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11 Nets of a Cube by Gunther Van den Berghe

11 Nets of a Cube by Gunther Van den Berghe
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2016-10-12 22:40
PolygonSolid GeometryGeometric Construction
This example demonstrates the 11 different nets of a cube. It involves core mathematical concepts such as solid geometry, polygonal combinations, and spatial visualization. Construction highlights include using GeoGebra commands like Cube, Rotate, Midpoint, OrthogonalLine, Segment, and Translate to create the cube and its nets. Suitable for high school geometry teaching or geometric construction learning.

GeoGebra Commands & Parameters

  1. Cube(A, B, xOyPlane) → a, C, faceBCGF, faceCDHG, faceEFGH, edgeAD, edgeAB, edgeBC, edgeCD, edgeAE, edgeDH, edgeEH, D, edgeEF, edgeBF, edgeFG, edgeCG, edgeGH, E, F, G, H, faceABCD, faceADHE, faceABFE
  2. Rotate(faceABFE, (90° * t), edgeAB) → faceABFE'
  3. Midpoint(A, C) → I
  4. OrthogonalLine(I, xOyPlane) → b
  5. Rotate(faceABFE', 90°, b) → faceABFE''
  6. Rotate(faceABFE', 180°, b) → faceABFE''_1
  7. Rotate(faceABFE', 270°, b) → faceABFE''_2
  8. Rotate(E, (90° * t), edgeAB) → E'
  9. Rotate(F, (90° * t), edgeAB) → F'
  10. Segment(E', F') → c
  11. Rotate(Rotate[faceEFGH, (90° * t), edgeAB], (90° * t), Segment[E', F']) → poly1
  12. Translate(J, Vector[A, B]) → K
  13. Translate(faceABFE''_1, Vector[A, J]) → poly2
  14. Translate(faceABCD, Vector[A, J]) → faceABCD'
  15. Translate(faceABFE', Vector[A, J]) → faceABFE''_3
  16. Translate(poly1, Vector[A, J]) → poly1'
  17. Translate(Rotate[Rotate[faceADHE, (90° * t), edgeAB], (-90° * t), Segment[Rotate[E, (90° * t), edgeAB], Rotate[A, (90° * t), edgeAB]]], Vector[A, J]) → poly4
  18. Translate(faceABFE'', Vector[A, J]) → faceABFE'''
  19. Translate(faceABCD, Vector[A, L]) → faceABCD'_1
  20. Translate(faceABFE''_1, Vector[A, L]) → poly3
  21. Translate(faceABFE', Vector[A, L]) → faceABFE''_4
  22. Translate(poly1, Vector[A, L]) → poly1'_1
  23. Translate(faceABFE'', Vector[A, L]) → faceABFE'''_1
  24. Midpoint(B, E') → M
  25. OrthogonalLine(M, faceABFE') → d
  26. Translate(Rotate[faceABFE'', 180°, d], Vector[A, L]) → poly5
  27. Translate(faceABCD, Vector[A, O]) → faceABCD'_2
  28. Translate(faceABFE''_1, Vector[A, O]) → poly6
  29. Translate(faceABFE', Vector[A, O]) → faceABFE''_5
  30. Translate(poly1, Vector[A, O]) → poly1'_2
  31. Translate(faceABFE'', Vector[A, O]) → faceABFE'''_2
  32. Translate(Rotate[Rotate[faceADHE, (90° * t), edgeCD], (90° * t), Segment[Rotate[H, (90° * t), edgeCD], Rotate[D, (90° * t), edgeCD]]], Vector[A, O]) → poly8
  33. Translate(faceABCD, Vector[A, N]) → faceABCD'_3
  34. Translate(faceABFE''_1, Vector[A, N]) → poly7
  35. Translate(faceABFE', Vector[A, N]) → faceABFE''_6
  36. Translate(poly1, Vector[A, N]) → poly1'_3
  37. Translate(Rotate[faceABFE'', 180°, d], Vector[A, N]) → poly9
  38. Rotate(Rotate[H, (90° * t), edgeAB], (90° * t), Segment[E', F']) → P
  39. Midpoint(P, F') → Q
  40. OrthogonalLine(Q, poly1) → e
  41. Translate(Rotate[Rotate[faceABFE'', 180°, d], 180°, e], Vector[A, N]) → poly10
  42. Translate(faceABCD, Vector[A, R]) → faceABCD'_4
  43. Translate(faceABFE''_1, Vector[A, R]) → poly11
  44. Translate(faceABFE', Vector[A, R]) → faceABFE''_7
  45. Translate(poly1, Vector[A, R]) → poly1'_4
  46. Translate(Rotate[faceABFE'', 180°, d], Vector[A, R]) → poly12
  47. Rotate(F', 180°, b) → F''
  48. Midpoint(C, F'') → S
  49. OrthogonalLine(S, faceABFE''_1) → g
  50. Translate(Rotate[Rotate[Rotate[faceADHE, (90° * t), edgeCD], (90° * t), Segment[Rotate[H, (90° * t), edgeCD], Rotate[D, (90° * t), edgeCD]]], 180°, g], Vector[A, R]) → poly15
  51. Translate(faceABCD, Vector[A, T]) → faceABCD'_5
