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Hexagon Loop Patterns Investigation

Hexagon Loop Patterns Investigation
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2014-07-14 23:06
PolygonTransformation
This example demonstrates the exploration of pattern formation in polygon tiling through rotating and combining hexagons. Key math concepts include polygon construction, rotational transformation, parametric equations, and geometric logical reasoning. The construction uses commands like Polygon, Segment, Point, and Rotate from GeoGebra. Suitable for high school mathematics classes to develop spatial visualization, geometric intuition, and investigative skills.

GeoGebra Commands & Parameters

  1. Polygon(A_1, B_1, 6) → poly1_1, a_1, F_1, b_1, c_1, d_1, e_1, f_1, C_1, D_1, E_1
  2. Segment(E_1, D_1) → g_1
  3. Segment(D_1, C_1) → h_1
  4. Segment(C_1, B_1) → i_1
  5. Segment(B_1, A_1) → j_1
  6. Segment(A_1, F_1) → k_1
  7. Segment(F_1, E_1) → l_1
  8. Point(Circle[A, a_1]) → B
  9. Rotate(B + (b_1, 0), Angle[C_1 - B_1] + Angle[B - A] - Angle[B_1 - A_1], B) → C
  10. Rotate(C + (c_1, 0), Angle[D_1 - C_1] + Angle[C - B] - Angle[C_1 - B_1], C) → D
  11. Rotate(D + (d_1, 0), Angle[E_1 - D_1] + Angle[D - C] - Angle[D_1 - C_1], D) → E
  12. Rotate(E + (e_1, 0), Angle[F_1 - E_1] + Angle[E - D] - Angle[E_1 - D_1], E) → F
  13. Polygon(A, B, C, D, E, F) → poly1, a, b, c, d, e, f
  14. Point(Circle[G, a_1]) → H
  15. Rotate(H + (b_1, 0), Angle[C_1 - B_1] + Angle[H - G] - Angle[B_1 - A_1], H) → I
  16. Rotate(I + (c_1, 0), Angle[D_1 - C_1] + Angle[I - H] - Angle[C_1 - B_1], I) → J
  17. Rotate(J + (d_1, 0), Angle[E_1 - D_1] + Angle[J - I] - Angle[D_1 - C_1], J) → K
  18. Rotate(K + (e_1, 0), Angle[F_1 - E_1] + Angle[K - J] - Angle[E_1 - D_1], K) → L
  19. Polygon(G, H, I, J, K, L) → poly2, g, h, i, j, k, l
  20. Point(Circle[M, a_1]) → N
  21. Rotate(N + (b_1, 0), Angle[C_1 - B_1] + Angle[N - M] - Angle[B_1 - A_1], N) → O
  22. Rotate(O + (c_1, 0), Angle[D_1 - C_1] + Angle[O - N] - Angle[C_1 - B_1], O) → P
  23. Rotate(P + (d_1, 0), Angle[E_1 - D_1] + Angle[P - O] - Angle[D_1 - C_1], P) → Q
  24. Rotate(Q + (e_1, 0), Angle[F_1 - E_1] + Angle[Q - P] - Angle[E_1 - D_1], Q) → R
  25. Polygon(M, N, O, P, Q, R) → poly3, m, n, o, p, q, r
  26. Point(Circle[S, a_1]) → T
  27. Rotate(T + (b_1, 0), Angle[C_1 - B_1] + Angle[T - S] - Angle[B_1 - A_1], T) → U
