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正八边形平面镶嵌

正八边形平面镶嵌
几何画板及绘图内容加载中，请稍等……
2013-11-01 05:31
多边形变换几何作图
所属专题 Geo&Alg:Area Volume Pythagorean Th.
①本范例演示正八边形是否能够无缝隙地镶嵌平面；②涉及的核心数学概念包括多边形性质、角度和镶嵌条件；③作图要点——使用了Polygon（多边形）、Point（点）、Rotate（旋转）等GeoGebra命令；④适合初中及以上学生学习平面几何和镶嵌图案的构造。

GeoGebra 命令及参数

  1. Polygon(A, B, 8) → poly1, a, D, E, F, G, H, b, c, d, e, f, g, h, C
  2. Point(Circle[I, a]) → J
  3. Rotate(J + (b, 0), Angle[C - B] + Angle[J - I] - Angle[B - A], J) → K
  4. Rotate(K + (c, 0), Angle[D - C] + Angle[K - J] - Angle[C - B], K) → L
  5. Rotate(L + (d, 0), Angle[E - D] + Angle[L - K] - Angle[D - C], L) → M
  6. Rotate(M + (e, 0), Angle[F - E] + Angle[M - L] - Angle[E - D], M) → N
  7. Rotate(N + (f, 0), Angle[G - F] + Angle[N - M] - Angle[F - E], N) → O
  8. Rotate(O + (g, 0), Angle[H - G] + Angle[O - N] - Angle[G - F], O) → P
  9. Polygon(I, J, K, L, M, N, O, P) → poly2, i, j, k, l, m, n, o, p
  10. Point(Circle[Q, a]) → R
  11. Rotate(R + (b, 0), Angle[C - B] + Angle[R - Q] - Angle[B - A], R) → S
  12. Rotate(S + (c, 0), Angle[D - C] + Angle[S - R] - Angle[C - B], S) → T
  13. Rotate(T + (d, 0), Angle[E - D] + Angle[T - S] - Angle[D - C], T) → U
  14. Rotate(U + (e, 0), Angle[F - E] + Angle[U - T] - Angle[E - D], U) → V
  15. Rotate(V + (f, 0), Angle[G - F] + Angle[V - U] - Angle[F - E], V) → W
  16. Rotate(W + (g, 0), Angle[H - G] + Angle[W - V] - Angle[G - F], W) → Z
  17. Polygon(Q, R, S, T, U, V, W, Z) → poly3, q, r, s, t, u, v, w, z
  18. Point(Circle[A_1, i]) → B_1
  19. Rotate(B_1 + (j, 0), Angle[K - J] + Angle[B_1 - A_1] - Angle[J - I], B_1) → C_1
  20. Rotate(C_1 + (k, 0), Angle[L - K] + Angle[C_1 - B_1] - Angle[K - J], C_1) → D_1
  21. Rotate(D_1 + (l, 0), Angle[M - L] + Angle[D_1 - C_1] - Angle[L - K], D_1) → E_1
  22. Rotate(E_1 + (m, 0), Angle[N - M] + Angle[E_1 - D_1] - Angle[M - L], E_1) → F_1
  23. Rotate(F_1 + (n, 0), Angle[O - N] + Angle[F_1 - E_1] - Angle[N - M], F_1) → G_1
  24. Rotate(G_1 + (o, 0), Angle[P - O] + Angle[G_1 - F_1] - Angle[O - N], G_1) → H_1
  25. Polygon(A_1, B_1, C_1, D_1, E_1, F_1, G_1, H_1) → poly4, a_1, b_1, c_1, d_1, e_1, f_1, g_1, h_1
  26. Point(Circle[I_1, a_1]) → J_1
  27. Rotate(J_1 + (b_1, 0), Angle[C_1 - B_1] + Angle[J_1 - I_1] - Angle[B_1 - A_1], J_1) → K_1
  28. Rotate(K_1 + (c_1, 0), Angle[D_1 - C_1] + Angle[K_1 - J_1] - Angle[C_1 - B_1], K_1) → L_1
