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Cube packing with fractional edge lengths

Cube packing with fractional edge lengths
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2018-01-05 00:25
SequenceSolid GeometryProbability & Statistics
This example demonstrates packing cubes with varying fractional edge lengths, involving sequences, probability & statistics, solid geometry, and expected value. It uses commands like RandomElement, Sequence, Random, Surface, and Prism in GeoGebra for construction. Suitable for high school mathematics education or geometric exploration.

GeoGebra Commands & Parameters

  1. RandomElement({3, 5, 7, 9, 11, 13}) → a
  2. Sequence(Plane[x + (0 * z) = k], k, 1, a, 1) → list2
  3. RandomElement({3, 5, 7, 9, 11, 13}) → b
  4. Sequence(Plane[y + (0 * z) = k], k, 1, b, 1) → list1
  5. Random(3, 10) → h
  6. Surface(j, k, h, j, 0, a, k, 0, b) → d
  7. Surface(j, k, 0, j, 0, a, k, 0, b) → A2
  8. Prism(A, B, C, D, A, E) → e, F, faceAAEI, faceEFGHI, edgeAB, edgeBC, edgeCD, edgeAD, edgeAA, edgeAE, edgeBF, edgeCG, G, edgeDH, edgeAI, edgeEF, edgeFG, edgeGH, edgeHI, edgeEI, H, I, faceAADCB, faceABFE, faceBCGF, faceCDHG, faceADHI
  9. Random(0, 10) → c
  10. Random(1, 13) → a_1
  11. Random(1, 13) → b_1
  12. Surface(j, k, C2 / n, j, -0.03, (u * row) + 0.03, k, -0.1, (u * col) + 0.03) → G2
  13. Surface(j, k, C3 / n, j, -0.03, (u * row) + 0.03, k, -0.1, (u * col) + 0.03) → G3
  14. Surface(j, k, C4 / n, j, -0.03, (u * row) + 0.03, k, -0.1, (u * col) + 0.03) → G4
  15. Surface(j, k, C5 / n, j, -0.03, (u * row) + 0.03, k, -0.1, (u * col) + 0.03) → G5
  16. Surface(j, k, C6 / n, j, -0.03, (u * row) + 0.03, k, -0.1, (u * col) + 0.03) → G6
  17. Surface(j, k, C7 / n, j, -0.03, (u * row) + 0.03, k, -0.1, (u * col) + 0.03) → G7
  18. Surface(j, k, C8 / n, j, -0.03, (u * row) + 0.03, k, -0.1, (u * col) + 0.03) → G8
  19. Surface(j, k, C9 / n, j, -0.03, (u * row) + 0.03, k, -0.1, (u * col) + 0.03) → G9
  20. Surface(j, k, C10 / n, j, -0.03, (u * row) + 0.03, k, -0.1, (u * col) + 0.03) → G10
  21. Surface(j, k, C11 / n, j, -0.03, (u * row) + 0.03, k, -0.1, (u * col) + 0.03) → G11
  22. Surface(j, k, C12 / n, j, -0.03, (u * row) + 0.03, k, -0.1, (u * col) + 0.03) → G12
  23. Surface(j, k, C13 / n, j, -0.03, (u * row) + 0.03, k, -0.1, (u * col) + 0.03) → G13
  24. Surface(j, k, C14 / n, j, -0.03, (u * row) + 0.03, k, -0.1, (u * col) + 0.03) → G14
  25. Surface(j, k, C15 / n, j, -0.03, (u * row) + 0.03, k, -0.1, (u * col) + 0.03) → G15
  26. Surface(j, k, C16 / n, j, -0.03, (u * row) + 0.03, k, -0.1, (u * col) + 0.03) → G16
  27. Surface(j, k, C17 / n, j, -0.03, (u * row) + 0.03, k, -0.1, (u * col) + 0.03) → G17
  28. Surface(j, k, C18 / n, j, -0.03, (u * row) + 0.03, k, -0.1, (u * col) + 0.03) → G18
  29. Surface(j, k, C19 / n, j, -0.03, (u * row) + 0.03, k, -0.1, (u * col) + 0.03) → G19
