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5x5x5涂色立方体：三维几何与计数探索

5x5x5涂色立方体：三维几何与计数探索
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2016-03-06 22:47
多边形数列立体几何几何作图
该范例展示了一个5x5x5的立方体，其表面被涂色，内部小立方体未被涂色，用于直观演示三维空间中的涂色计数问题。核心数学概念包括立体几何中的立方体结构、表面积与体积的关系，以及分类计数（如三面涂色、两面涂色、一面涂色和未涂色的小立方体数量）。作图要点：使用GeoGebra的Point命令创建顶点，Cube命令构建立方体，PointIn命令标记内部点，Segment命令绘制棱和辅助线段，并定义了多个面（faceABCD等）和棱（edgeAD等）对象，便于观察和交互。适合中学阶段（初中高年级至高中）的数学教学，用于空间想象能力培养、组合计数练习以及立体几何入门，也可作为探究性学习或竞赛辅导的素材。

GeoGebra 命令及参数

  1. Point(xAxis) → A
  2. Point(xAxis) → B
  3. Point(Circle[B, Distance[A, B], Segment[A, B]]) → C
  4. Cube(A, B, C) → a, 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(edgeBC) → I
  6. PointIn(faceBCGF) → J
  7. Segment(I, J) → b
  8. Point(edgeBC) → K
  9. Point(edgeFG) → L
  10. Segment(K, L) → c
  11. Point(edgeBC) → M
  12. Point(edgeFG) → N
  13. Segment(M, N) → d
  14. Point(edgeBC) → O
  15. PointIn(faceBCGF) → P
  16. Segment(O, P) → e
  17. Point(edgeCD) → Q
  18. Point(edgeGH) → R
  19. Segment(Q, R) → f
  20. Point(edgeCD) → S
  21. Point(edgeGH) → T
  22. Segment(S, T) → g
  23. Point(edgeCD) → U
  24. Point(edgeGH) → V
  25. Segment(U, V) → h
  26. Point(edgeCD) → W
  27. PointIn(faceCDHG) → Z
  28. Segment(W, Z) → i
  29. Point(yAxis) → A_1
  30. Point(edgeEH) → B_1
  31. Segment(A_1, B_1) → j
  32. Point(yAxis) → C_1
  33. Point(edgeEH) → D_1
  34. Segment(C_1, D_1) → k
  35. Point(yAxis) → E_1
  36. PointIn(faceADHE) → F_1
  37. Segment(E_1, F_1) → l
  38. Point(yAxis) → G_1
  39. PointIn(faceADHE) → H_1
  40. Segment(G_1, H_1) → m
  41. Point(xAxis) → I_1
  42. Point(edgeEF) → J_1
  43. Segment(I_1, J_1) → n
  44. Point(xAxis) → K_1
  45. Point(edgeEF) → L_1
  46. Segment(K_1, L_1) → p
  47. Point(xAxis) → M_1
  48. Point(edgeEF) → N_1
  49. Segment(M_1, N_1) → q
  50. Point(xAxis) → O_1
  51. Point(edgeEF) → P_1
  52. Segment(O_1, P_1) → r
  53. Segment(P, B_1) → s
  54. Segment(D_1, N) → t
  55. Segment(L, F_1) → a_1
  56. Segment(J, H_1) → b_1
  57. Segment(Z, P_1) → c_1
  58. Segment(N_1, V) → d_1
  59. Segment(T, L_1) → e_1
  60. Segment(R, J_1) → f_1
  61. Segment(Q, I_1) → g_1
  62. Segment(S, K_1) → h_1
  63. Segment(U, M_1) → i_1
  64. Segment(W, O_1) → j_1
  65. Segment(A_1, O) → k_1
  66. Segment(M, C_1) → l_1
  67. Segment(E_1, K) → m_1
  68. Segment(G_1, I) → n_1
  69. Point(zAxis) → Q_1
  70. Point(edgeBF) → R_1
  71. Segment(Q_1, R_1) → p_1
  72. Point(zAxis) → S_1
  73. Point(edgeBF) → T_1
  74. Segment(S_1, T_1) → q_1
  75. Point(zAxis) → U_1
  76. Point(edgeBF) → V_1
  77. Segment(U_1, V_1) → r_1
  78. Point(zAxis) → W_1
  79. PointIn(faceABFE) → Z_1
  80. Segment(W_1, Z_1) → s_1
  81. Point(edgeDH) → A_2
  82. Segment(Q_1, A_2) → t_1
  83. Point(edgeDH) → B_2
  84. Segment(S_1, B_2) → a_2
  85. Point(edgeDH) → C_2
  86. Segment(U_1, C_2) → b_2
  87. Point(edgeDH) → D_2
  88. Segment(W_1, D_2) → c_2
  89. Point(edgeCG) → E_2
  90. Segment(D_2, E_2) → d_2
  91. Point(edgeCG) → F_2
  92. Segment(C_2, F_2) → e_2
  93. Point(edgeCG) → G_2
  94. Segment(B_2, G_2) → f_2
  95. Point(edgeCG) → H_2
  96. Segment(A_2, H_2) → g_2
  97. Segment(Z_1, E_2) → h_2
  98. Segment(V_1, F_2) → i_2
  99. Segment(T_1, G_2) → j_2
  100. Segment(R_1, H_2) → k_2