  52. Translate(faceABFE''_1, Vector[A, T]) → poly13
  53. Translate(faceABFE', Vector[A, T]) → faceABFE''_8
  54. Translate(Rotate[faceABFE', 270°, b], Vector[A, T]) → poly14
  55. Translate(Rotate[poly1, -90°, b], Vector[A, T]) → poly16
  56. Translate(Rotate[Rotate[Rotate[faceADHE, (90° * t), edgeAB], (-90° * t), Segment[Rotate[E, (90° * t), edgeAB], Rotate[A, (90° * t), edgeAB]]], 180°, d], Vector[A, T]) → poly17
  57. Translate(faceABCD, Vector[A, U]) → faceABCD'_6
  58. Translate(faceABFE''_1, Vector[A, U]) → poly18
  59. Translate(faceABFE', Vector[A, U]) → faceABFE''_9
  60. Translate(Rotate[faceABFE', 270°, b], Vector[A, U]) → poly19
  61. Translate(Rotate[Rotate[Rotate[faceADHE, (90° * t), edgeAB], (-90° * t), Segment[Rotate[E, (90° * t), edgeAB], Rotate[A, (90° * t), edgeAB]]], 180°, d], Vector[A, U]) → poly20
  62. Translate(Rotate[faceABFE'', 270°, d], Vector[A, U]) → poly22
  63. Translate(faceABCD, Vector[A, V]) → faceABCD'_7
  64. Translate(faceABFE''_1, Vector[A, V]) → poly21
  65. Translate(Rotate[faceABFE, (90° * t), edgeAB], Vector[A, V]) → poly23
  66. Translate(Rotate[Rotate[faceADHE, (90° * t), edgeCD], (90° * t), Segment[Rotate[H, (90° * t), edgeCD], Rotate[D, (90° * t), edgeCD]]], Vector[A, V]) → poly24
  67. Rotate(Rotate[faceADHE, (90° * t), edgeCD], (90° * t), Segment[Rotate[H, (90° * t), edgeCD], Rotate[D, (90° * t), edgeCD]]) → poly25
  68. Rotate(Rotate[E, (90° * t), edgeCD], (90° * t), Segment[Rotate[H, (90° * t), edgeCD], Rotate[D, (90° * t), edgeCD]]) → W
  69. Midpoint(D, W) → Z
  70. OrthogonalLine(Z, poly25) → h
  71. Rotate(faceABFE''_1, 90°, h) → poly26
  72. Translate(Rotate[faceABFE''_1, 90°, h], Vector[A, V]) → poly27
  73. Rotate(C, 90°, h) → C'
  74. Midpoint(W, C') → A_1
  75. OrthogonalLine(A_1, poly26) → i
  76. Rotate(poly25, 180°, i) → poly25'
  77. Translate(Rotate[poly25, 180°, i], Vector[A, V]) → poly28
  78. Translate(faceABCD, Vector[A, B_1]) → faceABCD'_8
  79. Translate(faceABFE''_1, Vector[A, B_1]) → poly29
  80. Translate(faceABFE', Vector[A, B_1]) → faceABFE''_{10}
  81. Translate(Rotate[Rotate[Rotate[faceADHE, (90° * t), edgeAB], (-90° * t), Segment[Rotate[E, (90° * t), edgeAB], Rotate[A, (90° * t), edgeAB]]], 180°, d], Vector[A, B_1]) → poly30
  82. Translate(Rotate[Rotate[faceADHE, (90° * t), edgeCD], (90° * t), Segment[Rotate[H, (90° * t), edgeCD], Rotate[D, (90° * t), edgeCD]]], Vector[A, B_1]) → poly31
  83. Translate(Rotate[faceABFE''_1, 90°, h], Vector[A, B_1]) → poly32
  84. Translate(faceABCD, Vector[A, C_1]) → faceABCD'_9
  85. Translate(faceABFE''_1, Vector[A, C_1]) → poly33
  86. Translate(Rotate[Rotate[faceADHE, (90° * t), edgeCD], (90° * t), Segment[Rotate[H, (90° * t), edgeCD], Rotate[D, (90° * t), edgeCD]]], Vector[A, C_1]) → poly34
  87. Translate(Rotate[faceABFE''_1, 90°, h], Vector[A, C_1]) → poly35
  88. Translate(faceABFE'', Vector[A, C_1]) → faceABFE'''_3
  89. Rotate(G, 90°, b) → G'
  90. Rotate(F', 90°, b) → F''_1
  91. Midpoint(B, F''_1) → D_1
  92. OrthogonalLine(D_1, faceABFE'') → j
  93. Rotate(faceABCD, 90°, j) → faceABCD'_{10}
  94. Translate(Rotate[faceABCD, 90°, j], Vector[A, C_1]) → poly36

AI Drawing Prompt Reference

Draw the 11 nets of a cube. Start by creating a cube, then rotate face ABFE around edge AB to generate face ABFE'. Find the midpoint I of ABFE' and draw a perpendicular line b to the xOy plane through I. Rotate face ABFE' around b to generate face ABFE'', and then generate other nets by rotating and translating. For example, rotate face ABFE'' around b to generate face ABFE''_1, and then rotate to generate face ABFE''_2. Finally, generate the remaining nets through rotation and translation operations.
The prompt above was compiled from the original drawing commands and is for reference only.