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Icosahedron and Balloon

Icosahedron and Balloon
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2020-06-23 15:48
Solid GeometryCalculusGeometric Construction
This example demonstrates the geometric relationship between an icosahedron and a balloon, showing surface transformations. It involves core math concepts like solid geometry, integral calculus, and variance analysis. GeoGebra commands used include Intersect, Point, Sphere, etc. Suitable for high school students and math enthusiasts exploring spatial geometry and calculus applications.

GeoGebra Commands & Parameters

  1. Intersect(zAxis, xAxis) → A
  2. Point(zAxis) → B
  3. Sphere(A, B) → a
  4. IntersectPath(xOyPlane, a) → c
  5. Intersect(xAxis, c) → C, D
  6. Intersect(yAxis, c) → E, F
  7. Intersect(xAxis, zAxis) → H
  8. Rotate(C, 72°, A, xOyPlane) → C'
  9. Angle(C, A, C', xOyPlane) → α
  10. Rotate(C', 72°, A, xOyPlane) → C''
  11. Angle(C', A, C'', xOyPlane) → β
  12. Rotate(C'', 72°, A, xOyPlane) → C'''
  13. Angle(C'', A, C''', xOyPlane) → γ
  14. Rotate(C''', 72°, A, xOyPlane) → G
  15. Angle(C''', A, G, xOyPlane) → δ
  16. CircleArc(A, B, C''') → d
  17. CircleArc(A, B, G) → e
  18. CircleArc(A, B, C) → f
  19. CircleArc(A, B, C') → g
  20. CircleArc(A, B, C'') → h
  21. Intersect(zAxis, a) → I, J
  22. Rotate(C, 36°, A, xOyPlane) → C'_1
  23. Angle(C, A, C'_1, xOyPlane) → ε
  24. Rotate(C'_1, 72°, A, xOyPlane) → K
  25. Angle(C'_1, A, K, xOyPlane) → ζ
  26. Rotate(K, 72°, A, xOyPlane) → K'
  27. Angle(K, A, K', xOyPlane) → η
  28. Rotate(K', 72°, A, xOyPlane) → K''
  29. Angle(K', A, K'', xOyPlane) → θ
  30. Rotate(K'', 72°, A, xOyPlane) → K'''
  31. Angle(K'', A, K''', xOyPlane) → ι
  32. CircleArc(A, I, K'') → k
  33. Point(zAxis) → L
  34. CircleArc(L, K', I) → p
  35. CircleArc(L, K''', I) → q
  36. CircleArc(L, K, I) → r
  37. CircleArc(L, C'_1, I) → s
  38. Distance(A, B) → AB
  39. Sphere(B, i) → j
  40. IntersectConic(j, a) → t
  41. Intersect(f, t) → M, N, ,
  42. Intersect(g, t) → O, P, ,
  43. Intersect(h, t) → Q, R, ,
  44. Intersect(t, d) → S, T, ,
  45. Intersect(t, e) → U, V, ,
  46. Sphere(I, i) → l
  47. IntersectConic(l, a) → c_1
  48. Intersect(q, c_1) → W, Z, ,
  49. Intersect(c_1, s) → A_1, B_1, ,
  50. Intersect(c_1, k) → C_1, D_1, ,
  51. Intersect(c_1, r) → E_1, F_1, ,
  52. Intersect(p, c_1) → G_1, H_1, ,
  53. CircleArc(L, B, N) → d_1
  54. CircleArc(L, N, U) → e_1
  55. CircleArc(L, B, U) → f_1
  56. CircleArc(L, N, W) → g_1
  57. Point(g_1) → I_1
  58. CircleArc(L, U, I_1) → h_1
  59. CircleArc(L, S, U) → k_1
  60. CircleArc(L, N, P) → p_1
  61. CircleArc(L, B, P) → q_1
  62. CircleArc(L, S, B) → r_1
  63. CircleArc(L, B, R) → s_1
  64. CircleArc(L, R, S) → t_1
  65. CircleArc(L, R, P) → c_2
  66. CircleArc(L, P, F_1) → d_2
  67. CircleArc(L, R, F_1) → e_2
  68. CircleArc(H, R, G_1) → f_2
  69. CircleArc(H, S, G_1) → g_2
  70. CircleArc(H, F_1, G_1) → h_2
  71. Intersect(t, k_1, 2) → J_1
  72. CircleArc(H, J_1, C_1) → k_2
  73. CircleArc(H, U, C_1) → p_2
  74. CircleArc(H, C_1, I_1) → q_2
  75. CircleArc(H, C_1, G_1) → r_2
  76. CircleArc(H, I_1, B_1) → s_2
  77. CircleArc(H, I_1, I) → t_2
  78. CircleArc(H, C_1, I) → c_3
  79. CircleArc(H, I, B_1) → d_3
  80. CircleArc(H, G_1, I) → e_3
  81. CircleArc(H, B_1, F_1) → f_3
  82. CircleArc(H, F_1, I) → g_3
  83. CircleArc(H, B_1, P) → h_3
  84. CircleArc(H, B_1, N) → k_3
  85. Intersect(t_1, t, 1) → K_1
  86. Polygon(B, R, S) → poly1, s_3, b_1, r_3
  87. Intersect(t, k_1, 1) → L_1
  88. Polygon(B, S, U) → poly2, u, b_2, s_4
  89. Polygon(S, C_1, L_1) → poly3, l_1, s_5, c_4
  90. Polygon(S, R, G_1) → poly4, g_4, s_6, r_4
  91. Polygon(S, G_1, C_1) → poly5, c_5, s_7, g_5
  92. Polygon(P, K_1, B) → poly6, b_3, p_3, k_4
  93. Polygon(B, N, U) → poly7, u_1, b_4, n
  94. Polygon(U, I_1, C_1) → poly8, c_6, u_2, i_1
  95. Polygon(P, N, B) → poly9, b_5, p_4, n_1
  96. Polygon(I_1, B_1, N) → poly10, n_2, i_2, b_6
  97. Polygon(N, P, B_1) → poly11, b_7, n_3, p_5
  98. Polygon(B_1, F_1, P) → poly12, p_6, b_8, f_4
  99. Polygon(P, F_1, K_1) → poly13, k_5, p_7, f_5
  100. Polygon(F_1, G_1, R) → poly14, r_5, f_6, g_6
  101. Polygon(F_1, I, G_1) → poly15, g_7, f_7, i_3
  102. Intersect(c_1, h_3, 1) → M_1
  103. Polygon(B_1, F_1, I) → poly16, i_4, b_9, f_8
  104. Polygon(I_1, B_1, I) → poly17, i_5, i_6, b_{10}
  105. Polygon(G_1, I, C_1) → poly18, c_7, g_8, i_7
  106. Intersect(c_3, q_2, 1) → N_1
  107. Polygon(I_1, C_1, I) → poly19, i_8, i_9, c_8
  108. Polygon(L_1, N, I_1) → t1, j_1, m_1, n_4