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Dividing a Cube into 7 Parts

Dividing a Cube into 7 Parts
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2018-12-26 03:15
Solid GeometryGeometric Construction
This example demonstrates how to divide a cube into 7 equal parts. Key mathematical concepts include the division of a cube, volume, and surface area calculations. Using GeoGebra commands such as Intersect, Point, Cube, etc., the various faces and edges of the cube are constructed. Suitable for middle or high school students learning solid geometry and division problems.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, zAxis) → A
  2. Point(Circle[B, Distance[A, B], Segment[A, B]]) → C
  3. Cube(A, B, C) → a, D, faceCDHG, faceEFGH, edgeBD, edgeAB, edgeAC, edgeCD, edgeBE, edgeDH, edgeEH, edgeEF, E, edgeAF, edgeFG, edgeCG, edgeGH, F, G, H, faceABDC, faceBDHE, faceABEF, faceACGF
  4. Midpoint(H, B) → C_p
  5. Line(C_p, zAxis) → f
  6. Sphere(C_p, (0.25 * 7^(2 / 3))) → b
  7. Line(C_p, xAxis) → g
  8. Line(C_p, yAxis) → h
  9. Intersect(f, b) → J, K
  10. Intersect(g, b) → L, M
  11. Intersect(h, b) → N, O
  12. Plane(L, faceBDHE) → c
  13. Plane(M, faceACGF) → d_1
  14. Plane(K, faceEFGH) → e
  15. IntersectPath(c, e) → i
  16. IntersectPath(d_1, e) → j
  17. Plane(J, e) → k
  18. Plane(N, faceABEF) → l
  19. Plane(O, faceCDHG) → m
  20. IntersectPath(d_1, l) → n
  21. IntersectPath(d_1, m) → p
  22. Intersect(j, n) → F'
  23. Intersect(j, p) → G'_2
  24. IntersectPath(c, m) → q
  25. IntersectPath(c, l) → r
  26. IntersectPath(d_1, k) → s
  27. IntersectPath(c, k) → t
  28. Intersect(s, n) → B'
  29. Intersect(p, s) → C'
  30. Intersect(t, q) → D'
  31. Intersect(r, t) → A'
  32. Intersect(i, r) → E'_2
  33. Cube(A', B', C') → o, D'_2, faceC'RWU, faceSTUW, edgeA'R, edgeA'B', edgeB'C', edgeC'R, edgeA'S, edgeRW, edgeSW, edgeST, E', edgeB'T, edgeTU, edgeC'U, edgeUW, F'_2, G', H', faceA'B'C'R, faceA'RWS, faceA'B'TS, faceB'C'UT
  34. Sphere(C_p, d) → v
  35. Sphere(C_p, d) → u
  36. Intersect(h, u) → I, P
  37. Intersect(f, u) → Q, R
  38. Intersect(g, u) → S, T
  39. Vector(N, I) → y_2
  40. Vector(O, P) → y_1
  41. Vector(M, T) → x_1
  42. Vector(L, S) → x_2
  43. Vector(K, R) → z_1
  44. Vector(J, Q) → z_2
  45. Polygon(G, H, D, C) → q1_{y1}, g_1, h_1, d_2, c_1
  46. Polygon(G, G', C', C) → t2_{y1}, g_2, g', c', c_2
  47. Polygon(H, H', G', G) → t1_{y1}, h_2, h', g'_1, g_3
  48. Polygon(D, D', H', H) → t4_{y1}, d_3, d', h'_1, h_3
  49. Polygon(D, C, C', D') → t3_{y1}, d_4, c_3, c'_1, d'_1
  50. Translate(q1_{y1}, y_1) → q1'_{y1}
  51. Translate(t1_{y1}, y_1) → t1'_{y1}
  52. Translate(t2_{y1}, y_1) → t2'_{y1}
  53. Translate(t3_{y1}, y_1) → t3'_{y1}
  54. Translate(t4_{y1}, y_1) → t4'_{y1}
