14], ((2 * pi) * sin(angle14))], Vector[(0, 0), (posX14, posY14)]) → T14
  98. Translate(Rotate[Dilate[pic1, size15], ((2 * pi) * sin(angle15))], Vector[(0, 0), (posX15, posY15)]) → T15
  99. Translate(Rotate[Dilate[pic1, size16], ((2 * pi) * sin(angle16))], Vector[(0, 0), (posX16, posY16)]) → T16
  100. Translate(Rotate[Dilate[pic1, size17], ((2 * pi) * sin(angle17))], Vector[(0, 0), (posX17, posY17)]) → T17
  101. Translate(Rotate[Dilate[pic1, size18], ((2 * pi) * sin(angle18))], Vector[(0, 0), (posX18, posY18)]) → T18
  102. Translate(Rotate[Dilate[pic1, size19], ((2 * pi) * sin(angle19))], Vector[(0, 0), (posX19, posY19)]) → T19
  103. Translate(Rotate[Dilate[pic1, size20], ((2 * pi) * sin(angle20))], Vector[(0, 0), (posX20, posY20)]) → T20

AI Drawing Prompt Reference

Create a winter scene featuring a fractal tree and snow patterns. The tree branches grow recursively with randomized lengths and angles, shrinking in size with each iteration. Initial angles and positions of the branches are also randomized. Snowflakes overlay the branches, creating a snow-covered forest. The scene is controlled by parameters like initial angles, lengths, widths, and iteration counts, showcasing the details and randomness of fractal geometry.
The prompt above was compiled from the original drawing commands and is for reference only.