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Fathauer Kites: Colorful Geometric Spiral Construction

Fathauer Kites: Colorful Geometric Spiral Construction
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2020-11-23 05:20
PolygonTrigonometrySequenceTransformationGeometric Construction
① This example demonstrates a colorful spiral pattern composed of kite-shaped polygons, created through rotations, reflections, and sequences to produce an aesthetically pleasing geometric design. ② Core mathematical concepts include polygons, rotational transformations, mirror symmetry, trigonometric functions (tangent), and centroid/barycenter calculations, along with percentages and statistics for color or size distribution. ③ Construction highlights: using GeoGebra commands such as PointIn, Polygon, Mirror, Segment, Circle, Intersect, Line, Angle, Sequence, Element, and Textfield, with objects like points A, B, C, polygon q1, segments a, c, b, circle f, parameters R and n, mirrored points (B', C'), and auxiliary lines, finally generating a series A1 to A11 via sequence to form the spiral. ④ Suitable for high school to early college students, for exploring geometric transformations, trigonometric applications, and dynamic mathematical art, also serving as a cross-disciplinary teaching resource between math and art.

GeoGebra Commands & Parameters

  1. PointIn(Polygon[A, B, 4]) → C
  2. Polygon(A, C, B, Mirror[C, Line[A, B]]) → q1, a, c, b, f
  3. Mirror(C, Line[A, B]) → C'
  4. Segment(A, B) → g
  5. Circle(C', g / R) → d
  6. Circle(B, c / R) → e
  7. Intersect(d, e, 2) → D
  8. Segment(C', D) → h
  9. Mirror(B, h) → B'
  10. Polygon(D, B, C', B') → q2, d_1, b_1, c', b'
  11. Line(D, C') → i
  12. Line(B, A) → j
  13. Intersect(i, j) → E
  14. Angle(D, E, B) → ang
  15. Sequence(Rotate[Dilate[q1, R^(i), E], (i * ang), E], i, (-n), n) → l1
  16. Sequence(Rotate[l1, ((i * 2) * ang), E], i, 0, floor((180°) / ang) - 1) → l2
  17. Element(l2, A1) → B1
  18. Element(l2, A2) → B2
  19. Element(l2, A3) → B3
  20. Element(l2, A4) → B4
  21. Element(l2, A5) → B5
  22. Element(l2, A6) → B6
  23. Element(l2, A7) → B7
  24. Element(l2, A8) → B8
  25. Element(l2, A9) → B9
  26. Element(l2, A10) → B10
  27. Element(l2, A11) → B11
  28. Element(l2, A12) → B12
  29. Element(l2, A13) → B13
  30. Element(l2, A14) → B14
  31. Element(l2, A15) → B15
  32. Element(l2, A16) → B16
  33. Element(l2, A17) → B17
  34. Element(l2, A18) → B18
  35. Element(l2, A19) → B19
  36. Element(l2, A20) → B20
  37. Element(l2, A21) → B21
  38. Element(l2, A22) → B22
  39. Element(l2, A23) → B23
  40. Element(l2, A24) → B24
  41. Element(l2, A25) → B25
  42. Element(l2, A26) → B26
  43. Element(l2, A27) → B27
  44. Element(l2, A28) → B28
  45. Element(l2, A29) → B29
  46. Element(l2, A30) → B30
  47. Element(l2, A31) → B31
  48. Element(l2, A32) → B32
  49. Element(l2, A33) → B33
  50. Element(l2, A34) → B34
  51. Element(l2, A35) → B35
  52. Element(l2, A36) → B36
  53. Element(l2, A37) → B37
  54. Element(l2, A38) → B38
  55. Element(l2, A39) → B39
  56. Element(l2, A40) → B40
  57. Element(l2, A41) → B41
  58. Element(l2, A42) → B42
  59. Element(l2, A43) → B43
  60. Element(l2, A44) → B44
  61. Element(l2, A45) → B45
  62. Element(l2, A46) → B46
  63. Element(l2, A47) → B47
  64. Element(l2, A48) → B48
  65. Element(l2, A49) → B49
  66. Element(l2, A50) → B50
  67. Element(l2, A51) → B51
  68. Element(l2, A52) → B52
  69. Element(l2, A53) → B53
  70. Element(l2, A54) → B54
  71. Element(l2, A55) → B55
  72. Element(l2, A56) → B56
  73. Element(l2, A57) → B57
  74. Element(l2, A58) → B58
  75. Element(l2, A59) → B59
  76. Element(l2, A60) → B60
  77. Element(l2, A61) → B61
  78. Element(l2, A62) → B62
  79. Element(l2, A63) → B63
  80. Element(l2, A64) → B64
  81. Element(l2, A65) → B65
  82. Element(l2, A66) → B66
  83. Element(l2, A67) → B67
  84. Element(l2, A68) → B68
  85. Element(l2, A69) → B69
  86. Element(l2, A70) → B70
  87. Element(l2, A71) → B71
  88. Element(l2, A72) → B72
  89. Element(l2, A73) → B73
  90. Element(l2, A74) → B74
  91. Element(l2, A75) → B75
  92. Element(l2, A76) → B76
  93. Element(l2, A77) → B77
  94. Element(l2, A78) → B78
  95. Element(l2, A79) → B79
  96. Element(l2, A80) → B80
  97. Element(l2, A81) → B81
  98. Element(l2, A82) → B82
  99. Element(l2, A83) → B83
  100. Element(l2, A84) → B84
  101. Element(l2, A85) → B85
  102. Element(l2, A86) → B86
  103. Element(l2, A87) → B87
  104. Element(l2, A88) → B88
  105. Element(l2, A89) → B89
  106. Element(l2, A90) → B90
  107. Element(l2, A91) → B91
  108. Element(l2, A92) → B92
  109. Element(l2, A93) → B93
  110. Element(l2, A94) → B94
  111. Element(l2, A95) → B95
  112. Element(l2, A96) → B96
  113. Element(l2, A97) → B97
  114. Element(l2, A98) → B98
  115. Element(l2, A99) → B99
  116. Element(l2, A100) → B100
  117. Angle(B, A, C) → α
  118. Angle(A, C, B) → β
  119. Angle(C', E, C) → γ
  120. Intersect(Circle[E, C], j) → F, G
  121. Textfield(ang2) → InputBox1

AI Drawing Prompt Reference

On a coordinate plane, start with a segment AB and use the line AB as a mirror axis to build a kite-shaped polygon pattern. Define a point C inside a polygon with vertices including A, B and a fourth point, then reflect C across line AB to get C'. Construct circles centered at C' and at B using radii derived from a shared parameter R and intersect them to obtain a point D, then draw segment C'–D. Reflect the segment-related geometry about the line AB to get B', and form a kite polygon with vertices D, B, C', and B'. Finally, treat D and the reference angle from lines involving D, C', and B to iteratively rotate and scale this base polygon around an appropriate center using parameters R and an integer n, producing a spiral-like sequence of repeated kite polygons similar in count and placement to A1 through A11.
The prompt above was compiled from the original drawing commands and is for reference only.