  29. Rotate(L_1 + (d_1, 0), Angle[E_1 - D_1] + Angle[L_1 - K_1] - Angle[D_1 - C_1], L_1) → M_1
  30. Rotate(M_1 + (e_1, 0), Angle[F_1 - E_1] + Angle[M_1 - L_1] - Angle[E_1 - D_1], M_1) → N_1
  31. Rotate(N_1 + (f_1, 0), Angle[G_1 - F_1] + Angle[N_1 - M_1] - Angle[F_1 - E_1], N_1) → O_1
  32. Rotate(O_1 + (g_1, 0), Angle[H_1 - G_1] + Angle[O_1 - N_1] - Angle[G_1 - F_1], O_1) → P_1
  33. Polygon(I_1, J_1, K_1, L_1, M_1, N_1, O_1, P_1) → poly5, i_1, j_1, k_1, l_1, m_1, n_1, o_1, p_1
  34. Point(Circle[Q_1, i_1]) → R_1
  35. Rotate(R_1 + (j_1, 0), Angle[K_1 - J_1] + Angle[R_1 - Q_1] - Angle[J_1 - I_1], R_1) → S_1
  36. Rotate(S_1 + (k_1, 0), Angle[L_1 - K_1] + Angle[S_1 - R_1] - Angle[K_1 - J_1], S_1) → T_1
  37. Rotate(T_1 + (l_1, 0), Angle[M_1 - L_1] + Angle[T_1 - S_1] - Angle[L_1 - K_1], T_1) → U_1
  38. Rotate(U_1 + (m_1, 0), Angle[N_1 - M_1] + Angle[U_1 - T_1] - Angle[M_1 - L_1], U_1) → V_1
  39. Rotate(V_1 + (n_1, 0), Angle[O_1 - N_1] + Angle[V_1 - U_1] - Angle[N_1 - M_1], V_1) → W_1
  40. Rotate(W_1 + (o_1, 0), Angle[P_1 - O_1] + Angle[W_1 - V_1] - Angle[O_1 - N_1], W_1) → Z_1
  41. Polygon(Q_1, R_1, S_1, T_1, U_1, V_1, W_1, Z_1) → poly6, q_1, r_1, s_1, t_1, u_1, v_1, w_1, z_1
  42. Point(Circle[A_2, i]) → B_2
  43. Rotate(B_2 + (j, 0), Angle[K - J] + Angle[B_2 - A_2] - Angle[J - I], B_2) → C_2
  44. Rotate(C_2 + (k, 0), Angle[L - K] + Angle[C_2 - B_2] - Angle[K - J], C_2) → D_2
  45. Rotate(D_2 + (l, 0), Angle[M - L] + Angle[D_2 - C_2] - Angle[L - K], D_2) → E_2
  46. Rotate(E_2 + (m, 0), Angle[N - M] + Angle[E_2 - D_2] - Angle[M - L], E_2) → F_2
  47. Rotate(F_2 + (n, 0), Angle[O - N] + Angle[F_2 - E_2] - Angle[N - M], F_2) → G_2
  48. Rotate(G_2 + (o, 0), Angle[P - O] + Angle[G_2 - F_2] - Angle[O - N], G_2) → H_2
  49. Polygon(A_2, B_2, C_2, D_2, E_2, F_2, G_2, H_2) → poly7, a_2, b_2, c_2, d_2, e_2, f_2, g_2, h_2
  50. Point(Circle[I_2, a_2]) → J_2
  51. Rotate(J_2 + (b_2, 0), Angle[C_2 - B_2] + Angle[J_2 - I_2] - Angle[B_2 - A_2], J_2) → K_2
  52. Rotate(K_2 + (c_2, 0), Angle[D_2 - C_2] + Angle[K_2 - J_2] - Angle[C_2 - B_2], K_2) → L_2
  53. Rotate(L_2 + (d_2, 0), Angle[E_2 - D_2] + Angle[L_2 - K_2] - Angle[D_2 - C_2], L_2) → M_2
  54. Rotate(M_2 + (e_2, 0), Angle[F_2 - E_2] + Angle[M_2 - L_2] - Angle[E_2 - D_2], M_2) → N_2
  55. Rotate(N_2 + (f_2, 0), Angle[G_2 - F_2] + Angle[N_2 - M_2] - Angle[F_2 - E_2], N_2) → O_2
