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Folding Cube

Folding Cube
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2019-03-22 18:46
VectorTrigonometrySolid Geometry
This example demonstrates a foldable cube. It involves solid geometry and trigonometry. Using commands like Vector, Polygon, and Angle in GeoGebra, the faces of the cube can be constructed. Suitable for high school or university solid geometry and trigonometry courses.

GeoGebra Commands & Parameters

  1. Vector(O, X) → u
  2. Vector(O, Y) → v
  3. Vector(O, Z) → w
  4. If(0.0 ≤ θ ∧ θ < 3.141592653589793 / 2.0, v * 2.0 * cos(3.141592653589793 / 2.0 - θ) + w * 2.0 * sin(3.141592653589793 / 2.0 - θ) + H, v * 2.0 + H) → D
  5. If(0.0 ≤ θ ∧ θ < 3.141592653589793 / 2.0, v * 2.0 * cos(3.141592653589793 / 2.0 - θ) + w * 2.0 * sin(3.141592653589793 / 2.0 - θ) + G, v * 2.0 + G) → C
  6. If(0.0 ≤ θ ∧ θ < 3.141592653589793 / 2.0, v * 2.0 * cos(3.141592653589793 - θ) + w * 2.0 * sin(3.141592653589793 - θ) + C, w * 2.0 + C) → B
  7. If(0.0 ≤ θ ∧ θ < 3.141592653589793 / 2.0, v * 2.0 * cos(3.141592653589793 - θ) + w * 2.0 * sin(3.141592653589793 - θ) + D, w * 2.0 + D) → A
  8. If(0.0 ≤ θ ∧ θ < 3.141592653589793 / 2.0, v * 2.0 * cos(3.141592653589793 - θ) + w * 2.0 * sin(3.141592653589793 - θ) + G, w * 2.0 + G) → F2
  9. If(0.0 ≤ θ ∧ θ < 3.141592653589793 / 2.0, v * 2.0 * cos(3.141592653589793 - θ) + w * 2.0 * sin(3.141592653589793 - θ) + H, w * 2.0 + H) → E2
  10. If(3.141592653589793 / 2.0 ≤ θ ∧ θ < 3.141592653589793, u * 2.0 * cos(θ) + w * 2.0 * sin(θ) + D, -(u) * 2.0 + D) → A2
  11. If(3.141592653589793 / 2.0 ≤ θ ∧ θ < 3.141592653589793, u * 2.0 * cos(θ) + w * 2.0 * sin(θ) + H, -(u) * 2.0 + H) → E3
  12. If(3.141592653589793 / 2.0 ≤ θ ∧ θ < 3.141592653589793, u * 2.0 * cos(θ - 3.141592653589793 / 2.0) + w * 2.0 * sin(θ - 3.141592653589793 / 2.0) + A2, w * 2.0 + A2) → B2
  13. If(3.141592653589793 / 2.0 ≤ θ ∧ θ < 3.141592653589793, u * 2.0 * cos(θ - 3.141592653589793 / 2.0) + w * 2.0 * sin(θ - 3.141592653589793 / 2.0) + E3, w * 2.0 + E3) → F3
  14. If(3.141592653589793 / 2.0 ≤ θ ∧ θ < 3.141592653589793, u * 2.0 * cos(θ - 3.141592653589793) + w * 2.0 * sin(θ - 3.141592653589793) + B2, u * 2.0 + B2) → C2
  15. If(3.141592653589793 / 2.0 ≤ θ ∧ θ < 3.141592653589793, u * 2.0 * cos(θ - 3.141592653589793) + w * 2.0 * sin(θ - 3.141592653589793) + F3, u * 2.0 + F3) → G2
  16. Segment(G2, H) → h_1
  17. If(3.141592653589793 < θ ∧ θ < 3.0 * 3.141592653589793 / 2.0, v * 2.0 * cos(3.0 * 3.141592653589793 / 2.0 - θ) + w * 2.0 * sin(3.0 * 3.141592653589793 / 2.0 - θ) + D, v * 2.0 + D) → C3
