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Icosidodecahedron: 3D Construction in GeoGebra

Icosidodecahedron: 3D Construction in GeoGebra
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2017-11-21 15:37
PolygonAngleSolid GeometryTransformationGeometric Construction
This example demonstrates how to construct an icosidodecahedron, an Archimedean solid composed of 20 equilateral triangles and 12 regular pentagons, using GeoGebra's 3D capabilities. The core mathematical concepts involve solid geometry, polyhedron classification, symmetry, and the use of geometric transformations to build complex 3D structures. Key construction steps include using the Angle command to define rotation angles, Plane to create reference planes, OrthogonalLine to construct perpendicular lines, Rotate to rotate polygons in 3D space, and Polygon to generate faces. Multiple vertex objects (e.g., A1, B1, C1, etc.) are systematically combined to assemble the full solid. This example is suitable for high school or university students learning solid geometry, polyhedron properties, and 3D modeling, and serves as a valuable teaching resource for educators illustrating Archimedean solids.

GeoGebra Commands & Parameters

  1. Angle(A, B, D1) → α
  2. Plane(A, D1, B) → a
  3. OrthogonalLine(B, a) → b
  4. Rotate(D1, α, b) → E1
  5. Rotate(D2, α, b) → E2
  6. Rotate(D3, α, b) → E3
  7. Rotate(D4, α, b) → E4
  8. Rotate(D5, α, b) → E5
  9. Rotate(D6, α, b) → E6
  10. Rotate(D7, α, b) → E7
  11. Rotate(D8, α, b) → E8
  12. Rotate(D9, α, b) → E9
  13. Rotate(D10, α, b) → E10
  14. Rotate(D11, α, b) → E11
  15. Rotate(D12, α, b) → E12
  16. Rotate(D13, α, b) → E13
  17. Rotate(D14, α, b) → E14
  18. Rotate(D15, α, b) → E15
  19. Rotate(D16, α, b) → E16
  20. Rotate(D17, α, b) → E17
  21. Rotate(D18, α, b) → E18
  22. Rotate(D19, α, b) → E19
  23. Rotate(D20, α, b) → E20
  24. Rotate(D21, α, b) → E21
  25. Rotate(D22, α, b) → E22
  26. Rotate(D23, α, b) → E23
  27. Rotate(D24, α, b) → E24
  28. Rotate(D25, α, b) → E25
  29. Rotate(D26, α, b) → E26
  30. Rotate(D27, α, b) → E27
  31. Rotate(D28, α, b) → E28
  32. Rotate(D29, α, b) → E29
  33. Rotate(D30, α, b) → E30
  34. Polygon(E30, E18, E22, E4, E28) → wielokąt1, e30, e18, e22, e4, e28
  35. Rotate(E26, β, Line[E22, E18]) → E26'
  36. Polygon(E22, E18, E26') → wielokąt2, e26', e22_1, e18_1
  37. Rotate(E10, β, Line[E4, E22]) → E10'
  38. Polygon(E10', E4, E22) → wielokąt3, e22_2, e10', e4_1
  39. Rotate(E16, β, Line[E28, E4]) → E16'
  40. Polygon(E16', E28, E4) → wielokąt4, e4_2, e16', e28_1
  41. Rotate(E29, β, Line[E30, E28]) → E29'
  42. Polygon(E30, E28, E29') → wielokąt5, e29', e30_1, e28_2
  43. Rotate(E6, β, Line[E18, E30]) → E6'
  44. Polygon(E6', E18, E30) → wielokąt6, e30_2, e6', e18_2
  45. Rotate(Rotate[E24, β, Line[E6, E30]], β, Line[E18, E30]) → C
  46. Rotate(Rotate[E17, β, Line[E6, E30]], β, Line[E18, E30]) → D
  47. Rotate(Rotate[E29, β, Line[E6, E30]], β, Line[E18, E30]) → E
  48. Polygon(E30, E, D, C, E6') → wielokąt7, e30_3, e, d, c, e6'_1
  49. Rotate(Rotate[E14, β, Line[E26, E18]], β, Line[E22, E18]) → F
  50. Rotate(Rotate[E12, β, Line[E26, E18]], β, Line[E22, E18]) → G
  51. Rotate(Rotate[E6, β, Line[E26, E18]], β, Line[E22, E18]) → H
  52. Polygon(E18, H, G, F, E26') → wielokąt8, e18_3, h, g, f_1, e26'_1
