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Constructing Hilbert’s Cube with 3D Symmetry

Constructing Hilbert’s Cube with 3D Symmetry
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2023-11-13 03:37
CircleVectorSolid GeometryTransformationGeometric Construction
In collections Use Cases of GeoGebra
This GeoGebra example demonstrates how to build a Hilbert’s-cube-related 3D geometric configuration by applying a sequence of symmetry and placement operations. It highlights how local geometric constraints generate a larger spatial structure. The key mathematics includes solid geometry (cube/prism-like composition), spatial intersection of objects, and transformation geometry using translations, rotations, and reflections to systematically reproduce congruent parts. Construction-wise, the workflow leverages commands from the provided construct summary: Segment and Line to define structural edges and directions; Sphere to create a 3D constraint surface; Intersect to compute intersection points that anchor key vertices (e.g., A, B, C, …); Polygon and Prism to form faces and solid bodies; Vector together with Translate, Rotate, and Mirror to control movement and orientation; Midpoint to establish scale and reference; and Point/PointIn to define points and check their placement within the intended geometric objects. Suitable for advanced secondary school and early university learning, especially for lessons on 3D constructions and transformation-based reasoning.

GeoGebra Commands & Parameters

  1. Segment(C, E) → f
  2. Segment(E, F) → i
  3. Segment(F, D) → j
  4. Segment(D, C) → l
  5. Segment(H, G) → m
  6. Segment(G, A) → n
  7. Segment(A, B) → p
  8. Segment(B, E) → q
  9. Segment(C, A) → r
  10. Segment(D, G) → s
  11. Segment(F, H) → g
  12. Sphere(F, 1) → a
  13. Intersect(i, a, 1) → J
  14. Intersect(j, a, 1) → K
  15. Intersect(g, a, 1) → L
  16. Sphere(F, 4) → b
  17. Intersect(j, b, 1) → M
  18. Intersect(g, b, 1) → N
  19. Intersect(i, b, 1) → O
  20. Line(L, m) → k
  21. Line(M, g) → t
  22. Intersect(t, k) → P
  23. Line(L, i) → c
  24. Line(J, g) → d
  25. Intersect(c, d) → Q
  26. Polygon(Q, J, F, L) → q1, q_1, j_1, f_1, l_1
  27. Prism(q1, 4) → o, V, edgeQV, edgeJW, edgeFZ, edge1, edgeVW, edgeWZ, edge3, edge2, W, Z, A_1, faceJQVW, faceFJWZ, face1, face2, face3
  28. Sphere(O, 1) → e
  29. Intersect(i, e) → R, S
  30. Vector(F, S) → u
  31. Translate(o, u) → o'_{1}
  32. Sphere(Z, 1) → v
  33. Intersect(j, v) → T, U
  34. Line(T, t) → g_1
  35. Midpoint(W, T) → B_1
  36. Line(B_1, t) → h_1
  37. Rotate(o, 90°, h_1) → o'
  38. Point(c) → C_1
  39. Sphere(C_1, 1) → w
  40. Sphere(S, 1) → b_1
  41. PointIn(b_1) → D_1
  42. Midpoint(D_1, C_1) → E_1
  43. Segment(B, H) → h
  44. Line(E_1, h) → i_1
  45. Rotate(o'_{1}, -90°, i_1) → c_1
  46. Line(N, c) → k_1
  47. Line(S, d) → m_1
  48. Intersect(k_1, m_1) → I
  49. Line(I, p) → n_1
  50. Sphere(I, 1) → d_1
  51. Intersect(n_1, d_1) → F_1, G_1
  52. Intersect(m_1, d_1, 1) → H_1
  53. Intersect(k_1, d_1, 1) → J_1
  54. Midpoint(H_1, J_1) → I_1
  55. Line(I_1, j) → p_1
  56. Rotate(c_1, 90°, p_1) → e_1
  57. Intersect(d, k_1) → K_1
  58. Line(H_1, k_1) → r_1
  59. Intersect(g, r_1) → L_1
  60. Midpoint(L_1, K_1) → M_1
  61. Line(M_1, j) → s_1
  62. Rotate(e_1, -90°, s_1) → o_1
  63. Sphere(N, 3) → t_1
  64. Intersect(g, t_1) → N_1, O_1
  65. Line(O_1, h) → f_2
  66. Intersect(f_2, d) → P_1
  67. Mirror(c_1, eq1) → u_1
  68. Mirror(e_1, eq1) → v_1
  69. Mirror(o', eq1) → o''
  70. Sphere(H, 1) → w_1
  71. Intersect(g, w_1) → Q_1,
  72. Vector(O_1, H) → u_2
  73. Translate(v_1, u_2) → a_2
  74. Mirror(o, eq1) → o'_{2}
  75. Intersect(t, m) → R_1
  76. Line(R_1, n) → g_2
  77. Sphere(R_1, 4) → b_2