  30. Surface(j, k, C20 / n, j, -0.03, (u * row) + 0.03, k, -0.1, (u * col) + 0.03) → G20
  31. Surface(j, k, C21 / n, j, -0.03, (u * row) + 0.03, k, -0.1, (u * col) + 0.03) → G21
  32. Surface(j, k, C22 / n, j, -0.03, (u * row) + 0.03, k, -0.1, (u * col) + 0.03) → G22
  33. Surface(j, k, C23 / n, j, -0.03, (u * row) + 0.03, k, -0.1, (u * col) + 0.03) → G23
  34. Surface(j, k, C24 / n, j, -0.03, (u * row) + 0.03, k, -0.1, (u * col) + 0.03) → G24
  35. Surface(C1, k, j, j, -0.05, (u * layer) + 0.015, k, -0.05, (u * col) + 0.05) → H1
  36. Surface(C2, k, j, j, -0.0500000000000002, (u * layer) + 0.015, k, -0.0500000000000002, (u * col) + 0.0500000000000002) → H2
  37. Surface(C3, k, j, j, -0.0500000000000002, (u * layer) + 0.015, k, -0.0500000000000002, (u * col) + 0.0500000000000002) → H3
  38. Surface(C4, k, j, j, -0.0500000000000002, (u * layer) + 0.015, k, -0.0500000000000002, (u * col) + 0.0500000000000002) → H4
  39. Surface(C5, k, j, j, -0.0500000000000002, (u * layer) + 0.015, k, -0.0500000000000002, (u * col) + 0.0500000000000002) → H5
  40. Surface(C6, k, j, j, -0.0500000000000002, (u * layer) + 0.015, k, -0.0500000000000002, (u * col) + 0.0500000000000002) → H6
  41. Surface(C7, k, j, j, -0.0500000000000002, (u * layer) + 0.015, k, -0.0500000000000002, (u * col) + 0.0500000000000002) → H7
  42. Surface(C8, k, j, j, -0.0500000000000002, (u * layer) + 0.015, k, -0.0500000000000002, (u * col) + 0.0500000000000002) → H8
  43. Surface(C9, k, j, j, -0.0500000000000002, (u * layer) + 0.015, k, -0.0500000000000002, (u * col) + 0.0500000000000002) → H9
  44. Surface(C10, k, j, j, -0.0500000000000002, (u * layer) + 0.015, k, -0.0500000000000002, (u * col) + 0.0500000000000002) → H10
  45. Surface(C11, k, j, j, -0.0500000000000002, (u * layer) + 0.015, k, -0.0500000000000002, (u * col) + 0.0500000000000002) → H11
  46. Surface(C12, k, j, j, -0.0500000000000002, (u * layer) + 0.015, k, -0.0500000000000002, (u * col) + 0.0500000000000002) → H12
  47. Surface(C13, k, j, j, -0.0500000000000002, (u * layer) + 0.015, k, -0.0500000000000002, (u * col) + 0.0500000000000002) → H13
  48. Surface(C14, k, j, j, -0.0500000000000002, (u * layer) + 0.015, k, -0.0500000000000002, (u * col) + 0.0500000000000002) → H14
  49. Surface(C15, k, j, j, -0.0500000000000002, (u * layer) + 0.015, k, -0.0500000000000002, (u * col) + 0.0500000000000002) → H15
  50. Surface(C16, k, j, j, -0.0500000000000002, (u * layer) + 0.015, k, -0.0500000000000002, (u * col) + 0.0500000000000002) → H16
  51. Surface(C17, k, j, j, -0.0500000000000002, (u * layer) + 0.015, k, -0.0500000000000002, (u * col) + 0.0500000000000002) → H17
  52. Surface(C18, k, j, j, -0.0500000000000002, (u * layer) + 0.015, k, -0.0500000000000002, (u * col) + 0.0500000000000002) → H18
  53. Surface(C19, k, j, j, -0.0500000000000002, (u * layer) + 0.015, k, -0.0500000000000002, (u * col) + 0.0500000000000002) → H19
