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Triangle Tiling Patterns

Triangle Tiling Patterns
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2014-07-14 22:05
TrianglePolygonTrigonometryTransformationGeometric Construction
This example demonstrates exploring geometric patterns in tiling polygons by rotating equilateral triangles. It involves concepts like polygons, equilateral triangles, geometric transformations, and trigonometric functions. Constructed using GeoGebra commands such as Point, Polygon, Segment, Intersect, and Rotate. Suitable for middle school and above students for exploring geometric transformations and patterns.

GeoGebra Commands & Parameters

  1. Point(Circle[D_1, 3.9400507610943287]) → E_1
  2. Polygon(D_1, E_1, F_1) → poly2_1, f_1, d_1, e_1
  3. Segment(D_1, E_1) → a_1
  4. Intersect(d_1, e_1) → C_1
  5. Point(Circle[A, f_1]) → B
  6. Rotate(B + (d_1, 0), Angle[F_1 - E_1] + Angle[B - A] - Angle[E_1 - D_1], B) → C
  7. Polygon(A, B, C) → poly1, c, a, b
  8. Point(Circle[D, f_1]) → E
  9. Rotate(E + (d_1, 0), Angle[F_1 - E_1] + Angle[E - D] - Angle[E_1 - D_1], E) → F
  10. Polygon(D, E, F) → poly2, f, d, e
  11. Point(Circle[G, f_1]) → H
  12. Rotate(H + (d_1, 0), Angle[F_1 - E_1] + Angle[H - G] - Angle[E_1 - D_1], H) → I
  13. Polygon(G, H, I) → poly3, i, g, h
  14. Point(Circle[J, f_1]) → K
  15. Rotate(K + (d_1, 0), Angle[F_1 - E_1] + Angle[K - J] - Angle[E_1 - D_1], K) → L
  16. Polygon(J, K, L) → poly4, l, j, k
  17. Point(Circle[M, f_1]) → N
  18. Rotate(N + (d_1, 0), Angle[F_1 - E_1] + Angle[N - M] - Angle[E_1 - D_1], N) → O
  19. Polygon(M, N, O) → poly5, o, m, n
  20. Point(Circle[P, f_1]) → Q
  21. Rotate(Q + (d_1, 0), Angle[F_1 - E_1] + Angle[Q - P] - Angle[E_1 - D_1], Q) → R
  22. Polygon(P, Q, R) → poly6, r, p, q
  23. Point(Circle[S, f_1]) → T
  24. Rotate(T + (d_1, 0), Angle[F_1 - E_1] + Angle[T - S] - Angle[E_1 - D_1], T) → U
  25. Polygon(S, T, U) → poly7, u, s, t
  26. Point(Circle[V, f_1]) → W
  27. Rotate(W + (d_1, 0), Angle[F_1 - E_1] + Angle[W - V] - Angle[E_1 - D_1], W) → Z
  28. Polygon(V, W, Z) → poly8, z, v, w
  29. Point(Circle[A_1, f_1]) → B_1
  30. Rotate(B_1 + (d_1, 0), Angle[F_1 - E_1] + Angle[B_1 - A_1] - Angle[E_1 - D_1], B_1) → G_1
  31. Polygon(A_1, B_1, G_1) → poly9, g_1, a_2, b_1
  32. Point(Circle[H_1, f_1]) → I_1
  33. Rotate(I_1 + (d_1, 0), Angle[F_1 - E_1] + Angle[I_1 - H_1] - Angle[E_1 - D_1], I_1) → J_1
  34. Polygon(H_1, I_1, J_1) → poly10, j_1, h_1, i_1
  35. Point(Circle[K_1, f_1]) → L_1
  36. Rotate(L_1 + (d_1, 0), Angle[F_1 - E_1] + Angle[L_1 - K_1] - Angle[E_1 - D_1], L_1) → M_1
  37. Polygon(K_1, L_1, M_1) → poly11, m_1, k_1, l_1
  38. Point(Circle[N_1, f_1]) → O_1
  39. Rotate(O_1 + (d_1, 0), Angle[F_1 - E_1] + Angle[O_1 - N_1] - Angle[E_1 - D_1], O_1) → P_1
  40. Polygon(N_1, O_1, P_1) → poly12, p_1, n_1, o_1
  41. Point(Circle[Q_1, f_1]) → R_1
  42. Rotate(R_1 + (d_1, 0), Angle[F_1 - E_1] + Angle[R_1 - Q_1] - Angle[E_1 - D_1], R_1) → S_1
  43. Polygon(Q_1, R_1, S_1) → poly13, s_1, q_1, r_1
  44. Point(Circle[T_1, f_1]) → U_1
  45. Rotate(U_1 + (d_1, 0), Angle[F_1 - E_1] + Angle[U_1 - T_1] - Angle[E_1 - D_1], U_1) → V_1
  46. Polygon(T_1, U_1, V_1) → poly14, v_1, t_1, u_1
  47. Point(Circle[W_1, f_1]) → Z_1
  48. Rotate(Z_1 + (d_1, 0), Angle[F_1 - E_1] + Angle[Z_1 - W_1] - Angle[E_1 - D_1], Z_1) → A_2
  49. Polygon(W_1, Z_1, A_2) → poly15, a_3, w_1, z_1
  50. Point(Circle[B_2, f_1]) → C_2
  51. Rotate(C_2 + (d_1, 0), Angle[F_1 - E_1] + Angle[C_2 - B_2] - Angle[E_1 - D_1], C_2) → E_2
  52. Polygon(B_2, C_2, E_2) → poly16, e_2, b_2, c_2

AI Drawing Prompt Reference

Draw a geometric pattern composed of rotating equilateral triangles. Start by creating an equilateral triangle, then rotate and draw multiple identical triangles around its vertices in sequence to form a regular tiling pattern. Each new triangle is obtained by rotating the previous one by a certain angle, showcasing the beauty of rotational symmetry and geometric transformations.
The prompt above was compiled from the original drawing commands and is for reference only.