  53. Rotate(Rotate[E20, β, Line[E10, E22]], β, Line[E4, E22]) → I
  54. Rotate(Rotate[E2, β, Line[E10, E22]], β, Line[E4, E22]) → J
  55. Rotate(Rotate[E26, β, Line[E10, E22]], β, Line[E4, E22]) → K
  56. Polygon(E10', E22, K, J, I) → wielokąt9, e10'_1, e22_3, k, j, i
  57. Rotate(Rotate[E27, β, Line[E16, E4]], β, Line[E28, E4]) → L
  58. Rotate(Rotate[E15, β, Line[E16, E4]], β, Line[E28, E4]) → M
  59. Rotate(Rotate[E10, β, Line[E16, E4]], β, Line[E28, E4]) → N
  60. Polygon(E4, N, M, L, E16') → wielokąt10, e4_3, n, m, l, e16'_1
  61. Rotate(Rotate[E5, β, Line[E29, E28]], β, Line[E30, E28]) → O
  62. Rotate(Rotate[E23, β, Line[E29, E28]], β, Line[E30, E28]) → P
  63. Rotate(Rotate[E16, β, Line[E29, E28]], β, Line[E30, E28]) → Q
  64. Polygon(E28, Q, P, O, E29') → wielokąt11, e28_3, q, p, o, e29'_1
  65. Rotate(Rotate[Rotate[E24, β, Line[E12, E6]], β, Line[E26, E18]], β, Line[E22, E18]) → R
  66. Polygon(G, H, R) → wielokąt12, r, g_1, h_1
  67. Rotate(Rotate[Rotate[E14, β, Line[E2, E26]], β, Line[E10, E22]], β, Line[E4, E22]) → S
  68. Polygon(K, S, J) → wielokąt13, j_1, k_1, s_1
  69. Rotate(Rotate[Rotate[E20, β, Line[E15, E10]], β, Line[E16, E4]], β, Line[E28, E4]) → T
  70. Polygon(N, T, M) → wielokąt14, m_1, n_1, t
  71. Rotate(Rotate[Rotate[E27, β, Line[E23, E16]], β, Line[E29, E28]], β, Line[E30, E28]) → U
  72. Polygon(P, Q, U) → wielokąt15, u, p_1, q_1
  73. Rotate(Rotate[Rotate[E5, β, Line[E17, E29]], β, Line[E6, E30]], β, Line[E18, E30]) → V
  74. Polygon(E, V, D) → wielokąt16, d_1, e_1, v
  75. Rotate(Rotate[Rotate[E25, β, Line[E24, E17]], β, Line[E6, E30]], β, Line[E18, E30]) → W
  76. Rotate(Rotate[Rotate[E11, β, Line[E5, E23]], β, Line[E29, E28]], β, Line[E30, E28]) → Z
  77. Rotate(Rotate[Rotate[E3, β, Line[E27, E15]], β, Line[E16, E4]], β, Line[E28, E4]) → A_1
  78. Rotate(Rotate[Rotate[E8, β, Line[E20, E2]], β, Line[E10, E22]], β, Line[E4, E22]) → B_1
  79. Rotate(Rotate[Rotate[E19, β, Line[E14, E12]], β, Line[E26, E18]], β, Line[E22, E18]) → C_1
  80. Polygon(D, W, C) → wielokąt17, c_1, d_2, w
  81. Polygon(G, C_1, F) → wielokąt18, f_2, g_2, c_2
  82. Polygon(J, B_1, I) → wielokąt19, i_1, j_2, b_1
  83. Polygon(M, A_1, L) → wielokąt20, l_1, m_2, a_1
  84. Polygon(P, Z, O) → wielokąt21, o_1, p_2, z_1
  85. Rotate(Rotate[Rotate[Rotate[E9, β, Line[E3, E15]], β, Line[E27, E15]], β, Line[E16, E4]], β, Line[E28, E4]) → D_1
  86. Rotate(Rotate[Rotate[Rotate[E8, β, Line[E3, E15]], β, Line[E27, E15]], β, Line[E16, E4]], β, Line[E28, E4]) → E_1
  87. Rotate(Rotate[Rotate[Rotate[E20, β, Line[E3, E15]], β, Line[E27, E15]], β, Line[E16, E4]], β, Line[E28, E4]) → F_1
  88. Polygon(M, F_1, E_1, D_1, A_1) → wielokąt22, m_3, f_3, e_2, d_3, a_2
  89. Rotate(Rotate[Rotate[Rotate[E21, β, Line[E11, E23]], β, Line[E5, E23]], β, Line[E29, E28]], β, Line[E30, E28]) → G_1
  90. Rotate(Rotate[Rotate[Rotate[E3, β, Line[E11, E23]], β, Line[E5, E23]], β, Line[E29, E28]], β, Line[E30, E28]) → H_1
  91. Rotate(Rotate[Rotate[Rotate[E27, β, Line[E11, E23]], β, Line[E5, E23]], β, Line[E29, E28]], β, Line[E30, E28]) → I_1
  92. Polygon(P, I_1, H_1, G_1, Z) → wielokąt23, p_3, i_2, h_2, g_3, z_2
  93. Rotate(Rotate[Rotate[Rotate[E13, β, Line[E25, E17]], β, Line[E24, E17]], β, Line[E6, E30]], β, Line[E18, E30]) → J_1
  94. Rotate(Rotate[Rotate[Rotate[E11, β, Line[E25, E17]], β, Line[E24, E17]], β, Line[E6, E30]], β, Line[E18, E30]) → K_1
