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Can Octagons Tile the Plane?

Can Octagons Tile the Plane?
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2013-11-01 05:31
PolygonTransformationGeometric Construction
In collections Geo&Alg:Area Volume Pythagorean Th.
This example demonstrates whether regular octagons can tessellate the plane, involving core math concepts like polygon properties, angles, and tiling conditions. It is constructed using GeoGebra commands such as Polygon, Point, and Rotate, making it suitable for middle school and above students learning plane geometry and tessellations.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

Draw a plane tiling pattern composed of regular octagons to demonstrate whether they can tile the entire plane without gaps. Start by drawing a regular octagon, then use rotation and translation to generate multiple octagons that connect and form a continuous tessellation. Observe the angle relationships and positional connections between each octagon to ensure the pattern's continuity and seamless fit.
The prompt above was compiled from the original drawing commands and is for reference only.