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Russian Football Geometry Construction

Russian Football Geometry Construction
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2017-08-26 04:56
PolygonVectorSolid GeometryMatrixGeometric Construction
This example dynamically demonstrates the geometric construction of a Russian football, revealing its complex symmetry and spatial relationships. It involves core mathematical concepts like vectors, matrix transformations, rotations, reflections, polygon construction, and orthocenters. The construction uses commands and objects such as Point, Vector, Mirror, Polygon, Rotate, Segment, Sphere, OrthogonalPlane, Circle, Center, Ray, Intersect, Distance from GeoGebra. Suitable for high school and above learners to understand the practical applications of spatial geometry, linear algebra, and transformation theory.

GeoGebra Commands & Parameters

  1. Point(zAxis) → H
  2. Vector(O, H) → u
  3. Mirror(A, yAxis) → A'
  4. Polygon(A, A', 8) → poly1, f, C, D, E, F, G, g, h, i, j, k, l, m, B
  5. Rotate(Translate[Lo, u], 22.5°, zAxis) → Lo'
  6. Point(Lo') → I
  7. Point(Lo') → J
  8. Point(Lo') → K
  9. Rotate(Lo', 90°, xAxis) → Lo''
  10. Rotate(Rotate[Lo', 90°, xAxis], 22.5°, yAxis) → liste1
  11. Rotate(Translate[Lo, u], 45°, zAxis) → poly1'
  12. Point(Lo') → L
  13. Point(Lo'') → M
  14. Segment(L, M) → n
  15. Rotate(Lo'', 90°, zAxis) → Lb
  16. Rotate(Lb, 90°, zAxis) → Lc
  17. Rotate(Lc, 90°, zAxis) → Ld
  18. Point(Lo') → N
  19. Sphere(O, N) → a
  20. Point(Lo') → P
  21. Point(Lb) → Q
  22. Segment(P, Q) → p
  23. Point(Lo'') → R
  24. Point(Lo'') → S
  25. OrthogonalPlane(O, xAxis) → d
  26. Point(Lb) → U
  27. Circle(N, S, U) → e
  28. Center(e) → V
  29. Sphere(O, N) → b
  30. Point(Lo'') → T
  31. Point(Lb) → W
  32. Segment(T, W) → t
  33. Ray(O, V) → c
  34. Intersect(c, b) → Z,
  35. Polygon(P, Q, U, Z, N) → poly2, p_1, q, u_1, z_1, n_1
  36. Segment(S, Z) → r
  37. Rotate(Z, 90°, zAxis) → Z'
  38. Rotate(Z', 90°, zAxis) → Z''
  39. Rotate(Z'', 90°, zAxis) → Z'''
  40. Point(Lo') → A_1
  41. Segment(A_1, Z''') → s
  42. Segment(R, Z''') → f_1
  43. Point(Ld) → B_1
  44. Segment(Z''', B_1) → g_1
  45. Point(Ld) → C_1
  46. Point(Lo') → D_1
  47. Segment(C_1, D_1) → h_1
  48. Point(Lo') → E_1
  49. Segment(E_1, Z'') → i_1
  50. Point(Ld) → F_1
  51. Segment(Z'', F_1) → j_1
  52. Point(Lc) → G_1
  53. Segment(Z'', G_1) → k_1
  54. Point(Lo') → H_1
  55. Segment(H_1, Z') → l_1
  56. Point(Lc) → I_1
  57. Segment(Z', I_1) → m_1
  58. Point(Lb) → J_1
  59. Segment(Z', J_1) → q_1
  60. Point(Lb) → K_1
  61. Point(Lc) → L_1
  62. Segment(K_1, L_1) → r_1
  63. Point(Lc) → M_1
  64. Point(Ld) → N_1
  65. Segment(M_1, N_1) → s_1
  66. Point(Ld) → O_1
  67. Point(Lo'') → P_1
  68. Segment(O_1, P_1) → t_1
  69. Point(Lc) → R_1
  70. Point(Lo') → S_1
  71. Segment(R_1, S_1) → a_1
  72. Point(Lo'') → Q_1
  73. Point(Lo'') → T_1
  74. Point(Lo'') → U_1
  75. Point(Lo'') → V_1
  76. Point(Lo'') → W_1
  77. Point(Lo'') → Z_1
  78. Point(Lo'') → A_2
  79. Point(Lo'') → B_2
  80. Rotate(Translate[poly1, u], 22.5°, zAxis) → poly4
  81. Rotate(poly4, 90°, xAxis) → poly4'
  82. Rotate(poly4', 90°, zAxis) → poly4''
  83. Rotate(poly4'', 90°, zAxis) → poly4'''
  84. Rotate(poly4', -90°, zAxis) → poly4''_1
  85. Mirror(poly4, O) → poly4'_1
  86. Distance(O, H) → distanceOH