  18. If(3.141592653589793 < θ ∧ θ < 3.0 * 3.141592653589793 / 2.0, v * 2.0 * cos(2.0 * 3.141592653589793 - θ) + w * 2.0 * sin(2.0 * 3.141592653589793 - θ) + C3, w * 2.0 + C3) → G3
  19. If(3.141592653589793 < θ ∧ θ < 3.0 * 3.141592653589793 / 2.0, v * 2.0 * cos(3.0 * 3.141592653589793 / 2.0 - θ) + w * 2.0 * sin(3.0 * 3.141592653589793 / 2.0 - θ) + A2, v * 2.0 + A2) → B3
  20. If(3.141592653589793 < θ ∧ θ < 3.0 * 3.141592653589793 / 2.0, v * 2.0 * cos(2.0 * 3.141592653589793 - θ) + w * 2.0 * sin(2.0 * 3.141592653589793 - θ) + B3, w * 2.0 + B3) → F4
  21. If(3.141592653589793 < θ ∧ θ < 3.0 * 3.141592653589793 / 2.0, v * 2.0 * cos(2.0 * 3.141592653589793 - θ) + w * 2.0 * sin(2.0 * 3.141592653589793 - θ) + D, w * 2.0 + D) → H2
  22. If(3.141592653589793 < θ ∧ θ < 3.0 * 3.141592653589793 / 2.0, v * 2.0 * cos(2.0 * 3.141592653589793 - θ) + w * 2.0 * sin(2.0 * 3.141592653589793 - θ) + A2, w * 2.0 + A2) → E4
  23. Segment(H2, G3) → q_1
  24. If(3.0 * 3.141592653589793 / 2.0 < θ ∧ θ < 2.0 * 3.141592653589793, u * 2.0 * cos(θ - 3.141592653589793) + w * 2.0 * sin(θ - 3.141592653589793) + B3, -(u) * 2.0 + B3) → F5
  25. If(3.0 * 3.141592653589793 / 2.0 < θ ∧ θ < 2.0 * 3.141592653589793, u * 2.0 * cos(θ - 3.141592653589793) + w * 2.0 * sin(θ - 3.141592653589793) + A2, -(u) * 2.0 + A2) → E5
  26. If(3.0 * 3.141592653589793 / 2.0 < θ ∧ θ < 2.0 * 3.141592653589793, u * 2.0 * cos(θ - 3.0 * 3.141592653589793 / 2.0) + w * 2.0 * sin(θ - 3.0 * 3.141592653589793 / 2.0) + F5, w * 2.0 + F5) → G4
  27. If(3.0 * 3.141592653589793 / 2.0 < θ ∧ θ < 2.0 * 3.141592653589793, u * 2.0 * cos(θ - 3.0 * 3.141592653589793 / 2.0) + w * 2.0 * sin(θ - 3.0 * 3.141592653589793 / 2.0) + E5, w * 2.0 + E5) → H3
  28. If(3.0 * 3.141592653589793 / 2.0 < θ ∧ θ < 2.0 * 3.141592653589793, u * 2.0 * cos(θ - 3.0 * 3.141592653589793 / 2.0) + w * 2.0 * sin(θ - 3.0 * 3.141592653589793 / 2.0) + A2, w * 2.0 + A2) → D1
  29. If(3.0 * 3.141592653589793 / 2.0 < θ ∧ θ < 2.0 * 3.141592653589793, u * 2.0 * cos(θ - 3.0 * 3.141592653589793 / 2.0) + w * 2.0 * sin(θ - 3.0 * 3.141592653589793 / 2.0) + B3, w * 2.0 + B3) → C4
  30. Segment(B3, C4) → d_2
  31. Segment(A2, D1) → f_2
  32. Segment(E5, H3) → g_2
  33. Segment(D1, C4) → h_2
  34. Segment(G4, H3) → j_2
  35. Segment(H3, D1) → k_2
  36. If(2.0 * 3.141592653589793 < θ ∧ θ < 5.0 * 3.141592653589793 / 2.0, v * 2.0 * cos(5.0 * 3.141592653589793 / 2.0 - θ) + w * 2.0 * sin(5.0 * 3.141592653589793 / 2.0 - θ) + B3, v * 2.0 + B3) → C5
  37. If(2.0 * 3.141592653589793 < θ ∧ θ < 5.0 * 3.141592653589793 / 2.0, v * 2.0 * cos(5.0 * 3.141592653589793 / 2.0 - θ) + w * 2.0 * sin(5.0 * 3.141592653589793 / 2.0 - θ) + F5, v * 2.0 + F5) → G5
