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Hexagon Puzzle: Exploring Polygons and Symmetry in Geometric Construction

Hexagon Puzzle: Exploring Polygons and Symmetry in Geometric Construction
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2020-01-07 17:11
PolygonTransformationGeometric Construction
① This example demonstrates how to construct a complex geometric figure using a hexagon-based puzzle, combining multiple polygons, segments, and symmetry operations to create a visual puzzle effect. ② Core mathematical concepts include polygons (especially quadrilaterals and hexagons), centroid (barycenter), symmetry (reflection), translation, orthocenter, and function graphs, involving area, geometric transformations, and inequalities. ③ Construction highlights: Based on the construction summary, GeoGebra commands such as Point, Polygon, Centroid, Line, Mirror, and Segment are used, creating multiple points (A, B, C, D, E, F, G, H, I, J, K, L, M, N) and polygons (poly1, poly2, poly3, poly4), with mirroring and segment connections forming the puzzle structure. ④ Suitable for: Upper middle school to high school students, for teaching geometric transformations, symmetry, and polygon properties, as well as for mathematical exploration or puzzle games to stimulate interest in geometric structures.

GeoGebra Commands & Parameters

  1. Point(Circle[A, a]) → B
  2. Polygon(A, B, 6) → poly1, g, F, h, i, j, k, l, C, D, E
  3. Polygon(D, C, 6) → poly2, m, J, n, p, q, r, s, G, H, I
  4. Polygon(E, D, 6) → poly3, t, N, b, c, d, e, f_1, K, L, M
  5. Polygon(F, E, 6) → poly4, g_1, R, h_1, i_1, j_1, k_1, l_1, O, P, Q
  6. Polygon(A, F, 6) → poly5, m_1, V, n_1, p_1, q_1, r_1, s_1, S, T, U
  7. Polygon(B, A, 6) → poly6, t_1, B_1, a_1, b_1, c_1, d_1, e_1, W, Z, A_1
  8. Polygon(C, B, 6) → poly7, f_2, F_1, g_2, h_2, i_2, j_2, k_2, C_1, D_1, E_1
  9. Centroid(poly1) → G_1
  10. Centroid(poly2) → H_1
  11. Centroid(poly3) → I_1
  12. Centroid(poly4) → J_1
  13. Centroid(poly5) → K_1
  14. Centroid(poly6) → L_1
  15. Centroid(poly7) → M_1
  16. Line(N_1, O_1) → l_2
  17. Mirror(poly1, l_2) → poly1'
  18. Mirror(poly2, l_2) → poly2'
  19. Mirror(poly3, l_2) → poly3'
  20. Mirror(poly4, l_2) → poly4'
  21. Mirror(poly5, l_2) → poly5'
  22. Mirror(poly6, l_2) → poly6'
  23. Mirror(poly7, l_2) → poly7'
  24. Mirror(A, l_2) → A'
  25. Mirror(B, l_2) → B'
  26. Mirror(G_1, l_2) → P_1
  27. Mirror(H_1, l_2) → Q_1
  28. Mirror(I_1, l_2) → R_1
  29. Mirror(J_1, l_2) → S_1
  30. Mirror(K_1, l_2) → T_1
  31. Mirror(L_1, l_2) → U_1
  32. Mirror(M_1, l_2) → V_1
  33. Mirror(L, l_2) → L'
  34. Mirror(M, l_2) → M'
  35. Mirror(N, l_2) → N'
  36. Mirror(E_1, l_2) → W_1
  37. Mirror(E_1, l_2) → Z_1
  38. Mirror(E_1, l_2) → A_2
  39. Mirror(E_1, l_2) → B_2
  40. Mirror(E_1, l_2) → C_2
  41. Mirror(D_1, l_2) → D_2
  42. Mirror(A_1, l_2) → E_2
  43. Mirror(B_1, l_2) → F_2
  44. Mirror(U, l_2) → U'
  45. Mirror(V, l_2) → V'
  46. Mirror(Z, l_2) → Z'
  47. Mirror(T, l_2) → T'
  48. Mirror(R, l_2) → R'
  49. Mirror(Q, l_2) → Q'
  50. Mirror(P, l_2) → P'
  51. Mirror(H, l_2) → H'
  52. Mirror(I, l_2) → I'
  53. Mirror(J, l_2) → J'
  54. Mirror(N, l_2) → N'_1
  55. Mirror(G, l_2) → G'
  56. Mirror(D, l_2) → D'
  57. Mirror(C, l_2) → C'
  58. Mirror(B, l_2) → B'_1
  59. Mirror(F, l_2) → F'
  60. Mirror(D, l_2) → D'_1
  61. Mirror(E, l_2) → E'
  62. Line(N_1, G_2) → f
  63. Mirror(poly2, f) → poly2'_1
  64. Mirror(poly3, f) → poly3'_1
  65. Mirror(poly2', f) → poly2''
  66. Mirror(poly3', f) → poly3''
  67. Mirror(C, f) → C'_1
  68. Mirror(D, f) → D'_2
  69. Segment(C'_1, D'_2) → i'
  70. Mirror(D, f) → D'_3
  71. Mirror(E, f) → E'_1
  72. Segment(D'_3, E'_1) → j'
  73. Mirror(E, f) → E'_2
  74. Mirror(O, f) → O'
  75. Segment(E'_2, O') → m_2
  76. Mirror(O, f) → O'_1
  77. Mirror(F_1, f) → I_2
  78. Mirror(F_1, f) → J_2
  79. Mirror(C, f) → C'_2
  80. Segment(J_2, C'_2) → q_2
  81. Mirror(H_1, f) → K_2
  82. Mirror(I_1, f) → L_2
  83. Mirror(Q_1, f) → M_2
  84. Mirror(R_1, f) → N_2
  85. Mirror(L', f) → L''
  86. Mirror(M', f) → M''
  87. Mirror(N', f) → N''
  88. Mirror(Z_1, f) → P_2
  89. Mirror(A_2, f) → Q_2
  90. Mirror(B_2, f) → R_2
  91. Mirror(C_2, f) → S_2
  92. Mirror(H', f) → H''
  93. Mirror(I', f) → I''
  94. Mirror(J', f) → J''
  95. Mirror(N'_1, f) → T_2
  96. Mirror(G', f) → G''
  97. Mirror(D', f) → D''
  98. Mirror(C', f) → C''
  99. Mirror(D'_1, f) → U_2
  100. Mirror(E', f) → E''

AI Drawing Prompt Reference

Draw a puzzle-like geometric figure composed of seven regular hexagons arranged around a central point, each hexagon subdivided into smaller polygons, creating a honeycomb-like symmetric pattern. All hexagons share edges or vertices, radiating outward. Mark a point at the center of each hexagon to indicate its centroid. Then, choose a symmetry axis running through the entire figure, and reflect the whole structure—including all hexagons, inner polygons, and center points—across this axis, producing a mirrored copy that together with the original forms a complete puzzle. The drawing should clearly show segment connections and symmetry, emphasizing polygon properties and geometric transformations.
The prompt above was compiled from the original drawing commands and is for reference only.