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Dynamic Proof of Euler's Formula for Polyhedra

Dynamic Proof of Euler's Formula for Polyhedra
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2014-07-15 20:00
VectorTrigonometrySolid GeometryCalculusGeometric Construction
In collections Surface Area and Volume Surface Area and Volume Surface Area and Volume
This example dynamically demonstrates the topological proof of Euler's formula for polyhedra, involving core concepts like vertices(V), edges(E), faces(F), and the formula V-E+F=2. It uses GeoGebra commands such as Intersect, Polygon, Segment, Vector, Translate, Circle, and Rotate to construct polyhedral models. Suitable for high school students and math enthusiasts studying topology and geometric proofs.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Polygon(A, B, 4) → mnohoúhelník1, a, b, c, d, C, D
  4. Segment(A, C) → e
  5. Segment(B, D) → f
  6. Intersect(f, e) → E
  7. Vector(A, E) → u
  8. Translate(B, u) → B'
  9. Translate(C, u) → C'
  10. Translate(D, u) → D'
  11. Segment(D, D') → g
  12. Segment(D', C') → h
  13. Segment(C', B') → i
  14. Segment(B', B) → j
  15. Segment(C, C') → k
  16. Segment(D', E) → l
  17. Segment(B', E) → m
  18. Segment(E, A) → n
  19. Polygon(D, C, C', D') → mnohoúhelník2, d_1, c_1, c', d'
  20. Polygon(B, B', C', C) → mnohoúhelník3, b_1, b', c'_1, c_2
  21. Intersect(c, d_1) → F
  22. Intersect(l, c) → G
  23. Intersect(m, b) → H
  24. Segment(E, H) → s
  25. Segment(E, G) → p
  26. Segment(A, E) → q
  27. Circle(C, A) → r
  28. Rotate(A, 90*°, C) → A'
  29. Rotate(A', 90*°, C) → A''
  30. Rotate(A'', 90*°, C) → A'''
  31. Vector(D, A''') → v
  32. Vector(C, A'') → w
  33. Vector(B, A') → t
  34. Translate(D, Vector[((o - 0.1) * v)]) → D'_1
  35. Translate(C, Vector[((o - 0.1) * w)]) → C'_1
  36. Translate(B, Vector[((o - 0.1) * t)]) → B'_1
  37. Segment(D'_1, D') → a_1
  38. Segment(C', C'_1) → e_1
  39. Segment(D'_1, A) → f_1
  40. Segment(D'_1, C'_1) → g_1
  41. Segment(B', B'_1) → h_1
  42. Segment(A, B'_1) → i_1
  43. Segment(C'_1, B'_1) → j_1
  44. Segment(A''', A) → k_1
  45. Segment(A''', A'') → l_1
  46. Segment(A, A') → m_1
  47. Segment(A'', A') → n_1
  48. Segment(A''', D') → p_1
  49. Segment(C', A'') → q_1
  50. Segment(A', B') → r_1
  51. Polygon(D'_1, C'_1, C', D') → mnohoúhelník4, d'_1, c'_2, c'_3, d'_2
  52. Polygon(C'_1, B'_1, B', C') → mnohoúhelník5, c'_4, b'_1, b'_2, c'_5
  53. Intersect(g_1, l) → I
  54. Segment(E, I) → s_1
  55. Segment(D', E) → t_1
  56. Segment(E, B') → a_2
  57. Intersect(m, j_1) → J
  58. Segment(E, J) → b_2
  59. Segment(A''', E) → d_2
  60. Segment(E, C') → e_2
  61. Segment(D', A'') → f_2
  62. Segment(C', A') → g_2
  63. Segment(B', A) → h_2
  64. Translate(A'', Vector[((3.2 - o) * w)]) → A'''_1
  65. Translate(A'', Vector[(-w)]) → A'''_2
  66. Translate(A', Vector[((3.2 - o) * t)]) → A''_1
  67. Translate(A', Vector[(-t)]) → A''_2
  68. Translate(A''', Vector[((3.2 - o) * v)]) → K
  69. Translate(A''', Vector[(-v)]) → L
  70. Segment(K, A) → i_2
  71. Segment(A, A''_1) → j_2
  72. Segment(K, A'''_1) → k_2
  73. Segment(A'''_1, A''_1) → l_2
  74. Segment(K, D') → m_2
  75. Segment(C', A'''_1) → n_2
  76. Segment(B', A''_1) → p_2
  77. Segment(L, A) → q_2
  78. Segment(A, A''_2) → r_2
  79. Segment(A'''_2, A''_2) → s_2
  80. Segment(L, A'''_2) → t_2
  81. Segment(L, D') → a_3
  82. Segment(A'''_2, C') → b_3
  83. Segment(A''_2, B') → c_3
  84. Intersect(k_2, t_1) → M
  85. Segment(M, E) → d_3
  86. Segment(D', M) → e_3
  87. Segment(D', E) → f_3
  88. Intersect(a_2, l_2) → N
  89. Segment(E, N) → h_3
  90. Segment(E, B') → i_3
  91. Segment(E, B') → g_3
  92. Polygon(D', K, A'''_1, C') → mnohoúhelník6, d'_3, k_3, a'''_1, c'_6
  93. Polygon(A'''_1, A''_1, B', C') → mnohoúhelník7, a'''_2, a''_1, b'_3, c'_7
  94. Polygon(L, A'''_2, C', D') → mnohoúhelník8, l_3, a'''_3, c'_8, d'_4
  95. Polygon(A'''_2, A''_2, B', C') → mnohoúhelník9, a'''_4, a''_2, b'_4, c'_9
  96. Polygon(A, A''_2, A'''_2, L) → mnohoúhelník10, a_4, a''_3, a'''_5, l_4

AI Drawing Prompt Reference

Draw a dynamic topological proof of Euler's formula for polyhedra. Start with a quadrilateral base and draw its diagonals intersecting at the center. From the center, create vectors to each base vertex and translate the base vertices along these vectors to form a corresponding upper quadrilateral, connecting corresponding vertices to form a prism. Inside the prism, connect the center to each translated vertex and base vertex to form internal triangular faces. Then, using one base vertex as a center, rotate to generate three auxiliary points, and use these to define three translation vectors. Translate each base vertex along its corresponding vector to create offset vertices. Connect these offset vertices to the original vertices and auxiliary points, forming multiple triangles and quadrilaterals that illustrate the topological relationships among faces, edges, and vertices, ultimately demonstrating Euler's formula V-E+F=2.
The prompt above was compiled from the original drawing commands and is for reference only.