  54. Surface(C20, k, j, j, -0.0500000000000002, (u * layer) + 0.015, k, -0.0500000000000002, (u * col) + 0.0500000000000002) → H20
  55. Surface(C21, k, j, j, -0.0500000000000002, (u * layer) + 0.015, k, -0.0500000000000002, (u * col) + 0.0500000000000002) → H21
  56. Surface(C22, k, j, j, -0.0500000000000002, (u * layer) + 0.015, k, -0.0500000000000002, (u * col) + 0.0500000000000002) → H22
  57. Surface(C23, k, j, j, -0.0500000000000002, (u * layer) + 0.015, k, -0.0500000000000002, (u * col) + 0.0500000000000002) → H23
  58. Surface(C24, k, j, j, -0.0500000000000002, (u * layer) + 0.015, k, -0.0500000000000002, (u * col) + 0.0500000000000002) → H24
  59. Surface(j, C1 - 0.01, k, j, 0, (u * row) + 0.05, k, 0, (u * layer) + 0.05) → I1
  60. Surface(j, C2 - 0.01, k, j, 0, (u * row) + 0.0500000000000002, k, 0, (u * layer) + 0.0500000000000002) → I2
  61. Surface(j, C3 - 0.01, k, j, 0, (u * row) + 0.0500000000000002, k, 0, (u * layer) + 0.0500000000000002) → I3
  62. Surface(j, C4 - 0.01, k, j, 0, (u * row) + 0.0500000000000002, k, 0, (u * layer) + 0.0500000000000002) → I4
  63. Surface(j, C5 - 0.01, k, j, 0, (u * row) + 0.0500000000000002, k, 0, (u * layer) + 0.0500000000000002) → I5
  64. Surface(j, C6 - 0.01, k, j, 0, (u * row) + 0.0500000000000002, k, 0, (u * layer) + 0.0500000000000002) → I6
  65. Surface(j, C7 - 0.01, k, j, 0, (u * row) + 0.0500000000000002, k, 0, (u * layer) + 0.0500000000000002) → I7
  66. Surface(j, C8 - 0.01, k, j, 0, (u * row) + 0.0500000000000002, k, 0, (u * layer) + 0.0500000000000002) → I8
  67. Surface(j, C9 - 0.01, k, j, 0, (u * row) + 0.0500000000000002, k, 0, (u * layer) + 0.0500000000000002) → I9
  68. Surface(j, C10 - 0.01, k, j, 0, (u * row) + 0.0500000000000002, k, 0, (u * layer) + 0.0500000000000002) → I10
  69. Surface(j, C11 - 0.01, k, j, 0, (u * row) + 0.0500000000000002, k, 0, (u * layer) + 0.0500000000000002) → I11
  70. Surface(j, C12 - 0.01, k, j, 0, (u * row) + 0.0500000000000002, k, 0, (u * layer) + 0.0500000000000002) → I12
  71. Surface(j, C13 - 0.01, k, j, 0, (u * row) + 0.0500000000000002, k, 0, (u * layer) + 0.0500000000000002) → I13
  72. Surface(j, C14 - 0.01, k, j, 0, (u * row) + 0.0500000000000002, k, 0, (u * layer) + 0.0500000000000002) → I14
  73. Surface(j, C15 - 0.01, k, j, 0, (u * row) + 0.0500000000000002, k, 0, (u * layer) + 0.0500000000000002) → I15
  74. Surface(j, C16 - 0.01, k, j, 0, (u * row) + 0.0500000000000002, k, 0, (u * layer) + 0.0500000000000002) → I16
  75. Surface(j, C17 - 0.01, k, j, 0, (u * row) + 0.0500000000000002, k, 0, (u * layer) + 0.0500000000000002) → I17
  76. Surface(j, C18 - 0.01, k, j, 0, (u * row) + 0.0500000000000002, k, 0, (u * layer) + 0.0500000000000002) → I18
  77. Surface(j, C19 - 0.01, k, j, 0, (u * row) + 0.0500000000000002, k, 0, (u * layer) + 0.0500000000000002) → I19
