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Dome 4: Symmetry and Rotation in 3D Construction

Dome 4: Symmetry and Rotation in 3D Construction
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2020-05-22 20:14
CirclePolygonSolid GeometryTransformationGeometric Construction
① This example demonstrates a 3D dome-like geometric structure named 'dome 4', built through symmetry, rotation, and mirroring in space. ② Core mathematical concepts include solid geometry (polyhedra and curved surfaces), axial and central symmetry, rotation and reflection transformations, and the projection of quadratic curves (e.g., ellipses) in 3D. ③ Construction highlights: from the summary, the GeoGebra commands used include Segment, Mirror, LineBisector, Polygon, Intersect, Midpoint, Rotate, CircleArc, Point, Plane, Sequence, Distance, and Line, involving objects such as points A to Z, polygon vierhoek1, segments a to h, and plane p. The dome is generated by rotating and mirroring base elements to create a symmetric framework. ④ Suitable for high school or college-level solid geometry lessons, to understand symmetry transformations, rotational solids, and comprehensive use of GeoGebra commands, also useful as an advanced practice exercise.

GeoGebra Commands & Parameters

  1. Segment(A, C) → f
  2. Mirror(E, f) → I
  3. Mirror(I, yAxis) → F
  4. LineBisector(E, F) → g
  5. Polygon(A, B, C, D) → vierhoek1, a, b, c, d
  6. Intersect(a, g) → H
  7. Intersect(xAxis, yAxis) → O
  8. Mirror(H, O) → K
  9. Mirror(E, O) → L
  10. Mirror(H, f) → J
  11. Mirror(J, yAxis) → G
  12. Midpoint(E, F) → M
  13. Rotate(F, 90°, g) → F'
  14. CircleArc(M, F, F') → e
  15. Point(e) → N
  16. Point(e) → P
  17. Point(e) → Q
  18. Point(e) → R
  19. Point(e) → S
  20. CircleArc(M, F', E) → h
  21. Plane(K, H, M + (0, 0, 1)) → p
  22. Mirror(T, p) → T'
  23. Mirror(V, p) → V'
  24. Mirror(W, p) → W'
  25. Mirror(Z, p) → Z'
  26. Mirror(A_1, p) → B_1
  27. Segment(E, B_1) → i
  28. Segment(B_1, Z') → j
  29. Segment(Z', W') → k
  30. Segment(W', V') → l
  31. Segment(V', T') → m
  32. Segment(T', U) → n
  33. Segment(U, T) → q
  34. Segment(T, V) → r
  35. Segment(V, W) → s
  36. Segment(W, Z) → t
  37. Segment(Z, A_1) → f_1
  38. Segment(A_1, F) → g_1
  39. Mirror(l1, sym) → l1'
  40. Sequence(Rotate[l1, (nn * 90°), zAxis], nn, 1, 3) → l2
  41. Sequence(Rotate[l1', (nn * 90°), zAxis], nn, 1, 3) → l3
  42. Midpoint(T, V) → C_1
  43. Mirror(C_1, p) → D_1
  44. Mirror(E_1, p) → F_1
  45. Segment(E, F) → h_1
  46. Distance(H, g) → o
  47. Rotate(E_1, (360°) / 8, zAxis) → G_1
  48. Rotate(C_1, (360°) / 8, zAxis) → H_1
  49. Polygon(F_1, E_1, 8, xOyPlane) → veelhoek1, i_1, J_1, K_1, L_1, M_1, N_1, j_1, k_1, l_1, m_1, n_1, p_1, q_1, I_1
  50. Polygon(D_1, T', U, T, C_1, E_1, F_1) → veelhoek2, d_1, t', u, t_1, c_1, e_1, r_1
  51. Line(K_1, F_1) → s_1
  52. CircleArc(P_1, F_1, Q_1) → c_2
  53. CircleArc(P_1, Q_1, K_1) → d_2
  54. Point(c_2) → S_1
  55. Point(c_2) → T_1
  56. Point(c_2) → U_1
  57. Point(c_2) → V_1
  58. Segment(R_1, W_1) → a_1
  59. Segment(W_1, Z_1) → b_1
  60. Segment(Z_1, A_2) → f_2
  61. Segment(A_2, B_2) → g_2
  62. Segment(B_2, F_1) → h_2
  63. Rotate(l5, (360°) / 8, zAxis) → l5'
  64. Rotate(W_1, (360°) / 8, zAxis) → C_2
  65. Rotate(Z_1, (360°) / 8, zAxis) → D_2
  66. Rotate(A_2, (360°) / 8, zAxis) → E_2
  67. Rotate(B_2, (360°) / 8, zAxis) → F_2
  68. Rotate(F_1, (360°) / 8, zAxis) → G_2
  69. Polygon(R_1, W_1, C_2) → driehoek1, i_2, j_2, w_1
  70. Polygon(W_1, Z_1, D_2, C_2) → vierhoek2, k_2, z_1, l_2, m_2
  71. Polygon(Z_1, A_2, E_2, D_2) → vierhoek3, n_2, a_2, e_2, p_2
  72. Polygon(A_2, B_2, F_2, E_2) → vierhoek4, q_2, b_2, r_2, s_2
  73. Polygon(B_2, F_1, G_2, F_2) → vierhoek5, t_2, f_3, g_3, h_3
  74. Sequence(Rotate[l6, (nn * (360°) / 8), zAxis], nn, 1, 11) → l7
  75. Sequence(Rotate[l5, (nn * (360°) / 8), zAxis], nn, 2, 11) → l8
  76. Sequence(Rotate[veelhoek2, (nn * (360°) / 8), zAxis], nn, 1, 7) → l4
  77. Segment(B, C) → i_3
  78. Segment(A, B) → j_3
  79. Segment(C, D) → k_3
  80. Segment(D, A) → l_3
  81. Segment(E, F) → m_3
  82. Mirror(E, yAxis) → E'
  83. Mirror(K, yAxis) → K'
  84. Segment(F, K') → n_3
  85. Segment(K', J) → p_3
  86. Segment(J, E) → q_3
  87. Segment(E', G) → r_3
  88. Segment(G, K) → s_3
  89. Segment(K, I) → t_3
  90. Segment(I, E') → a_3

AI Drawing Prompt Reference

Draw a 3D dome structure generated by symmetry and rotation. Start with a quadrilateral base and one diagonal as an axis of symmetry. Use mirroring and angle bisectors to determine symmetric points, forming the basic framework. Construct circular arcs on both sides of the axis and place several points on them. Mirror some points across an inclined plane to the other side, then connect these points to form the dome's edges. Finally, rotate a set of edges around the vertical axis three times at 90-degree intervals to complete the symmetric dome, adding auxiliary segments and distance marks.
The prompt above was compiled from the original drawing commands and is for reference only.