  28. Rotate(U + (c_1, 0), Angle[D_1 - C_1] + Angle[U - T] - Angle[C_1 - B_1], U) → V
  29. Rotate(V + (d_1, 0), Angle[E_1 - D_1] + Angle[V - U] - Angle[D_1 - C_1], V) → W
  30. Rotate(W + (e_1, 0), Angle[F_1 - E_1] + Angle[W - V] - Angle[E_1 - D_1], W) → Z
  31. Polygon(S, T, U, V, W, Z) → poly4, s, t, u, v, w, z
  32. Point(Circle[G_1, a_1]) → H_1
  33. Rotate(H_1 + (b_1, 0), Angle[C_1 - B_1] + Angle[H_1 - G_1] - Angle[B_1 - A_1], H_1) → I_1
  34. Rotate(I_1 + (c_1, 0), Angle[D_1 - C_1] + Angle[I_1 - H_1] - Angle[C_1 - B_1], I_1) → J_1
  35. Rotate(J_1 + (d_1, 0), Angle[E_1 - D_1] + Angle[J_1 - I_1] - Angle[D_1 - C_1], J_1) → K_1
  36. Rotate(K_1 + (e_1, 0), Angle[F_1 - E_1] + Angle[K_1 - J_1] - Angle[E_1 - D_1], K_1) → L_1
  37. Polygon(G_1, H_1, I_1, J_1, K_1, L_1) → poly5, g_2, h_2, i_2, j_2, k_2, l_2
  38. Point(Circle[M_1, a_1]) → N_1
  39. Rotate(N_1 + (b_1, 0), Angle[C_1 - B_1] + Angle[N_1 - M_1] - Angle[B_1 - A_1], N_1) → O_1
  40. Rotate(O_1 + (c_1, 0), Angle[D_1 - C_1] + Angle[O_1 - N_1] - Angle[C_1 - B_1], O_1) → P_1
  41. Rotate(P_1 + (d_1, 0), Angle[E_1 - D_1] + Angle[P_1 - O_1] - Angle[D_1 - C_1], P_1) → Q_1
  42. Rotate(Q_1 + (e_1, 0), Angle[F_1 - E_1] + Angle[Q_1 - P_1] - Angle[E_1 - D_1], Q_1) → R_1
  43. Polygon(M_1, N_1, O_1, P_1, Q_1, R_1) → poly6, m_1, n_1, o_1, p_1, q_1, r_1
  44. Point(Circle[S_1, a_1]) → T_1
  45. Rotate(T_1 + (b_1, 0), Angle[C_1 - B_1] + Angle[T_1 - S_1] - Angle[B_1 - A_1], T_1) → U_1
  46. Rotate(U_1 + (c_1, 0), Angle[D_1 - C_1] + Angle[U_1 - T_1] - Angle[C_1 - B_1], U_1) → V_1
  47. Rotate(V_1 + (d_1, 0), Angle[E_1 - D_1] + Angle[V_1 - U_1] - Angle[D_1 - C_1], V_1) → W_1
  48. Rotate(W_1 + (e_1, 0), Angle[F_1 - E_1] + Angle[W_1 - V_1] - Angle[E_1 - D_1], W_1) → Z_1
  49. Polygon(S_1, T_1, U_1, V_1, W_1, Z_1) → poly7, s_1, t_1, u_1, v_1, w_1, z_1
  50. Point(Circle[A_2, a_1]) → B_2
  51. Rotate(B_2 + (b_1, 0), Angle[C_1 - B_1] + Angle[B_2 - A_2] - Angle[B_1 - A_1], B_2) → C_2
  52. Rotate(C_2 + (c_1, 0), Angle[D_1 - C_1] + Angle[C_2 - B_2] - Angle[C_1 - B_1], C_2) → D_2
  53. Rotate(D_2 + (d_1, 0), Angle[E_1 - D_1] + Angle[D_2 - C_2] - Angle[D_1 - C_1], D_2) → E_2
  54. Rotate(E_2 + (e_1, 0), Angle[F_1 - E_1] + Angle[E_2 - D_2] - Angle[E_1 - D_1], E_2) → F_2
  55. Polygon(A_2, B_2, C_2, D_2, E_2, F_2) → poly8, a_2, b_2, c_2, d_2, e_2, f_2
  56. Point(Circle[G_2, a_1]) → H_2
