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Two Parts of a Cube (3) – 3D Division and Geometric Construction

Two Parts of a Cube (3) – 3D Division and Geometric Construction
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2019-04-28 02:58
VectorSolid GeometryTransformationOptimizationGeometric Construction
In collections TWO PARTS OF A CUBE
This example demonstrates the division of a cube into two parts in three-dimensional space, highlighting the relationships between cutting planes, edges, and faces. Core mathematical concepts include solid geometry properties of cubes, midpoints, vectors, translations, rotations, and prism construction. Construction details: using GeoGebra commands Point, Cube, Midpoint, Line, Vector, Translate, Rotate, and Prism, with objects such as vertices A-H, faces like faceABCD, edges like edgeAD, and auxiliary points L, M, N. The division is determined via midpoints and vectors, then the two parts are formed as prisms through rotation and translation. Suitable for high school or university solid geometry instruction, as well as spatial reasoning and geometric construction practice.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Point(xAxis) → B
  3. Point(Circle[B, Distance[A, B], Segment[A, B]]) → C
  4. Cube(A, B, C) → c, D, faceCDHG, faceEFGH, edgeAD, edgeAB, edgeBC, edgeCD, edgeAE, edgeDH, edgeEH, edgeEF, E, edgeBF, edgeFG, edgeCG, edgeGH, F, G, H, faceABCD, faceADHE, faceABFE, faceBCGF
  5. Point(xAxis) → L
  6. Point(xAxis) → M
  7. Point(Circle[M, Distance[L, M], Segment[L, M]]) → N
  8. Cube(L, M, N) → k, O, faceNOSR, facePQRS, edgeLO, edgeLM, edgeMN, edgeNO, edgeLP, edgeOS, edgePS, edgePQ, P, edgeMQ, edgeQR, edgeNR, edgeRS, Q, R, S, faceLMNO, faceLOSP, faceLMQP, faceMNRQ
  9. Midpoint(H, B) → S_2
  10. Midpoint(S, M) → S_1
  11. Line(S_1, S_2) → p_1
  12. Line(S_2, yAxis) → f
  13. Line(S_1, yAxis) → g
  14. Line(S_2, zAxis) → h
  15. Line(S_1, zAxis) → i
  16. Point(p_1) → I
  17. Point(p_1) → J
  18. Vector(I, J) → u
  19. Translate(i, Vector[(p * u)]) → i'
  20. Translate(g, Vector[(p * u)]) → g'
  21. Rotate(A, α, p_1) → A'
  22. Rotate(B, α, p_1) → B'
  23. Rotate(C, α, p_1) → C'
  24. Rotate(D, α, p_1) → D'
  25. Rotate(E, α, p_1) → E'
  26. Rotate(F, α, p_1) → F'
  27. Rotate(G, α, p_1) → G'
  28. Rotate(H, α, p_1) → H'
  29. Rotate(A', β, f) → A''
  30. Rotate(B', β, f) → B''
  31. Rotate(C', β, f) → C''
  32. Rotate(D', β, f) → D''
  33. Rotate(E', β, f) → E''
  34. Rotate(F', β, f) → F''
  35. Rotate(G', β, f) → G''
  36. Rotate(H', β, f) → H''
  37. Translate(k, Vector[(p * u)]) → k'
  38. Translate(L, Vector[(p * u)]) → L'
  39. Translate(M, Vector[(p * u)]) → M'
  40. Translate(N, Vector[(p * u)]) → N'
  41. Translate(O, Vector[(p * u)]) → O'
  42. Translate(P, Vector[(p * u)]) → P'
  43. Translate(Q, Vector[(p * u)]) → Q'
  44. Translate(R, Vector[(p * u)]) → R'
  45. Translate(S, Vector[(p * u)]) → S'
  46. Rotate(c, α, p_1) → c'
  47. Rotate(c', β, f) → c''
  48. Rotate(G'', γ, h) → G'''
  49. Rotate(H'', γ, h) → H'''
  50. Rotate(E'', γ, h) → E'''
  51. Rotate(F'', γ, h) → F'''
  52. Rotate(D'', γ, h) → D'''
  53. Rotate(C'', γ, h) → C'''
  54. Rotate(A'', γ, h) → A'''
  55. Rotate(B'', γ, h) → B'''
  56. Rotate(c'', γ, h) → c'''
  57. Rotate(k', δ, p_1) → k''
  58. Rotate(L', δ, p_1) → L''
  59. Rotate(M', δ, p_1) → M''
  60. Rotate(N', δ, p_1) → N''
  61. Rotate(O', δ, p_1) → O''
  62. Rotate(P', δ, p_1) → P''
  63. Rotate(Q', δ, p_1) → Q''
  64. Rotate(R', δ, p_1) → R''
  65. Rotate(S', δ, p_1) → S''
  66. Rotate(k'', ε, g') → k'''
  67. Rotate(L'', ε, g') → L'''
  68. Rotate(M'', ε, g') → M'''
  69. Rotate(N'', ε, g') → N'''
  70. Rotate(O'', ε, g') → O'''
  71. Rotate(P'', ε, g') → P'''
  72. Rotate(Q'', ε, g') → Q'''
  73. Rotate(R'', ε, g') → R'''
  74. Rotate(S'', ε, g') → S'''
  75. Rotate(k''', ζ, i') → a
  76. Rotate(L''', ζ, i') → K
