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Rhombic Tori

Rhombic Tori
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2017-06-23 05:12
PolygonAngleSolid GeometryTransformationGeometric Construction
This example demonstrates the construction of rhombic tori and the patterns of their angle measures. It involves core mathematical concepts like polygons, angles, solid geometry, and transformations. The construction uses commands such as Rotate, Line, Mirror, Polygon, RemoveUndefined, Length, and First in GeoGebra. It is suitable for high school students and above to explore geometric transformations and properties of solid figures.

GeoGebra Commands & Parameters

  1. Rotate(B, (360 * ° / N), A) → B'
  2. Rotate(B', (360 * ° / N), A) → B''
  3. Rotate(B, first, B') → B'_1
  4. Line(B'_1, B) → f
  5. Mirror(B', f) → B''_1
  6. Polygon(B', B, B''_1, B'_1) → poly1, b', b, b''_1, b'_1
  7. Rotate(B', first, B'') → B''_2
  8. Rotate(B''_1, (360 * ° / N), A) → C
  9. Line(C, B'_1) → g
  10. Mirror(B', g) → B''_3
  11. Line(B''_3, B''_1) → h
  12. Mirror(B'_1, h) → D
  13. Polygon(B'', B', C, B''_2) → poly2, b'', b'_2, c, b''_2
  14. Polygon(B', B'_1, B''_3, C) → poly3, b'_3, b'_4, b''_3, c_1
  15. Polygon(B'_1, B''_1, D, B''_3) → poly4, b'_5, b''_4, d, b''_5
  16. Rotate(B', (-((360 * ° / N))), A) → B''_4
  17. Rotate(B'_1, (-((360 * ° / N))), A) → E
  18. Rotate(B''_3, (-((360 * ° / N))), A) → F
  19. Rotate(C, (-((360 * ° / N))), A) → C'
  20. Polygon(B''_4, E, F, C') → poly3', b''_6, e, f_1, c'
  21. Line(D, F) → i
  22. Mirror(B''_1, i) → G
  23. Polygon(F, B''_1, D, G) → poly5, f_2, b''_7, d_1, g_1
  24. Rotate(B'_1, (-((360 * ° / N))), A) → H
  25. Rotate(B''_1, (-((360 * ° / N))), A) → I
  26. Rotate(D, (-((360 * ° / N))), A) → D'
  27. Rotate(B''_3, (-((360 * ° / N))), A) → J
  28. Polygon(H, I, D', J) → poly4', h_1, i_1, d', j
  29. Line(G, D') → k
  30. Mirror(F, k) → F'
  31. Polygon(D', F', G, F) → poly6, d'_1, f', g_2, f_3
  32. Rotate(F, (-((360 * ° / N))), A) → F'_1
  33. Rotate(B''_1, (-((360 * ° / N))), A) → K
  34. Rotate(D, (-((360 * ° / N))), A) → D'_1
  35. Rotate(G, (-((360 * ° / N))), A) → G'
  36. Polygon(F'_1, K, D'_1, G') → poly5', f'_1, k_1, d'_2, g'
  37. Line(F', G') → l
  38. Mirror(D', l) → D''
  39. Polygon(G', D'', F', D') → poly7, g'_1, d'', f'_2, d'_3
  40. Rotate(D', (-((360 * ° / N))), A) → D''_1
  41. Rotate(F', (-((360 * ° / N))), A) → F''
  42. Rotate(G, (-((360 * ° / N))), A) → G'_1
  43. Rotate(F, (-((360 * ° / N))), A) → F'_2
  44. Polygon(D''_1, F'', G'_1, F'_2) → poly6', d''_1, f'', g'_2, f'_3
  45. Line(D'', F'') → m
  46. Mirror(G', m) → G''
  47. Polygon(F'', G', D'', G'') → poly8, f''_1, g'_3, d''_2, g''
  48. Rotate(G', (-((360 * ° / N))), A) → G''_1
  49. Rotate(D'', (-((360 * ° / N))), A) → D'''
  50. Rotate(F', (-((360 * ° / N))), A) → F''_1
  51. Rotate(D', (-((360 * ° / N))), A) → D''_2
  52. Polygon(G''_1, D''', F''_1, D''_2) → poly7', g''_1, d''', f''_2, d''_3
  53. Line(G'', D''') → n
  54. Mirror(F'', n) → F'''
  55. Polygon(D''', F'', G'', F''') → poly9, d'''_1, f''_3, g''_2, f'''
  56. Rotate(F'', (-((360 * ° / N))), A) → F'''_1
  57. Rotate(G', (-((360 * ° / N))), A) → G''_2
  58. Rotate(D'', (-((360 * ° / N))), A) → D'''_1
  59. Rotate(G'', (-((360 * ° / N))), A) → G'''
  60. Polygon(F'''_1, G''_2, D'''_1, G''') → poly8', f'''_1, g''_3, d'''_2, g'''
  61. Line(G''', F''') → p
  62. Mirror(D''', p) → L
  63. Polygon(G''', D''', F''', L) → poly10, g'''_1, d'''_3, f'''_2, l_1
  64. Rotate(D''', (-((360 * ° / N))), A) → M_1
  65. Rotate(F'', (-((360 * ° / N))), A) → F'''_2
  66. Rotate(G'', (-((360 * ° / N))), A) → G'''_1
  67. Rotate(F''', (-((360 * ° / N))), A) → O
  68. Polygon(M_1, F'''_2, G'''_1, O) → poly9', m_1, f'''_3, g'''_2, o
  69. Line(O, L) → q
  70. Mirror(G''', q) → P
  71. Polygon(O, P, L, G''') → poly11, o_1, p_1, l_2, g'''_3
  72. RemoveUndefined(Sequence[If[Element[list1, i] < 180°, Element[list1, i]], i, 1, Length[list1]]) → list2
  73. Length(list2) → M
  74. First({poly2, poly3, poly4, poly5, poly6, poly7, poly8, poly9, poly10, poly11}, M) → list3
  75. Rotate(list3, (360 * ° / N), A) → list3'
  76. Rotate(list3', (360 * ° / N), A) → list3''
  77. Rotate(list3'', (360 * ° / N), A) → list3'''
  78. Rotate(list3''', (360 * ° / N), A) → list4
  79. Rotate(list4, (360 * ° / N), A) → list4'
  80. Rotate(list4', (360 * ° / N), A) → list4''
  81. Rotate(list4'', (360 * ° / N), A) → list4'''
  82. Rotate(list4''', (360 * ° / N), A) → list5
  83. Rotate(list5, (360 * ° / N), A) → list5'
  84. Rotate(list5', (360 * ° / N), A) → list5''
  85. Rotate(list5'', (360 * ° / N), A) → list5'''