  57. Rotate(H_2 + (b_1, 0), Angle[C_1 - B_1] + Angle[H_2 - G_2] - Angle[B_1 - A_1], H_2) → I_2
  58. Rotate(I_2 + (c_1, 0), Angle[D_1 - C_1] + Angle[I_2 - H_2] - Angle[C_1 - B_1], I_2) → J_2
  59. Rotate(J_2 + (d_1, 0), Angle[E_1 - D_1] + Angle[J_2 - I_2] - Angle[D_1 - C_1], J_2) → K_2
  60. Rotate(K_2 + (e_1, 0), Angle[F_1 - E_1] + Angle[K_2 - J_2] - Angle[E_1 - D_1], K_2) → L_2
  61. Polygon(G_2, H_2, I_2, J_2, K_2, L_2) → poly9, g_3, h_3, i_3, j_3, k_3, l_3
  62. Point(Circle[M_2, a_1]) → N_2
  63. Rotate(N_2 + (b_1, 0), Angle[C_1 - B_1] + Angle[N_2 - M_2] - Angle[B_1 - A_1], N_2) → O_2
  64. Rotate(O_2 + (c_1, 0), Angle[D_1 - C_1] + Angle[O_2 - N_2] - Angle[C_1 - B_1], O_2) → P_2
  65. Rotate(P_2 + (d_1, 0), Angle[E_1 - D_1] + Angle[P_2 - O_2] - Angle[D_1 - C_1], P_2) → Q_2
  66. Rotate(Q_2 + (e_1, 0), Angle[F_1 - E_1] + Angle[Q_2 - P_2] - Angle[E_1 - D_1], Q_2) → R_2
  67. Polygon(M_2, N_2, O_2, P_2, Q_2, R_2) → poly10, m_2, n_2, o_2, p_2, q_2, r_2
  68. Point(Circle[S_2, a_1]) → T_2
  69. Rotate(T_2 + (b_1, 0), Angle[C_1 - B_1] + Angle[T_2 - S_2] - Angle[B_1 - A_1], T_2) → U_2
  70. Rotate(U_2 + (c_1, 0), Angle[D_1 - C_1] + Angle[U_2 - T_2] - Angle[C_1 - B_1], U_2) → V_2
  71. Rotate(V_2 + (d_1, 0), Angle[E_1 - D_1] + Angle[V_2 - U_2] - Angle[D_1 - C_1], V_2) → W_2
  72. Rotate(W_2 + (e_1, 0), Angle[F_1 - E_1] + Angle[W_2 - V_2] - Angle[E_1 - D_1], W_2) → Z_2
  73. Polygon(S_2, T_2, U_2, V_2, W_2, Z_2) → poly11, s_2, t_2, u_2, v_2, w_2, z_2
  74. Point(Circle[A_3, a_1]) → B_3
  75. Rotate(B_3 + (b_1, 0), Angle[C_1 - B_1] + Angle[B_3 - A_3] - Angle[B_1 - A_1], B_3) → C_3
  76. Rotate(C_3 + (c_1, 0), Angle[D_1 - C_1] + Angle[C_3 - B_3] - Angle[C_1 - B_1], C_3) → D_3
  77. Rotate(D_3 + (d_1, 0), Angle[E_1 - D_1] + Angle[D_3 - C_3] - Angle[D_1 - C_1], D_3) → E_3
  78. Rotate(E_3 + (e_1, 0), Angle[F_1 - E_1] + Angle[E_3 - D_3] - Angle[E_1 - D_1], E_3) → F_3
  79. Polygon(A_3, B_3, C_3, D_3, E_3, F_3) → poly12, a_3, b_3, c_3, d_3, e_3, f_3
  80. Point(Circle[G_3, a_1]) → H_3
  81. Rotate(H_3 + (b_1, 0), Angle[C_1 - B_1] + Angle[H_3 - G_3] - Angle[B_1 - A_1], H_3) → I_3
  82. Rotate(I_3 + (c_1, 0), Angle[D_1 - C_1] + Angle[I_3 - H_3] - Angle[C_1 - B_1], I_3) → J_3
  83. Rotate(J_3 + (d_1, 0), Angle[E_1 - D_1] + Angle[J_3 - I_3] - Angle[D_1 - C_1], J_3) → K_3
  84. Rotate(K_3 + (e_1, 0), Angle[F_1 - E_1] + Angle[K_3 - J_3] - Angle[E_1 - D_1], K_3) → L_3