  95. Rotate(Rotate[Rotate[Rotate[E5, β, Line[E25, E17]], β, Line[E24, E17]], β, Line[E6, E30]], β, Line[E18, E30]) → L_1
  96. Polygon(W, D, L_1, K_1, J_1) → wielokąt24, w_1, d_4, l_2, k_2, j_3
  97. Rotate(Rotate[Rotate[Rotate[E1, β, Line[E19, E12]], β, Line[E14, E12]], β, Line[E26, E18]], β, Line[E22, E18]) → M_1
  98. Rotate(Rotate[Rotate[Rotate[E25, β, Line[E19, E12]], β, Line[E14, E12]], β, Line[E26, E18]], β, Line[E22, E18]) → N_1
  99. Rotate(Rotate[Rotate[Rotate[E24, β, Line[E19, E12]], β, Line[E14, E12]], β, Line[E26, E18]], β, Line[E22, E18]) → O_1
  100. Polygon(C_1, G, O_1, N_1, M_1) → wielokąt25, c_3, g_4, o_2, n_2, m_4
  101. Rotate(Rotate[Rotate[Rotate[E7, β, Line[E8, E2]], β, Line[E20, E2]], β, Line[E10, E22]], β, Line[E4, E22]) → P_1
  102. Rotate(Rotate[Rotate[Rotate[E19, β, Line[E8, E2]], β, Line[E20, E2]], β, Line[E10, E22]], β, Line[E4, E22]) → Q_1
  103. Rotate(Rotate[Rotate[Rotate[E14, β, Line[E8, E2]], β, Line[E20, E2]], β, Line[E10, E22]], β, Line[E4, E22]) → R_1
  104. Polygon(J, R_1, Q_1, P_1, B_1) → wielokąt26, j_4, r_1, q_2, p_4, b_2
  105. Rotate(Rotate[Rotate[Rotate[Rotate[E7, β, Line[E9, E8]], β, Line[E3, E15]], β, Line[E27, E15]], β, Line[E16, E4]], β, Line[E28, E4]) → S_1
  106. Polygon(S_1, D_1, E_1) → wielokąt27, e_3, s_2, d_5
  107. Rotate(Rotate[Rotate[Rotate[Rotate[E9, β, Line[E21, E3]], β, Line[E11, E23]], β, Line[E5, E23]], β, Line[E29, E28]], β, Line[E30, E28]) → T_1
  108. Polygon(G_1, H_1, T_1) → wielokąt28, t_1, g_5, h_3
  109. Rotate(Rotate[Rotate[Rotate[Rotate[E21, β, Line[E13, E11]], β, Line[E25, E17]], β, Line[E24, E17]], β, Line[E6, E30]], β, Line[E18, E30]) → U_1
  110. Polygon(J_1, K_1, U_1) → wielokąt29, u_1, j_5, k_3
  111. Rotate(Rotate[Rotate[Rotate[Rotate[E13, β, Line[E1, E25]], β, Line[E19, E12]], β, Line[E14, E12]], β, Line[E26, E18]], β, Line[E22, E18]) → V_1
  112. Polygon(M_1, N_1, V_1) → wielokąt30, v_1, m_5, n_3
  113. Rotate(Rotate[Rotate[Rotate[Rotate[E1, β, Line[E7, E19]], β, Line[E8, E2]], β, Line[E20, E2]], β, Line[E10, E22]], β, Line[E4, E22]) → W_1
  114. Polygon(P_1, Q_1, W_1) → wielokąt31, w_2, p_5, q_3
  115. Rotate(Rotate[Rotate[Rotate[Rotate[Rotate[E9, β, Line[E7, E1]], β, Line[E7, E19]], β, Line[E8, E2]], β, Line[E20, E2]], β, Line[E10, E22]], β, Line[E4, E22]) → Z_1
  116. Rotate(Rotate[Rotate[Rotate[Rotate[Rotate[E21, β, Line[E7, E1]], β, Line[E7, E19]], β, Line[E8, E2]], β, Line[E20, E2]], β, Line[E10, E22]], β, Line[E4, E22]) → A_2
  117. Rotate(Rotate[Rotate[Rotate[Rotate[Rotate[E13, β, Line[E7, E1]], β, Line[E7, E19]], β, Line[E8, E2]], β, Line[E20, E2]], β, Line[E10, E22]], β, Line[E4, E22]) → B_2
  118. Polygon(B_2, W_1, P_1, Z_1, A_2) → wielokąt32, b_3, w_3, p_6, z_3, a_3

AI Drawing Prompt Reference

Draw a 3D icosidodecahedron, an Archimedean solid with 20 equilateral triangles and 12 regular pentagons. Start with a base triangle and its plane, then construct a line perpendicular to the plane through one vertex. Rotate the triangle around this line by a specific angle to generate a set of new vertices. Use these vertices to form pentagonal and triangular faces, then apply further rotations to build additional faces, finally assembling all faces into a symmetric polyhedron.
The prompt above was compiled from the original drawing commands and is for reference only.