  78. Intersect(g_2, b_2) → S_1, T_1
  79. Line(T_1, p) → h_2
  80. Intersect(h, h_2) → U_1
  81. Intersect(f_2, h_2) → V_1
  82. Sphere(T_1, 1) → c_2
  83. Intersect(h_2, c_2) → W_1, Z_1
  84. Vector(U_1, Z_1) → v_2
  85. Translate(u_1, v_2) → d_2
  86. Line(T_1, r) → i_2
  87. Sphere(T_1, 4) → e_2
  88. Intersect(i_2, e_2) → A_2, B_2
  89. Intersect(t, e_2) → C_2, D_2
  90. Sphere(A_2, 1) → j_2
  91. Intersect(i_2, j_2) → E_2, F_2
  92. Line(A_2, g_2) → k_2
  93. Intersect(k_2, j_2) → G_2, H_2
  94. Line(F_2, k_2) → l_2
  95. Point(k_2) → I_2
  96. Line(I_2, i_2) → m_2
  97. Intersect(m_2, l_2) → J_2
  98. Polygon(A_2, I_2, J_2, F_2) → q2, n_2, p_2, q_2, r_2
  99. Prism(q2, 3) → s_2, O_2, edge12, edge14, edge15, edge13, edge16, edge18, edge19, edge17, P_2, Q_2, R_2, face10, face12, face11, face9, face13
  100. Vector(J_2, Q_2) → w_2
  101. Translate(d_2, w_2) → o_2
  102. Line(O_2, s) → t_2
  103. Intersect(h_2, t_2) → K_2
  104. Line(K_2, n) → f_3
  105. Sphere(K_2, 1) → a_3
  106. Intersect(f_3, a_3) → L_2, M_2
  107. Intersect(t_2, a_3) → N_2, S_2
  108. Midpoint(N_2, L_2) → T_2
  109. Line(T_2, p) → g_3
  110. Rotate(o_2, -90°, g_3) → b_3
  111. Vector(C_2, G) → u_3
  112. Translate(o'_{2}, u_3) → c_3
  113. Translate(o'', u_3) → o'''
  114. Translate(d_2, u_3) → d_3
  115. Intersect(m, w_1) → U_2,
  116. Vector(U_2, G) → v_3
  117. Translate(v_1, v_3) → e_3
  118. Translate(o_1, v_3) → h_3
  119. Sphere(D, 3) → i_3
  120. Intersect(s, i_3) → V_2,
  121. Vector(V_2, D) → w_3
  122. Translate(h_3, w_3) → j_3
  123. Intersect(k, t_2) → W_2
  124. Intersect(s, k) → Z_2
  125. Vector(G, Z_2) → k_3
  126. Translate(c_3, k_3) → l_3
  127. Intersect(j, i_3) → A_3,
  128. Vector(D, A_3) → m_3
  129. Translate(j_3, m_3) → n_3
  130. Intersect(s_1, t_2) → B_3
  131. Point(s_1) → C_3
  132. Vector(D_3, C_3) → p_3
  133. Translate(e_1, p_3) → o_3
  134. Translate(c_1, p_3) → q_3
  135. Translate(o'_{1}, p_3) → r_3
  136. Intersect(l, i_3) → E_3,
  137. Vector(D, E_3) → s_3
  138. Translate(j_3, s_3) → t_3
  139. Translate(o', u) → o''_{1}
  140. Translate(o''_{1}, p_3) → a_4
  141. Translate(a_4, u_2) → b_4
  142. Mirror(t_3, f_4) → c_4
  143. Mirror(d_3, f_4) → d_4
  144. Mirror(e_3, f_4) → e_4
  145. Mirror(o''', f_4) → g_4
  146. Mirror(r_3, f_4) → h_4
  147. Mirror(q_3, f_4) → i_4
  148. Mirror(n_3, f_4) → j_4
  149. Mirror(o_3, f_4) → k_4
  150. Mirror(l_3, f_4) → l_4
  151. Mirror(j_3, f_4) → m_4
  152. Mirror(h_3, f_4) → n_4
  153. Mirror(e_3, f_4) → o_4
  154. Mirror(d_3, f_4) → p_4
  155. Mirror(o''', f_4) → q_4
  156. Mirror(c_3, f_4) → r_4
  157. Mirror(b_3, f_4) → s_4
  158. Mirror(o_2, f_4) → t_4
  159. Mirror(d_2, f_4) → u_4
  160. Mirror(o'', f_4) → o'''_{1}
  161. Mirror(o'_{2}, f_4) → v_4
  162. Mirror(a_2, f_4) → w_4
  163. Mirror(u_1, f_4) → a_5
  164. Mirror(v_1, f_4) → b_5
  165. Mirror(o_1, f_4) → c_5
  166. Mirror(e_1, f_4) → d_5
  167. Mirror(c_1, f_4) → e_5
  168. Mirror(o'_{1}, f_4) → f_5
  169. Mirror(o', f_4) → o''_{2}
  170. Mirror(o, f_4) → o'_{3}
  171. Mirror(u_1, f_4) → g_5
  172. Mirror(s_2, f_4) → h_5

AI Drawing Prompt Reference

Draw a 3D geometric configuration demonstrating the construction of a Hilbert's cube-related structure through symmetry and placement operations. Begin by defining a series of line segments to form the basic framework, then use intersections between spheres and lines to determine key vertices. Form faces and solid bodies using polygons and prisms. Systematically replicate congruent parts by applying vectors combined with translation, rotation, and reflection, resulting in a complex spatial structure with 3D symmetry.
The prompt above was compiled from the original drawing commands and is for reference only.