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Module 7: Rhombi and Trapezoids from Hexagonal Cuts

Module 7: Rhombi and Trapezoids from Hexagonal Cuts
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2020-09-15 19:57
PolygonSolid GeometryTransformationGeometric Construction
This example demonstrates how three hexagonal cross-sections (cuts) in solid geometry produce recognizable quadrilateral patterns such as rhombi and trapezoids. By organizing vertices through consistent geometric transformations, it reveals the underlying symmetry and structural constraints that make these quadrilaterals appear as the cuts vary. Key concepts include 3D cross-sections, relationships tied to regular polygon geometry, classification of quadrilaterals (rhombus and trapezoid), and the use of rigid motions (rotation/translation) to build a coherent configuration. Construction highlights in GeoGebra: Circle and Point define foundational circles and key centers (e.g., r and O), while Line and Segment mark edges and intersections. Rotate and Translate generate the required transformed positions; Midpoint provides midpoint constraints needed for the cross-section geometry. A Cube is used to create the 3D object, and Polygon connects the computed cross-section vertices (e.g., faces labeled like caraABCD, caraADHE, caraABFE, caraBCGF, caraCDHG, caraEFGH). The corresponding “arista*” edges then visually form the rhombus/trapezoid structures. Suitable for advanced middle school through introductory university geometry, especially for teaching and exploring solid geometry sections and transformation-based reasoning.

GeoGebra Commands & Parameters

  1. Circle(O, r, xOyPlane) → c
  2. Point(c) → A1
  3. Rotate(A1, π / 2, O) → A2
  4. Midpoint(A1, A2) → B5
  5. Cube(A1, A2, zAxis) → a, A3, caraBCGF, caraCDHG, caraEFGH, aristaAD_1, aristaAB, aristaBC_1, aristaCD, aristaAE_1, aristaDH, arista, A4, aristaEF, aristaBF, aristaFG_1, aristaCG_1, aristaGH_1, A5, A6, A7, A8, caraABCD, caraADHE, caraABFE
  6. Midpoint(A6, A2) → B2
  7. Polygon(B5, B2, A2) → t3, a2, s_2, t_2
  8. Midpoint(A4, A8) → B4
  9. Midpoint(A1, A7) → centro
  10. Midpoint(A3, A7) → B3
  11. Midpoint(A1, A5) → B1
  12. Polygon(B5, B1, A1) → t1, a1, n_2, p_2
  13. Midpoint(A3, A4) → B7
  14. Polygon(A3, B3, B7) → t4, i_3, j_3, k_3
  15. Midpoint(A7, A8) → BC
  16. Polygon(BC, B4, A8) → t5, l_3, m_3, n_3
  17. Midpoint(A4, A1) → B8
  18. Polygon(A3, A2, A1, B8, B7) → p2, j_4, k_4, l_4, m_4, n_4
  19. Polygon(centro, BC, B4, B8) → c8, s_3, t_3, a_3, b_3
  20. Midpoint(A8, A5) → BD
  21. Polygon(B8, A1, B1, BD, A8, B4) → p3, c_3, d_3, e_3, f_4, g_4, h_4
  22. Polygon(BD, A8, BC) → t2, q_2, r_2, a8
  23. Polygon(B1, BD, BC, B5) → c5, j_2, k_2, l_2, m_2
  24. Midpoint(A6, A7) → BB
  25. Polygon(B8, BB, B3, B7) → c7, p_3, q_3, r_3, i_4
  26. Polygon(A2, A3, B3, BB, B2) → p1, e_2, a3, f_3, g_3, h_3
  27. Polygon(B5, B2, BB, centro) → c6, a_2, b_2, c_2, d_2
  28. Midpoint(A2, A3) → B6
  29. Vector(A5, A8) → u4
  30. Polygon(A2, A3, A8, A5) → c4, k_1, l_1, m_1, n_1
  31. Midpoint(A5, A6) → BA
  32. Polygon(A4, A1, A6, A7) → c3, g_1, h_1, i_1, j_1
  33. Polygon(A1, A3, A7, A5) → c2, q, s, e, f_1
  34. Vector(B1, B3) → u1
  35. Vector(B1, B3) → u3
  36. Sequence(Max[0, Min[1, t - s]], s, 0, 10) → sl
  37. Line(A5, A3) → k
  38. Segment(A7, A1) → h
  39. Midpoint(A1, A6) → O1
  40. Midpoint(A2, A7) → O2
  41. Midpoint(A3, A8) → O3
  42. Midpoint(A8, A1) → O4
  43. Midpoint(A5, A7) → O5
  44. Vector(B8, BB) → u2
  45. Polygon(A4, A2, A6, A8) → c1, l, m, n, p
  46. Polygon(A4, BD, A6, B6) → c3_1, e_1, f_2, g_2, h_2
  47. Translate(c1, u1) → c1'
  48. Polygon(A7, BD, A1, B6) → c2_1, f, g, i, j
  49. Translate(c3, u3) → c3'
  50. Translate(c2, u2) → c2'
  51. Translate(c1', Vector[((-sl(1)) * u1)]) → c1''
  52. Translate(c2', Vector[((-sl(2)) * u2)]) → c2''
  53. Translate(c3', Vector[((-sl(3)) * u3)]) → c3''
  54. Line(B1, B2) → t_1
  55. Translate(c4, u4) → c4'
  56. Translate(c4', Vector[((-sl(4)) * u4)]) → c4''
  57. Vector(A1, A8) → v
  58. Line(O1, O3) → bc
  59. Line(O5, O) → bb
  60. Line(O2, O4) → bd
  61. Vector(A2, A8) → v3
  62. Vector(BC, B5) → v1_1
  63. Vector(A6, A1) → v2_1
  64. Polygon(BC, B3, B6, B5, B1, BD) → po1, p_1, b3, b6, b5, b1, q_1
  65. Polygon(BC, B4, B8, B5, B2, BB) → po2, r_1, b4, b8, s_1, b2, a_1
  66. Polygon(BB, B3, B7, B8, B1, BA) → po3, b_1, c_1, b7, d_1, i_2, ba
  67. Translate(po1, u1) → po1'
  68. Translate(po2, u2) → po2'
  69. Translate(po3, u1) → po3'
  70. Translate(po1', Vector[((-sl(1)) * u1)]) → po1''
  71. Translate(po2', Vector[((-sl(2)) * u2)]) → po2''
  72. Translate(po3', Vector[((-sl(3)) * u3)]) → po3''
  73. Translate(ll1, u5) → ll2
  74. Rotate(ll2, (sl(6) * π), bd) → ll3
  75. Rotate(ll3, (sl(7) * π / 2), bb) → ll4
  76. Translate(ll2, Vector[((-sl(10)) * v)]) → ll5
  77. Translate(ll4, Vector[(sl(10) * v)]) → ll6

AI Drawing Prompt Reference

Draw a hexagonal prism and cut out rhombi and trapezoids from three different hexagonal cross-sections. Start by drawing the hexagonal base and sides of the prism, then use rotation and midpoints to construct the vertices of the rhombi and trapezoids. Finally, connect these vertices to form the rhombi and trapezoids.
The prompt above was compiled from the original drawing commands and is for reference only.