  78. Surface(j, C20 - 0.01, k, j, 0, (u * row) + 0.0500000000000002, k, 0, (u * layer) + 0.0500000000000002) → I20
  79. Surface(j, C21 - 0.01, k, j, 0, (u * row) + 0.0500000000000002, k, 0, (u * layer) + 0.0500000000000002) → I21
  80. Surface(j, C22 - 0.01, k, j, 0, (u * row) + 0.0500000000000002, k, 0, (u * layer) + 0.0500000000000002) → I22
  81. Surface(j, C23 - 0.01, k, j, 0, (u * row) + 0.0500000000000002, k, 0, (u * layer) + 0.0500000000000002) → I23
  82. Surface(j, C24 - 0.01, k, j, 0, (u * row) + 0.0500000000000002, k, 0, (u * layer) + 0.0500000000000002) → I24
  83. Surface(j, k, (u * C2) - 0.05, j, -0.1, (u * row) + 0.1, k, -0.1, (u * col) + 0.1) → D2
  84. Surface(j, k, (u * C3) - 0.0500000000000002, j, -0.1, (u * row) + 0.1, k, -0.1, (u * col) + 0.1) → D3
  85. Surface(j, k, (u * C4) - 0.0500000000000002, j, -0.1, (u * row) + 0.1, k, -0.1, (u * col) + 0.1) → D4
  86. Surface(j, k, (u * C5) - 0.0500000000000002, j, -0.1, (u * row) + 0.1, k, -0.1, (u * col) + 0.1) → D5
  87. Surface(j, k, (u * C6) - 0.0500000000000002, j, -0.1, (u * row) + 0.1, k, -0.1, (u * col) + 0.1) → D6
  88. Surface(j, k, (u * C7) - 0.0500000000000002, j, -0.1, (u * row) + 0.1, k, -0.1, (u * col) + 0.1) → D7
  89. Surface(j, k, (u * C8) - 0.0500000000000002, j, -0.1, (u * row) + 0.1, k, -0.1, (u * col) + 0.1) → D8
  90. Surface(j, k, (u * C9) - 0.0500000000000002, j, -0.1, (u * row) + 0.1, k, -0.1, (u * col) + 0.1) → D9
  91. Surface(j, k, (u * C10) - 0.0500000000000002, j, -0.1, (u * row) + 0.1, k, -0.1, (u * col) + 0.1) → D10
  92. Surface(j, k, (u * C11) - 0.0500000000000002, j, -0.1, (u * row) + 0.1, k, -0.1, (u * col) + 0.1) → D11
  93. Surface(j, k, (u * C12) - 0.0500000000000002, j, -0.1, (u * row) + 0.1, k, -0.1, (u * col) + 0.1) → D12
  94. Surface(j, k, (u * C13) - 0.0500000000000002, j, -0.1, (u * row) + 0.1, k, -0.1, (u * col) + 0.1) → D13
  95. Surface(j, k, (u * C14) - 0.0500000000000002, j, -0.1, (u * row) + 0.1, k, -0.1, (u * col) + 0.1) → D14
  96. Surface(j, k, (u * C15) - 0.0500000000000002, j, -0.1, (u * row) + 0.1, k, -0.1, (u * col) + 0.1) → D15
  97. Surface(j, k, (u * C16) - 0.0500000000000002, j, -0.1, (u * row) + 0.1, k, -0.1, (u * col) + 0.1) → D16
  98. Surface(j, k, (u * C17) - 0.0500000000000002, j, -0.1, (u * row) + 0.1, k, -0.1, (u * col) + 0.1) → D17
  99. Surface(j, k, (u * C18) - 0.0500000000000002, j, -0.1, (u * row) + 0.1, k, -0.1, (u * col) + 0.1) → D18
  100. Surface(j, k, (u * C19) - 0.0500000000000002, j, -0.1, (u * row) + 0.1, k, -0.1, (u * col) + 0.1) → D19
  101. Surface(j, k, (u * C20) - 0.0500000000000002, j, -0.1, (u * row) + 0.1, k, -0.1, (u * col) + 0.1) → D20
  102. Surface(j, k, (u * C21) - 0.0500000000000002, j, -0.1, (u * row) + 0.1, k, -0.1, (u * col) + 0.1) → D21
  103. Surface(j, k, (u * C22) - 0.0500000000000002, j, -0.1, (u * row) + 0.1, k, -0.1, (u * col) + 0.1) → D22