  55. Translate(faceC'RWU, y_1) → faceC'RWU'
  56. OrthogonalLine(H', g_1, space) → f_1
  57. Intersect(g_1, f_1) → U
  58. Segment(U, H') → i_1
  59. Midpoint(V, W) → Z
  60. Segment(V, W) → j_1
  61. OrthogonalLine(Z, j_1) → k_1
  62. Circle(Z, Distance[H', U]) → e_1
  63. Intersect(e_1, k_1, 2) → A_1
  64. Line(A_1, j_1) → l_1
  65. Circle(A_1, Distance[G', H'] / 2) → p_1
  66. Intersect(p_1, l_1) → B_1, C_1
  67. Segment(V, B_1) → m_1
  68. Segment(C_1, W) → n_1
  69. Polygon(B_1, V, W, C_1) → flat_{trap1}, b_1, v_1, w, c_4
  70. Polygon(B_1, C_1, 4) → poly1, q_1, r_1, s_1, t_1, D_1, E_1
  71. Polygon(W, V, 4) → poly2, a_1, f_2, i_2, j_2, F_1, G_1
  72. Midpoint(F_1, W) → H_1
  73. Rotate(B_1, 90°, H_1) → I_1
  74. Rotate(V, 90°, H_1) → V'
  75. Rotate(W, 90°, H_1) → W'
  76. Rotate(C_1, 90°, H_1) → J_1
  77. Polygon(I_1, V', W', J_1) → flat_{trap4}, i_3, v', w', j_3
  78. Rotate(I_1, 90°, H_1) → K_1
  79. Rotate(V', 90°, H_1) → V''
  80. Rotate(W', 90°, H_1) → W''
  81. Rotate(J_1, 90°, H_1) → L_1
  82. Polygon(K_1, V'', W'', L_1) → flat_{trap3}, k_2, v'', w'', l_2
  83. Rotate(K_1, 90°, H_1) → M_1
  84. Rotate(V'', 90°, H_1) → V'''
  85. Rotate(W'', 90°, H_1) → W'''
  86. Rotate(L_1, 90°, H_1) → N_1
  87. Polygon(M_1, V''', W''', N_1) → flat_{trap2}, m_2, v''', w''', n_2
  88. Rotate(faceC'RWU', 90°, f) → faceC'RWU'_{90}
  89. Rotate(faceC'RWU'_{90}, 90°, f) → faceC'RWU'_{180}
  90. Rotate(q1'_{y1}, 90°, f) → q1'_{y1}90
  91. Rotate(q1'_{y1}90, 90°, f) → q1'_{y1}180
  92. Rotate(t1'_{y1}, 90°, f) → t1'_{y1}90
  93. Rotate(t2'_{y1}, 90°, f) → t2'_{y1}90
  94. Rotate(t3'_{y1}, 90°, f) → t3'_{y1}90
  95. Rotate(t4'_{y1}, 90°, f) → t4'_{y1}90
  96. Rotate(t2'_{y1}, 180°, f) → t2'_{y1}180
  97. Rotate(t1'_{y1}, 180°, f) → t1'_{y1}180
  98. Rotate(t3'_{y1}, 180°, f) → q5
  99. Rotate(t4'_{y1}, 180°, f) → q6
  100. Rotate(t1'_{y1}90, 180°, f) → t1'_{y1}270
  101. Rotate(t2'_{y1}90, 180°, f) → t2'_{y1}270
  102. Rotate(q1'_{y1}90, 180°, f) → q1'_{y1}270
  103. Rotate(t3'_{y1}90, 180°, f) → t3'_{y1}270
  104. Rotate(t4'_{y1}90, 180°, f) → t4'_{y1}270
  105. Rotate(faceC'RWU'_{90}, 180°, f) → faceC'RWU'_{270}
  106. Rotate(faceC'RWU', (-(90°)), g) → faceC'RWU'_{bottom}
  107. Rotate(q1'_{y1}, (-(90°)), g) → q1'_{y1}bottom
  108. Rotate(t1'_{y1}, (-(90°)), g) → t1'_{y1}bottom
  109. Rotate(t2'_{y1}, (-(90°)), g) → t2'_{y1}bottom
  110. Rotate(t3'_{y1}, (-(90°)), g) → t3'_{y1}bottom
  111. Rotate(t4'_{y1}, (-(90°)), g) → t4'_{y1}bottom
  112. Rotate(q1'_{y1}bottom, (-(180°)), g) → q1'_{y1}top
  113. Rotate(faceC'RWU'_{bottom}, 180°, g) → faceC'RWU'_{top}
  114. Rotate(t1'_{y1}bottom, 180°, g) → t1'_1
  115. Rotate(t2'_{y1}bottom, 180°, g) → t2'_{y1}top
  116. Rotate(t3'_{y1}bottom, 180°, g) → t3'_{y1}top
  117. Rotate(t4'_{y1}bottom, 180°, g) → t1'_{y1}top