  38. If(2.0 * 3.141592653589793 < θ ∧ θ < 5.0 * 3.141592653589793 / 2.0, v * 2.0 * cos(3.0 * 3.141592653589793 - θ) + w * 2.0 * sin(3.0 * 3.141592653589793 - θ) + C5, w * 2.0 + C5) → D2
  39. If(2.0 * 3.141592653589793 < θ ∧ θ < 5.0 * 3.141592653589793 / 2.0, v * 2.0 * cos(3.0 * 3.141592653589793 - θ) + w * 2.0 * sin(3.0 * 3.141592653589793 - θ) + G5, w * 2.0 + G5) → H4
  40. If(2.0 * 3.141592653589793 < θ ∧ θ < 5.0 * 3.141592653589793 / 2.0, v * 2.0 * cos(3.0 * 3.141592653589793 - θ) + w * 2.0 * sin(3.0 * 3.141592653589793 - θ) + F5, w * 2.0 + F5) → E6
  41. If(2.0 * 3.141592653589793 < θ ∧ θ < 5.0 * 3.141592653589793 / 2.0, v * 2.0 * cos(3.0 * 3.141592653589793 - θ) + w * 2.0 * sin(3.0 * 3.141592653589793 - θ) + B3, w * 2.0 + B3) → A3
  42. Segment(D2, C5) → p_2
  43. Segment(H4, G5) → q_2
  44. Segment(A3, B3) → r_2
  45. Segment(E6, F5) → s_2
  46. Segment(A3, D2) → t_2
  47. Segment(D2, H4) → a_3
  48. Segment(E6, H4) → b_3
  49. Segment(E6, A3) → c_3
  50. Polygon(E, F, G, H) → 多邊形1, e, f, g, h
  51. Polygon(A, B, C, D) → 多邊形2, a, b, c, d
  52. Polygon(A, E2, F2, B) → 多邊形3, a_4, e2, f2, b_4
  53. Polygon(B, C, G, F2) → 多邊形4, b_5, c_4, g_3, f2_1
  54. Polygon(C, D, H, G) → 多邊形5, c_5, d_3, h_3, g_4
  55. Polygon(A, D, H, E2) → 多邊形6, a_5, d_4, h_4, e2_1
  56. Polygon(A2, E3, F3, B2) → 多邊形7, a2, e3, f3, b2
  57. Polygon(B2, F3, G2, C2) → 多邊形8, b2_1, f3_1, g2, c2
  58. Polygon(A2, B2, C2, D) → 多邊形9, a2_1, b2_2, c2_1, d_1
  59. Polygon(A2, E3, H, D) → 多邊形10, a2_2, e3_1, h_5, d_5
  60. Polygon(E3, H, D, A2) → 多邊形11, e3_2, h_6, d_6, a2_3
  61. Polygon(F4, G3, C3, B3) → 多邊形12, f4, g3, c3, b3
  62. Polygon(C3, D, A2, B3) → 多邊形13, c3_1, d_7, a2_4, b3_1
  63. Polygon(F4, E4, A2, B3) → 多邊形14, f4_1, e4, a2_5, b3_2
  64. Polygon(E5, F5, B3, A2) → 多邊形15, e5, f5, b3_3, a2_6
  65. Polygon(F5, G4, C4, B3) → 多邊形16, f5_1, g4, c4, b3_4
  66. Polygon(B3, A2, D, C3) → 多邊形17, b3_5, a2_7, d_8, c3_2
  67. Segment(H2, D) → i
  68. Polygon(E5, A2, B3, F5) → 多邊形18, e5_1, a2_8, b3_6, f5_2
  69. Polygon(G5, C5, B3, F5) → 多邊形19, g5, c5, b3_7, f5_3
  70. Segment(E4, H2) → j
  71. Segment(F, Path1) → k
  72. Segment(Path1, Path2) → l
  73. Segment(Path2, Path3) → m
  74. Segment(Path3, Path4) → n
  75. Segment(Path4, Path5) → p
  76. Segment(Path5, G5) → q
  77. Segment(Path2, Path6) → s
  78. Segment(Path7, Path3) → t
  79. Segment(Path8, Path6) → a_1
  80. Segment(Path4, Path9) → b_1
  81. Segment(Path10, Path7) → c_1
  82. Segment(Path8, Path11) → e_1
  83. Segment(Path5, G4) → f_1
  84. Segment(G3, Path9) → g_1
  85. Segment(Path10, G2) → i_1
  86. Segment(Path11, G) → j_1