  77. Rotate(M''', ζ, i') → T
  78. Rotate(N''', ζ, i') → U
  79. Rotate(O''', ζ, i') → V
  80. Rotate(P''', ζ, i') → W
  81. Rotate(Q''', ζ, i') → Z
  82. Rotate(R''', ζ, i') → A_1
  83. Rotate(S''', ζ, i') → B_1
  84. Midpoint(H''', G''') → C_1
  85. Midpoint(E''', F''') → D_1
  86. Midpoint(D''', C''') → E_1
  87. Midpoint(A''', B''') → F_1
  88. Midpoint(A''', D''') → G_1
  89. Midpoint(E''', H''') → H_1
  90. Midpoint(F''', G''') → I_1
  91. Midpoint(B''', C''') → J_1
  92. Midpoint(C_1, D_1) → K_1
  93. Midpoint(C_1, E_1) → L_1
  94. Midpoint(G_1, H_1) → M_1
  95. Midpoint(D_1, F_1) → N_1
  96. Midpoint(J_1, I_1) → O_1
  97. Midpoint(G_1, J_1) → P_1
  98. Midpoint(V, T) → Q_1
  99. Midpoint(B_1, Z) → R_1
  100. Midpoint(A_1, T) → T_1
  101. Midpoint(B_1, U) → U_1
  102. Midpoint(B_1, K) → V_1
  103. Midpoint(Z, K) → W_1
  104. Midpoint(B_1, A_1) → Z_1
  105. Midpoint(A_1, Z) → A_2
  106. Midpoint(Z, W) → B_2
  107. Midpoint(V, U) → C_2
  108. Midpoint(K, T) → D_2
  109. Midpoint(T, U) → E_2
  110. Midpoint(V, K) → F_2
  111. Midpoint(B_1, W) → G_2
  112. Midpoint(Z, T) → H_2
  113. Midpoint(K, W) → I_2
  114. Midpoint(B_1, V) → J_2
  115. Midpoint(U, A_1) → K_2
  116. Midpoint(G''', C''') → L_2
  117. Midpoint(D''', H''') → M_2
  118. Midpoint(A''', E''') → N_2
  119. Midpoint(F''', B''') → O_2
  120. Prism(D''', A''', B''', M_2) → b, P_2, edgeB'''D''', edge1, edge2, edge3, edge4, edge6, edge5, Q_2, faceA'''B'''D''', face2, face4, face3, face1, edgeA'''D''', edgeA'''B'''
  121. Prism(M_2, L_2, P_2, H''') → d, R_2, edge13, edge7, edge12, edge14, edge8, edge15, edge9, T_2, face6, face7, face9, face8, face5, edge10, edge11
  122. Prism(V, T, U, J_2) → e, U_2, edgeUV, edge18, edge16, edge17, edge19, edge21, edge20, V_2, faceTUV, face12, face11, face13, face10, edgeTV, edgeTU
  123. Point(edge21) → W_2
  124. Prism(K, Q_1, T, I_2) → j, Z_2, edgeKT, edge23, edge26, edge25, edge27, edge29, edge28, A_3, face14, face17, face18, face16, face15, edge22, edge24
  125. Prism(I_2, V_2, A_3, W) → l, B_3, edge34, edge30, edge36, edge37, edge31, edge38, edge32, C_3, face20, face21, face23, face22, face19, edge33, edge35
  126. Prism(W, V_1, B_1, Z) → n, D_3, edge39, edgeWZ, edge45, edge44, edge41, edge46, edge42, E_3, face24, face26, face28, face27, face25, edge40, edge43
  127. Prism(V, B_1, V_1, C_2) → o, F_3, edge48, edge49, edge51, edge52, edge53, edge55, edge54, G_3, face29, face31, face33, face32, face30, edge47, edge50
  128. Prism(W, V_1, K, B_2) → q, H_3, edgeKW, edge59, edge60, edge57, edge61, edge63, edge62, I_3, face34, face38, face37, face36, face35, edge58, edge56
  129. Prism(K, B_1, V, D_2) → s, J_3, edgeKV, edge65, edge68, edge67, edge69, edge71, edge70, K_3, face39, face42, face43, face41, face40, edge64, edge66
  130. Prism(B_1, V_1, W, A_1) → t, L_3, edge72, edge75, edge79, edge74, edge76, edge80, edge77, M_3, face44, face48, face47, face46, face45, edge78, edge73
  131. Prism(W, V, B_1, B_2) → w, N_3, edge83, edge84, edge82, edge85, edge86, edge88, edge87, O_3, face49, face51, face52, face53, face50, edgeVW, edge81
  132. Prism(O_3, N_3, D_2, A_1) → a_1, P_3, edge93, edge89, edge96, edge94, edge90, edge97, edge91, Q_3, face55, face57, face58, face56, face54, edge95, edge92

AI Drawing Prompt Reference

Visualize the division of a cube into two parts in 3D space, highlighting the relationships between cutting planes, edges, and faces. Start by constructing two cubes, then find midpoints of key vertices and draw lines connecting them. Use vectors and midpoints to determine the cutting direction, and split the cube into two parts through rotation and translation, forming new prisms. Finally, display the two resulting parts and their relationship with the original cube, emphasizing geometric construction and spatial geometry properties.
The prompt above was compiled from the original drawing commands and is for reference only.