  85. Polygon(G_3, H_3, I_3, J_3, K_3, L_3) → poly13, g_4, h_4, i_4, j_4, k_4, l_4
  86. Point(Circle[M_3, a_1]) → N_3
  87. Rotate(N_3 + (b_1, 0), Angle[C_1 - B_1] + Angle[N_3 - M_3] - Angle[B_1 - A_1], N_3) → O_3
  88. Rotate(O_3 + (c_1, 0), Angle[D_1 - C_1] + Angle[O_3 - N_3] - Angle[C_1 - B_1], O_3) → P_3
  89. Rotate(P_3 + (d_1, 0), Angle[E_1 - D_1] + Angle[P_3 - O_3] - Angle[D_1 - C_1], P_3) → Q_3
  90. Rotate(Q_3 + (e_1, 0), Angle[F_1 - E_1] + Angle[Q_3 - P_3] - Angle[E_1 - D_1], Q_3) → R_3
  91. Polygon(M_3, N_3, O_3, P_3, Q_3, R_3) → poly14, m_3, n_3, o_3, p_3, q_3, r_3
  92. Point(Circle[S_3, a_1]) → T_3
  93. Rotate(T_3 + (b_1, 0), Angle[C_1 - B_1] + Angle[T_3 - S_3] - Angle[B_1 - A_1], T_3) → U_3
  94. Rotate(U_3 + (c_1, 0), Angle[D_1 - C_1] + Angle[U_3 - T_3] - Angle[C_1 - B_1], U_3) → V_3
  95. Rotate(V_3 + (d_1, 0), Angle[E_1 - D_1] + Angle[V_3 - U_3] - Angle[D_1 - C_1], V_3) → W_3
  96. Rotate(W_3 + (e_1, 0), Angle[F_1 - E_1] + Angle[W_3 - V_3] - Angle[E_1 - D_1], W_3) → Z_3
  97. Polygon(S_3, T_3, U_3, V_3, W_3, Z_3) → poly15, s_3, t_3, u_3, v_3, w_3, z_3
  98. Point(Circle[A_4, a_1]) → B_4
  99. Rotate(B_4 + (b_1, 0), Angle[C_1 - B_1] + Angle[B_4 - A_4] - Angle[B_1 - A_1], B_4) → C_4
  100. Rotate(C_4 + (c_1, 0), Angle[D_1 - C_1] + Angle[C_4 - B_4] - Angle[C_1 - B_1], C_4) → D_4
  101. Rotate(D_4 + (d_1, 0), Angle[E_1 - D_1] + Angle[D_4 - C_4] - Angle[D_1 - C_1], D_4) → E_4
  102. Rotate(E_4 + (e_1, 0), Angle[F_1 - E_1] + Angle[E_4 - D_4] - Angle[E_1 - D_1], E_4) → F_4
  103. Polygon(A_4, B_4, C_4, D_4, E_4, F_4) → poly16, a_4, b_4, c_4, d_4, e_4, f_4
  104. Point(Circle[G_4, a_1]) → H_4
  105. Rotate(H_4 + (b_1, 0), Angle[C_1 - B_1] + Angle[H_4 - G_4] - Angle[B_1 - A_1], H_4) → I_4
  106. Rotate(I_4 + (c_1, 0), Angle[D_1 - C_1] + Angle[I_4 - H_4] - Angle[C_1 - B_1], I_4) → J_4
  107. Rotate(J_4 + (d_1, 0), Angle[E_1 - D_1] + Angle[J_4 - I_4] - Angle[D_1 - C_1], J_4) → K_4
  108. Rotate(K_4 + (e_1, 0), Angle[F_1 - E_1] + Angle[K_4 - J_4] - Angle[E_1 - D_1], K_4) → L_4
  109. Polygon(G_4, H_4, I_4, J_4, K_4, L_4) → poly17, g_5, h_5, i_5, j_5, k_5, l_5
  110. Point(Circle[M_4, a_1]) → N_4
  111. Rotate(N_4 + (b_1, 0), Angle[C_1 - B_1] + Angle[N_4 - M_4] - Angle[B_1 - A_1], N_4) → O_4
  112. Rotate(O_4 + (c_1, 0), Angle[D_1 - C_1] + Angle[O_4 - N_4] - Angle[C_1 - B_1], O_4) → P_4