  56. Rotate(O_2 + (g_2, 0), Angle[H_2 - G_2] + Angle[O_2 - N_2] - Angle[G_2 - F_2], O_2) → P_2
  57. Polygon(I_2, J_2, K_2, L_2, M_2, N_2, O_2, P_2) → poly8, i_2, j_2, k_2, l_2, m_2, n_2, o_2, p_2
  58. Point(Circle[Q_2, q]) → R_2
  59. Rotate(R_2 + (r, 0), Angle[S - R] + Angle[R_2 - Q_2] - Angle[R - Q], R_2) → S_2
  60. Rotate(S_2 + (s, 0), Angle[T - S] + Angle[S_2 - R_2] - Angle[S - R], S_2) → T_2
  61. Rotate(T_2 + (t, 0), Angle[U - T] + Angle[T_2 - S_2] - Angle[T - S], T_2) → U_2
  62. Rotate(U_2 + (u, 0), Angle[V - U] + Angle[U_2 - T_2] - Angle[U - T], U_2) → V_2
  63. Rotate(V_2 + (v, 0), Angle[W - V] + Angle[V_2 - U_2] - Angle[V - U], V_2) → W_2
  64. Rotate(W_2 + (w, 0), Angle[Z - W] + Angle[W_2 - V_2] - Angle[W - V], W_2) → Z_2
  65. Polygon(Q_2, R_2, S_2, T_2, U_2, V_2, W_2, Z_2) → poly9, q_2, r_2, s_2, t_2, u_2, v_2, w_2, z_2
  66. Point(Circle[A_3, a_1]) → B_3
  67. 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
  68. 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
  69. 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
  70. 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
  71. Rotate(F_3 + (f_1, 0), Angle[G_1 - F_1] + Angle[F_3 - E_3] - Angle[F_1 - E_1], F_3) → G_3
  72. Rotate(G_3 + (g_1, 0), Angle[H_1 - G_1] + Angle[G_3 - F_3] - Angle[G_1 - F_1], G_3) → H_3
  73. Polygon(A_3, B_3, C_3, D_3, E_3, F_3, G_3, H_3) → poly10, a_3, b_3, c_3, d_3, e_3, f_3, g_3, h_3
  74. Point(Circle[I_3, a_2]) → J_3
  75. Rotate(J_3 + (b_2, 0), Angle[C_2 - B_2] + Angle[J_3 - I_3] - Angle[B_2 - A_2], J_3) → K_3
  76. Rotate(K_3 + (c_2, 0), Angle[D_2 - C_2] + Angle[K_3 - J_3] - Angle[C_2 - B_2], K_3) → L_3
  77. Rotate(L_3 + (d_2, 0), Angle[E_2 - D_2] + Angle[L_3 - K_3] - Angle[D_2 - C_2], L_3) → M_3
  78. Rotate(M_3 + (e_2, 0), Angle[F_2 - E_2] + Angle[M_3 - L_3] - Angle[E_2 - D_2], M_3) → N_3
  79. Rotate(N_3 + (f_2, 0), Angle[G_2 - F_2] + Angle[N_3 - M_3] - Angle[F_2 - E_2], N_3) → O_3
  80. Rotate(O_3 + (g_2, 0), Angle[H_2 - G_2] + Angle[O_3 - N_3] - Angle[G_2 - F_2], O_3) → P_3
  81. Polygon(I_3, J_3, K_3, L_3, M_3, N_3, O_3, P_3) → poly11, i_3, j_3, k_3, l_3, m_3, n_3, o_3, p_3

AI 绘图提示语参考

画一个由正八边形构成的平面镶嵌图案，展示正八边形是否能够无缝拼接填满整个平面。首先绘制一个正八边形，然后通过旋转和位移操作，生成多个正八边形，使它们相互连接并形成连续的镶嵌结构。注意观察每个正八边形之间的角度关系和位置关系，确保图案的连续性和无缝性。
以上提示语为根据原始绘图命令整理而来，仅供参考。