  104. Surface(j, k, (u * C23) - 0.0500000000000002, j, -0.1, (u * row) + 0.1, k, -0.1, (u * col) + 0.1) → D23
  105. Surface(j, k, (u * C24) - 0.0500000000000002, j, -0.1, (u * row) + 0.1, k, -0.1, (u * col) + 0.1) → D24
  106. Surface((u * (C1 + 1)) + 0.05, k, j, j, -0.1, (u * layer) + 0.015, k, -0.1, (u * col) + 0.1) → E2
  107. Surface((u * (C2 + 1)) + 0.0500000000000002, k, j, j, -0.1, (u * layer) + 0.015, k, -0.1, (u * col) + 0.1) → E3
  108. Surface((u * (C3 + 1)) + 0.0500000000000002, k, j, j, -0.1, (u * layer) + 0.015, k, -0.1, (u * col) + 0.1) → E4
  109. Surface((u * (C4 + 1)) + 0.0500000000000002, k, j, j, -0.1, (u * layer) + 0.015, k, -0.1, (u * col) + 0.1) → E5
  110. Surface((u * (C5 + 1)) + 0.0500000000000002, k, j, j, -0.1, (u * layer) + 0.015, k, -0.1, (u * col) + 0.1) → E6
  111. Surface((u * (C6 + 1)) + 0.0500000000000002, k, j, j, -0.1, (u * layer) + 0.015, k, -0.1, (u * col) + 0.1) → E7
  112. Surface((u * (C7 + 1)) + 0.0500000000000002, k, j, j, -0.1, (u * layer) + 0.015, k, -0.1, (u * col) + 0.1) → E8
  113. Surface((u * (C8 + 1)) + 0.0500000000000002, k, j, j, -0.1, (u * layer) + 0.015, k, -0.1, (u * col) + 0.1) → E9
  114. Surface((u * (C9 + 1)) + 0.0500000000000002, k, j, j, -0.1, (u * layer) + 0.015, k, -0.1, (u * col) + 0.1) → E10
  115. Surface((u * (C10 + 1)) + 0.0500000000000002, k, j, j, -0.1, (u * layer) + 0.015, k, -0.1, (u * col) + 0.1) → E11
  116. Surface((u * (C11 + 1)) + 0.0500000000000002, k, j, j, -0.1, (u * layer) + 0.015, k, -0.1, (u * col) + 0.1) → E12
  117. Surface((u * (C12 + 1)) + 0.0500000000000002, k, j, j, -0.1, (u * layer) + 0.015, k, -0.1, (u * col) + 0.1) → E13
  118. Surface((u * (C13 + 1)) + 0.0500000000000002, k, j, j, -0.1, (u * layer) + 0.015, k, -0.1, (u * col) + 0.1) → E14
  119. Surface((u * (C14 + 1)) + 0.0500000000000002, k, j, j, -0.1, (u * layer) + 0.015, k, -0.1, (u * col) + 0.1) → E15
  120. Surface((u * (C15 + 1)) + 0.0500000000000002, k, j, j, -0.1, (u * layer) + 0.015, k, -0.1, (u * col) + 0.1) → E16
  121. Surface((u * (C16 + 1)) + 0.0500000000000002, k, j, j, -0.1, (u * layer) + 0.015, k, -0.1, (u * col) + 0.1) → E17
  122. Surface((u * (C17 + 1)) + 0.0500000000000002, k, j, j, -0.1, (u * layer) + 0.015, k, -0.1, (u * col) + 0.1) → E18
  123. Surface((u * (C18 + 1)) + 0.0500000000000002, k, j, j, -0.1, (u * layer) + 0.015, k, -0.1, (u * col) + 0.1) → E19
  124. Surface((u * (C19 + 1)) + 0.0500000000000002, k, j, j, -0.1, (u * layer) + 0.015, k, -0.1, (u * col) + 0.1) → E20
  125. Surface((u * (C20 + 1)) + 0.0500000000000002, k, j, j, -0.1, (u * layer) + 0.015, k, -0.1, (u * col) + 0.1) → E21
  126. Surface((u * (C21 + 1)) + 0.0500000000000002, k, j, j, -0.1, (u * layer) + 0.015, k, -0.1, (u * col) + 0.1) → E22
  127. Surface((u * (C22 + 1)) + 0.0500000000000002, k, j, j, -0.1, (u * layer) + 0.015, k, -0.1, (u * col) + 0.1) → E23
