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Dome 6: Geometric Dome Construction via Rotation and Mirroring

Dome 6: Geometric Dome Construction via Rotation and Mirroring
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2020-05-22 20:23
PolygonAngleTrigonometryTransformationGeometric Construction
This example demonstrates a dome-shaped geometric structure formed by combining polygons, line segments, circular arcs, and mirror transformations, creating a regular pattern through rotation and symmetry operations. The core mathematical concepts include median lines, natural numbers, angles, trigonometric functions, and right triangle properties, showcasing the integration of geometric transformations with trigonometric relationships. Construction highlights: using GeoGebra commands such as Polygon, Intersect, Segment, Point, Midpoint, Rotate, CircleArc, Line, Plane, Mirror, and Sequence, the construction involves numerous points (O, A, B, C, D, etc.), a polygon named vierhoek1, and segments a, b, c, d, etc. Rotations and mirroring generate symmetric arcs and lines, ultimately forming the dome outline. Suitable for high school or early college students to explore geometric transformations, trigonometry, and constructive geometry, and also useful as a reference for mathematical modeling or artistic geometric design.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C, D) → vierhoek1, a, b, c, d
  2. Intersect(d, xAxis) → E
  3. Intersect(a, yAxis) → F
  4. Segment(E, F) → f
  5. Intersect(b, xAxis) → G
  6. Segment(F, G) → g
  7. Point(c) → H
  8. Segment(G, H) → h
  9. Segment(H, E) → i
  10. Midpoint(f) → I
  11. Midpoint(g) → J
  12. Midpoint(h) → K
  13. Midpoint(i) → L
  14. Segment(D, B) → j
  15. Rotate(C, 90°, j) → C'
  16. CircleArc(O, C, C') → e
  17. Point(e) → N
  18. Point(e) → P
  19. Point(e) → Q
  20. Point(e) → R
  21. Line(K, zAxis) → k
  22. Intersect(k, e, 1) → S
  23. Line(U, b) → l
  24. Line(V, b) → m
  25. Line(W, b) → n
  26. Plane(G, H, T) → p
  27. Intersect(n, p) → Z
  28. Intersect(m, p) → A_1
  29. Intersect(l, p) → B_1
  30. Polygon(G, C, W, Z) → vierhoek2, g_1, c_1, w, z_1
  31. Polygon(Z, W, V, A_1) → vierhoek3, z_2, w_1, v, a_1
  32. Polygon(B_1, A_1, V, U) → vierhoek4, b_1, q, v_1, u
  33. Polygon(B_1, U, T) → driehoek1, t, r, u_1
  34. Plane(C, A, M) → p_1
  35. Mirror(vierhoek2, p_1) → vierhoek2'
  36. Mirror(vierhoek3, p_1) → vierhoek3'
  37. Mirror(vierhoek4, p_1) → vierhoek4'
  38. Mirror(driehoek1, p_1) → driehoek1'
  39. Polygon(G, Z, A_1, B_1, T, C_1) → veelhoek1, g_2, z_3, s, f_1, t_1, h_1
  40. Mirror(veelhoek1, p_1) → veelhoek1'
  41. Sequence(Rotate[l1, (nn * 90°), zAxis], nn, 1, 3) → l2
  42. Sequence(Rotate[l3, (nn * 90°), zAxis], nn, 1, 3) → l4
  43. Mirror(C_1, D_1) → E_1
  44. Rotate(C_1, 90°, zAxis) → F_1
  45. Line(C_1, E_1) → i_1
  46. Rotate(F_1, (-(90°)), i_1) → G_1
  47. CircleArc(D_1, F_1, G_1) → d_1
  48. Point(d_1) → I_1
  49. Point(d_1) → J_1
  50. Point(d_1) → K_1
  51. Point(d_1) → L_1
  52. Point(d_1) → M_1
  53. Mirror(N_1, p_1) → S_1
  54. Mirror(O_1, p_1) → T_1
  55. Mirror(P_1, p_1) → U_1
  56. Mirror(Q_1, p_1) → V_1
  57. Mirror(R_1, p_1) → W_1
  58. Polygon(C_1, F_1, R_1, W_1) → vierhoek5, j_1, k_1, r_1, l_1
  59. Polygon(W_1, R_1, Q_1, V_1) → vierhoek6, m_1, n_1, q_1, s_1
  60. Polygon(V_1, Q_1, P_1, U_1) → vierhoek7, e_1, f_2, h_2, i_2
  61. Polygon(U_1, P_1, O_1, T_1) → vierhoek8, j_2, k_2, o_1, l_2
  62. Polygon(T_1, O_1, N_1, S_1) → vierhoek9, m_2, n_2, p_2, q_2
  63. Polygon(S_1, N_1, H_1) → driehoek2, r_2, s_2, t_2
  64. Segment(C, W) → a_2
  65. Segment(W, V) → b_2
  66. Segment(V, U) → c_2
  67. Segment(U, T) → d_2
  68. Sequence(Rotate[l7, (nn * 90°), zAxis], nn, 1, 3) → l8
  69. Segment(C_1, W_1) → e_2
  70. Segment(W_1, V_1) → f_3
  71. Segment(V_1, U_1) → g_3
  72. Segment(U_1, T_1) → h_3
  73. Segment(T_1, S_1) → i_3
  74. Segment(S_1, H_1) → j_3
  75. Sequence(Rotate[l9, (nn * 90°), zAxis], nn, 1, 3) → l10
  76. Segment(G, Z) → k_3
  77. Segment(Z, A_1) → l_3
  78. Segment(A_1, B_1) → m_3
  79. Segment(B_1, T) → n_3
  80. Sequence(Rotate[l11, (nn * 90°), zAxis], nn, 1, 3) → l12
  81. Mirror(l11, p_1) → l11'
  82. Sequence(Rotate[l11', (nn * 90°), zAxis], nn, 1, 3) → l13
  83. Segment(F_1, H) → p_3
  84. Sequence(Segment[Rotate[F_1, (nn * 90°), zAxis], Rotate[H, (nn * 90°), zAxis]], nn, 1, 3) → l14
  85. Segment(C_1, F_1) → q_3
  86. Sequence(Segment[Rotate[C_1, (nn * 90°), zAxis], Rotate[F_1, (nn * 90°), zAxis]], nn, 1, 3) → l15
  87. Sequence(Rotate[l5, (nn * 90°), zAxis], nn, 1, 3) → l6
  88. Segment(A, C) → r_3