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Squares within Square v.3: Similarity Transformations and Vector Construction

Squares within Square v.3: Similarity Transformations and Vector Construction
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2016-07-17 10:16
PolygonVectorIntegralTransformationGeometric Construction
This example demonstrates the recursive nesting of smaller squares inside a larger square through similarity transformations and vector translations, creating a fractal-like visual pattern. It explores core mathematical concepts including perimeter and area relationships of squares, incircles and inscribed circles, similarity transformations, 3D vectors projected onto a plane, exponential growth patterns, and definite integrals as a way to sum accumulated areas. The construction uses GeoGebra commands such as Circle, Point, Ray, Intersect, Segment, OrthogonalLine, Line, Vector, Translate, and Polygon. Starting with a base square and its incircle, key points are determined via orthogonal lines and intersections, then vector translations and similarity transformations generate successive nested squares, involving auxiliary objects like l_1 and n_3. This is suitable for high school and university students exploring geometric transformations, vector operations, and area accumulation, as well as for mathematical modeling or artistic pattern design.

GeoGebra Commands & Parameters

  1. Circle(A, 6) → c
  2. Point(c) → B
  3. Ray(B, A) → a
  4. Intersect(c, a, 2) → D
  5. Segment(B, D) → q
  6. Point(q) → F
  7. Segment(F, D) → j
  8. OrthogonalLine(A, a) → b
  9. Intersect(c, b, 2) → E
  10. Segment(E, B) → g
  11. Intersect(c, b, 1) → C
  12. Segment(C, D) → e
  13. Line(B, e) → l_1
  14. Point(j) → G
  15. Ray(G, D) → m
  16. Point(m) → H
  17. OrthogonalLine(H, g) → n_3
  18. Segment(B, C) → d
  19. Segment(D, E) → f
  20. OrthogonalLine(F, g) → h
  21. OrthogonalLine(F, d) → i
  22. OrthogonalLine(G, g) → k
  23. OrthogonalLine(G, d) → l
  24. OrthogonalLine(H, d) → p
  25. Intersect(i, k) → I
  26. Intersect(h, l) → J
  27. Intersect(l, n_3) → K
  28. Intersect(k, p) → L
  29. Intersect(h, g) → M
  30. Intersect(i, d) → N
  31. Segment(M, J) → r
  32. Segment(N, I) → s
  33. Segment(I, L) → t
  34. Segment(J, K) → a_1
  35. Segment(K, H) → b_1
  36. Segment(L, H) → c_1
  37. Segment(B, M) → d_1
  38. Segment(F, M) → f_1
  39. Point(f_1) → O
  40. Intersect(k, g) → P
  41. Segment(G, P) → g_1
  42. Point(g_1) → Q
  43. Line(O, g) → e_1
  44. Intersect(e_1, g_1) → R
  45. Segment(O, R) → h_1
  46. Intersect(n_3, l_1) → S
  47. Segment(K, S) → i_1
  48. Line(Q, g) → j_1
  49. Intersect(j_1, i_1) → T
  50. Segment(Q, T) → k_1
  51. Segment(B, N) → m_1
  52. Segment(F, N) → n_1
  53. Point(n_1) → U
  54. OrthogonalLine(J, f) → p_1
  55. OrthogonalLine(U, e) → q_1
  56. Intersect(p_1, q_1) → V
  57. Segment(U, V) → r_1
  58. Intersect(p_1, d) → W
  59. Segment(G, W) → s_1
  60. Point(s_1) → Z
  61. OrthogonalLine(H, f) → t_1
  62. Line(Z, f) → a_2
  63. Intersect(t_1, a_2) → A_1
  64. Segment(Z, A_1) → b_2
  65. Vector(B, C) → u
  66. Translate(X, u) → X'
  67. Vector(X, X') → v
  68. Point(v) → Y
  69. Polygon(X, Y, 4) → poly1, c_2, d_2, e_2, f_2, C_1, D_1
  70. Vector(X, Y) → w
  71. Translate(X, z) → X'_1
  72. Vector(X, X'_1) → u_1
  73. Point(u_1) → E_1
  74. OrthogonalLine(E_1, e_2) → g_2
  75. OrthogonalLine(D_1, f_2) → h_2
  76. Intersect(h_2, g_2) → F_1
  77. Segment(E_1, F_1) → i_2
  78. Circle(X, Length[u_1]) → k_2
  79. Line(D_1, X) → j_2
  80. Intersect(k_2, j_2, 2) → G_1
  81. Segment(X, G_1) → l_2
  82. Point(l_2) → H_1
  83. OrthogonalLine(H_1, f_2) → m_2
  84. OrthogonalLine(Y, c_2) → n_2
  85. Intersect(m_2, n_2) → I_1
  86. Segment(H_1, I_1) → p_2

AI Drawing Prompt Reference

Draw a large square and recursively draw smaller squares inside it using similarity transformations and vector translations. Start by defining the incircle of the large square, then use orthogonal lines and intersections to find key points, and finally apply vector translations and similarity transformations to generate successive nested squares, forming a fractal-like pattern.
The prompt above was compiled from the original drawing commands and is for reference only.