  128. Surface((u * (C23 + 1)) + 0.0500000000000002, k, j, j, -0.1, (u * layer) + 0.015, k, -0.1, (u * col) + 0.1) → E24
  129. Surface(j, (u * C1), k, j, 0, (u * row) + 0.1, k, 0, (u * layer) + 0.1) → F2
  130. Surface(j, (u * C2), k, j, 0, (u * row) + 0.1, k, 0, (u * layer) + 0.1) → F3
  131. Surface(j, (u * C3), k, j, 0, (u * row) + 0.1, k, 0, (u * layer) + 0.1) → F4
  132. Surface(j, (u * C4), k, j, 0, (u * row) + 0.1, k, 0, (u * layer) + 0.1) → F5
  133. Surface(j, (u * C5), k, j, 0, (u * row) + 0.1, k, 0, (u * layer) + 0.1) → F6
  134. Surface(j, (u * C6), k, j, 0, (u * row) + 0.1, k, 0, (u * layer) + 0.1) → F7
  135. Surface(j, (u * C7), k, j, 0, (u * row) + 0.1, k, 0, (u * layer) + 0.1) → F8
  136. Surface(j, (u * C8), k, j, 0, (u * row) + 0.1, k, 0, (u * layer) + 0.1) → F9
  137. Surface(j, (u * C9), k, j, 0, (u * row) + 0.1, k, 0, (u * layer) + 0.1) → F10
  138. Surface(j, (u * C10), k, j, 0, (u * row) + 0.1, k, 0, (u * layer) + 0.1) → F11
  139. Surface(j, (u * C11), k, j, 0, (u * row) + 0.1, k, 0, (u * layer) + 0.1) → F12
  140. Surface(j, (u * C12), k, j, 0, (u * row) + 0.1, k, 0, (u * layer) + 0.1) → F13
  141. Surface(j, (u * C13), k, j, 0, (u * row) + 0.1, k, 0, (u * layer) + 0.1) → F14
  142. Surface(j, (u * C14), k, j, 0, (u * row) + 0.1, k, 0, (u * layer) + 0.1) → F15
  143. Surface(j, (u * C15), k, j, 0, (u * row) + 0.1, k, 0, (u * layer) + 0.1) → F16
  144. Surface(j, (u * C16), k, j, 0, (u * row) + 0.1, k, 0, (u * layer) + 0.1) → F17
  145. Surface(j, (u * C17), k, j, 0, (u * row) + 0.1, k, 0, (u * layer) + 0.1) → F18
  146. Surface(j, (u * C18), k, j, 0, (u * row) + 0.1, k, 0, (u * layer) + 0.1) → F19
  147. Surface(j, (u * C19), k, j, 0, (u * row) + 0.1, k, 0, (u * layer) + 0.1) → F20
  148. Surface(j, (u * C20), k, j, 0, (u * row) + 0.1, k, 0, (u * layer) + 0.1) → F21
  149. Surface(j, (u * C21), k, j, 0, (u * row) + 0.1, k, 0, (u * layer) + 0.1) → F22
  150. Surface(j, (u * C22), k, j, 0, (u * row) + 0.1, k, 0, (u * layer) + 0.1) → F23
  151. Surface(j, (u * C23), k, j, 0, (u * row) + 0.1, k, 0, (u * layer) + 0.1) → F24

AI Drawing Prompt Reference

Create a 3D geometric scene showing randomly stacked cubes with fractional edge lengths. Start with a rectangular base grid divided by a random number of parallel planes in both horizontal directions, where the counts are chosen from odd numbers. Above this grid, place a horizontal top surface at a random height to form a large cuboid frame. Then, within each grid cell, generate multiple small cubes with random fractional heights, stacking them vertically in layers to create an uneven columnar structure. The overall effect is a probabilistic solid packing, suitable for exploring geometry and expected value in education.
The prompt above was compiled from the original drawing commands and is for reference only.