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IMPOSSIBLE! The Illusion of 3D Cuboid and Reflection

IMPOSSIBLE! The Illusion of 3D Cuboid and Reflection
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2013-12-08 04:56
VectorSolid GeometryTransformationGeometric Construction
This GeoGebra example demonstrates a striking visual illusion constructed from a 3D cuboid, reflection, and translation, creating a seemingly impossible geometric configuration. The core mathematical concepts involve solid geometry (cuboid), reflection transformations, vector translation, and subtraction, with a normal distribution curve as an auxiliary element. The construction uses GeoGebra commands such as Segment, Point, Polygon, Line, Vector, Translate, and Intersect, linking numerous objects (A, B, C, etc.) to achieve precise geometric relationships that trick the eye. This example is suitable for high school or early college students, serving as a bridge between geometric transformations and visual perception. It can be used in math-art demonstrations or to stimulate spatial reasoning and geometric inquiry.

GeoGebra Commands & Parameters

  1. Segment(A, B) → a
  2. Segment(B, C) → b_6
  3. Segment(C, D) → c_6
  4. Segment(D, E) → d
  5. Segment(E, F) → e
  6. Segment(F, G) → f
  7. Segment(G, H) → g_6
  8. Segment(G, I) → h
  9. Segment(A, I) → i
  10. Point(f) → J
  11. Segment(J, K) → j
  12. Segment(E, K) → k_4
  13. Segment(K, L) → l
  14. Segment(L, D) → m_5
  15. Segment(I, M) → n_5
  16. Segment(M, H) → p
  17. Segment(A, N) → q_4
  18. Segment(N, M) → r_5
  19. Segment(O, P) → s_4
  20. Segment(O, Q) → t
  21. Segment(Q, R) → a_1
  22. Segment(R, P) → b_1
  23. Segment(O, S) → c_1
  24. Segment(S, T) → d_1
  25. Segment(P, T) → e_1
  26. Segment(R, U) → f_1
  27. Segment(V, W) → g_1
  28. Segment(V, Z) → h_1
  29. Segment(W, A_1) → i_1
  30. Segment(W, B_1) → j_1
  31. Segment(B_1, C_1) → k_1
  32. Point(f_1) → D_1
  33. Segment(Z, D_1) → l_1
  34. Segment(U, A_1) → m_1
  35. Segment(T, C_1) → n_1
  36. Polygon(E, D, L, K) → πολύγ1, e_2, d_2, l_2, k_2
  37. Polygon(B, E, D, C) → πολύγ2, b_2, e_3, d_3, c_2
  38. Polygon(B, A, F, E) → πολύγ3, b_3, a_2, f_2, e_4
  39. Polygon(F, J, K, E) → πολύγ4, f_3, j_2, k_3, e_5
  40. Polygon(A, I, G, F) → πολύγ5, a_3, i_2, g_2, f_4
  41. Polygon(M, H, G, I) → πολύγ6, m_2, h_2, g_3, i_3
  42. Polygon(A, N, M, I) → πολύγ7, a_4, n_2, m_3, i_4
  43. Polygon(Q, O, P, R) → πολύγ8, q_1, o, p_1, r_1
  44. Polygon(O, S, T, P) → πολύγ9, o_1, s_1, t_1, p_2
  45. Polygon(R, U, T, P) → πολύγ10, r_2, u_3, t_2, p_3
  46. Polygon(D_1, U, A_1, Z) → πολύγ11, d_4, u_1, a_5, z
  47. Polygon(U, T, C_1, A_1) → πολύγ12, u_2, t_3, c_3, a_6
  48. Polygon(Z, V, W, A_1) → πολύγ13, z_1, v, w, a_7
  49. Polygon(W, B_1, C_1, A_1) → πολύγ14, w_1, b_4, c_4, a_8
  50. Line(B, S) → q_2
  51. Line(E, T) → s_2
  52. Line(K, U) → h_3
  53. Line(F, Z) → j_3
  54. Line(A, A_1) → l_3
  55. Line(N, C_1) → n_3
  56. Vector(E_1, F_1) → u_4
  57. Translate(D, Vector[(k * u_4)]) → D'
  58. Translate(C, Vector[(k * u_4)]) → C'
  59. Segment(D', C') → q_3
  60. Translate(D, Vector[(k * u_4)]) → D'_1
  61. Translate(L, Vector[(k * u_4)]) → L'
  62. Segment(D'_1, L') → r_3
  63. Polygon(C, C', D', D) → πολύγ15, c_5, c', d', d_5
  64. Polygon(D, D', L', L) → πολύγ16, d_6, d'_1, l', l_4
  65. Translate(V, Vector[(((-k)) / 1.4 * u_4)]) → V'
  66. Translate(W, Vector[(((-k)) / 1.4 * u_4)]) → W'
  67. Segment(V', W') → v'
  68. Translate(B_1, Vector[(((-k)) / 1.4 * u_4)]) → G_1
  69. Segment(W', G_1) → s_3
  70. Segment(V', V) → g_4
  71. Segment(G_1, B_1) → h_4
  72. Segment(W', W) → j_4
  73. Polygon(V', V, W, W') → πολύγ17, v'_1, v_1, w_2, w'
  74. Polygon(W', W, B_1, G_1) → πολύγ18, w'_1, w_3, b_5, g_5
  75. Intersect(q_1, c') → Α
  76. Intersect(d'_1, r_1) → Β
  77. Segment(Α, O) → m_4
  78. Segment(Β, R) → n_4
  79. Intersect(s_3, m_2) → Γ
  80. Intersect(v', h_2) → Δ
  81. Segment(M, Γ) → p_4
  82. Segment(Δ, G) → r_4
  83. Vector(Ε, Ζ) → u
  84. Translate(εικ1, Vector[(c * u)]) → εικ1'

AI Drawing Prompt Reference

Draw a 3D cuboid illusion composed of multiple polygonal faces, including a main cuboid and its reflected and mirrored parts. Start by outlining the main cuboid with line segments, then add internal auxiliary lines and intersection points to form several quadrilateral faces such as top, side, and bottom faces. Next, construct a reflected cuboid by translating and mirroring corresponding vertices, and add an extra polygonal face to enhance the 3D effect. The overall composition is symmetric and interlocking, creating a visual illusion of an impossible geometric shape.
The prompt above was compiled from the original drawing commands and is for reference only.