  113. Rotate(P_4 + (d_1, 0), Angle[E_1 - D_1] + Angle[P_4 - O_4] - Angle[D_1 - C_1], P_4) → Q_4
  114. Rotate(Q_4 + (e_1, 0), Angle[F_1 - E_1] + Angle[Q_4 - P_4] - Angle[E_1 - D_1], Q_4) → R_4
  115. Polygon(M_4, N_4, O_4, P_4, Q_4, R_4) → poly18, m_4, n_4, o_4, p_4, q_4, r_4
  116. Point(Circle[S_4, a_1]) → T_4
  117. Rotate(T_4 + (b_1, 0), Angle[C_1 - B_1] + Angle[T_4 - S_4] - Angle[B_1 - A_1], T_4) → U_4
  118. Rotate(U_4 + (c_1, 0), Angle[D_1 - C_1] + Angle[U_4 - T_4] - Angle[C_1 - B_1], U_4) → V_4
  119. Rotate(V_4 + (d_1, 0), Angle[E_1 - D_1] + Angle[V_4 - U_4] - Angle[D_1 - C_1], V_4) → W_4
  120. Rotate(W_4 + (e_1, 0), Angle[F_1 - E_1] + Angle[W_4 - V_4] - Angle[E_1 - D_1], W_4) → Z_4
  121. Polygon(S_4, T_4, U_4, V_4, W_4, Z_4) → poly19, s_4, t_4, u_4, v_4, w_4, z_4
  122. Point(Circle[A_5, a_1]) → B_5
  123. Rotate(B_5 + (b_1, 0), Angle[C_1 - B_1] + Angle[B_5 - A_5] - Angle[B_1 - A_1], B_5) → C_5
  124. Rotate(C_5 + (c_1, 0), Angle[D_1 - C_1] + Angle[C_5 - B_5] - Angle[C_1 - B_1], C_5) → D_5
  125. Rotate(D_5 + (d_1, 0), Angle[E_1 - D_1] + Angle[D_5 - C_5] - Angle[D_1 - C_1], D_5) → E_5
  126. Rotate(E_5 + (e_1, 0), Angle[F_1 - E_1] + Angle[E_5 - D_5] - Angle[E_1 - D_1], E_5) → F_5
  127. Polygon(A_5, B_5, C_5, D_5, E_5, F_5) → poly20, a_5, b_5, c_5, d_5, e_5, f_5
  128. Point(Circle[G_5, a_5]) → H_5
  129. Rotate(H_5 + (b_5, 0), Angle[C_5 - B_5] + Angle[H_5 - G_5] - Angle[B_5 - A_5], H_5) → I_5
  130. Rotate(I_5 + (c_5, 0), Angle[D_5 - C_5] + Angle[I_5 - H_5] - Angle[C_5 - B_5], I_5) → J_5
  131. Rotate(J_5 + (d_5, 0), Angle[E_5 - D_5] + Angle[J_5 - I_5] - Angle[D_5 - C_5], J_5) → K_5
  132. Rotate(K_5 + (e_5, 0), Angle[F_5 - E_5] + Angle[K_5 - J_5] - Angle[E_5 - D_5], K_5) → L_5
  133. Polygon(G_5, H_5, I_5, J_5, K_5, L_5) → poly21, g_6, h_6, i_6, j_6, k_6, l_6

AI Drawing Prompt Reference

Draw a pattern formed by six hexagons, each created by rotating and combining others. Start by constructing a regular hexagon and connect its vertices to form the sides. Then, rotate the hexagon around one of its vertices by a certain angle and repeat this process to generate multiple rotated hexagons. Finally, combine these hexagons to form a complex loop pattern, showcasing the geometric relationships between rotation and combination.
The prompt above was compiled from the original drawing commands and is for reference only.