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Square Loops Exploration

Square Loops Exploration
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2014-07-14 22:49
CirclePolygonGeometric ConstructionGraphing
This example demonstrates the investigation of patterns in polygon tiling by using square loops. It involves core math concepts like polygons, squares, circumcircle, kites, bar charts, and histograms, along with equilateral triangles. The construction uses commands such as Polygon, Intersect, Point, and Rotate in GeoGebra. Suitable for middle school and above for geometric exploration and data analysis education.

GeoGebra Commands & Parameters

  1. Polygon(A_1, B_1, 4) → poly1_1, a_1, b_1, c_1, d_1, C_1, D_1
  2. Intersect(c_1, d_1) → E_1
  3. Intersect(b_1, c_1) → F_1
  4. Intersect(a_1, b_1) → G_1
  5. Intersect(a_1, d_1) → H_1
  6. Point(Circle[A, a_1]) → B
  7. Rotate(B + (b_1, 0), Angle[C_1 - B_1] + Angle[B - A] - Angle[B_1 - A_1], B) → C
  8. Rotate(C + (c_1, 0), Angle[D_1 - C_1] + Angle[C - B] - Angle[C_1 - B_1], C) → D
  9. Polygon(A, B, C, D) → poly1, a, b, c, d
  10. Point(Circle[E, a_1]) → F
  11. Rotate(F + (b_1, 0), Angle[C_1 - B_1] + Angle[F - E] - Angle[B_1 - A_1], F) → G
  12. Rotate(G + (c_1, 0), Angle[D_1 - C_1] + Angle[G - F] - Angle[C_1 - B_1], G) → H
  13. Polygon(E, F, G, H) → poly2, e, f, g, h
  14. Point(Circle[I, a_1]) → J
  15. Rotate(J + (b_1, 0), Angle[C_1 - B_1] + Angle[J - I] - Angle[B_1 - A_1], J) → K
  16. Rotate(K + (c_1, 0), Angle[D_1 - C_1] + Angle[K - J] - Angle[C_1 - B_1], K) → L
  17. Polygon(I, J, K, L) → poly3, i, j, k, l
  18. Point(Circle[M, a_1]) → N
  19. Rotate(N + (b_1, 0), Angle[C_1 - B_1] + Angle[N - M] - Angle[B_1 - A_1], N) → O
  20. Rotate(O + (c_1, 0), Angle[D_1 - C_1] + Angle[O - N] - Angle[C_1 - B_1], O) → P
  21. Polygon(M, N, O, P) → poly4, m, n, o, p
  22. Point(Circle[Q, a_1]) → R
  23. Rotate(R + (b_1, 0), Angle[C_1 - B_1] + Angle[R - Q] - Angle[B_1 - A_1], R) → S
  24. Rotate(S + (c_1, 0), Angle[D_1 - C_1] + Angle[S - R] - Angle[C_1 - B_1], S) → T
  25. Polygon(Q, R, S, T) → poly5, q, r, s, t
  26. Point(Circle[U, a_1]) → V
  27. Rotate(V + (b_1, 0), Angle[C_1 - B_1] + Angle[V - U] - Angle[B_1 - A_1], V) → W
  28. Rotate(W + (c_1, 0), Angle[D_1 - C_1] + Angle[W - V] - Angle[C_1 - B_1], W) → Z
  29. Polygon(U, V, W, Z) → poly6, u, v, w, z
  30. Point(Circle[I_1, a_1]) → J_1
  31. 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
  32. 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
  33. Polygon(I_1, J_1, K_1, L_1) → poly7, i_1, j_1, k_1, l_1
  34. Point(Circle[M_1, a_1]) → N_1
  35. 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
  36. 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
  37. Polygon(M_1, N_1, O_1, P_1) → poly8, m_1, n_1, o_1, p_1
  38. Point(Circle[Q_1, a_1]) → R_1
  39. Rotate(R_1 + (b_1, 0), Angle[C_1 - B_1] + Angle[R_1 - Q_1] - Angle[B_1 - A_1], R_1) → S_1
  40. Rotate(S_1 + (c_1, 0), Angle[D_1 - C_1] + Angle[S_1 - R_1] - Angle[C_1 - B_1], S_1) → T_1
  41. Polygon(Q_1, R_1, S_1, T_1) → poly9, q_1, r_1, s_1, t_1
  42. Point(Circle[U_1, a_1]) → V_1
  43. Rotate(V_1 + (b_1, 0), Angle[C_1 - B_1] + Angle[V_1 - U_1] - Angle[B_1 - A_1], V_1) → W_1
  44. Rotate(W_1 + (c_1, 0), Angle[D_1 - C_1] + Angle[W_1 - V_1] - Angle[C_1 - B_1], W_1) → Z_1
  45. Polygon(U_1, V_1, W_1, Z_1) → poly10, u_1, v_1, w_1, z_1
  46. Point(Circle[B_2, a_1]) → C_2
  47. Rotate(C_2 + (b_1, 0), Angle[C_1 - B_1] + Angle[C_2 - B_2] - Angle[B_1 - A_1], C_2) → D_2
  48. Rotate(D_2 + (c_1, 0), Angle[D_1 - C_1] + Angle[D_2 - C_2] - Angle[C_1 - B_1], D_2) → E_2
  49. Polygon(B_2, C_2, D_2, E_2) → poly11, b_2, c_2, d_2, e_2
  50. Point(Circle[F_2, a_1]) → G_2
  51. Rotate(G_2 + (b_1, 0), Angle[C_1 - B_1] + Angle[G_2 - F_2] - Angle[B_1 - A_1], G_2) → H_2
  52. Rotate(H_2 + (c_1, 0), Angle[D_1 - C_1] + Angle[H_2 - G_2] - Angle[C_1 - B_1], H_2) → I_2
  53. Polygon(F_2, G_2, H_2, I_2) → poly12, f_2, g_2, h_2, i_2
  54. Point(Circle[J_2, a_1]) → K_2
  55. Rotate(K_2 + (b_1, 0), Angle[C_1 - B_1] + Angle[K_2 - J_2] - Angle[B_1 - A_1], K_2) → L_2
  56. Rotate(L_2 + (c_1, 0), Angle[D_1 - C_1] + Angle[L_2 - K_2] - Angle[C_1 - B_1], L_2) → M_2
  57. Polygon(J_2, K_2, L_2, M_2) → poly13, j_2, k_2, l_2, m_2
  58. Point(Circle[N_2, a_1]) → O_2
  59. Rotate(O_2 + (b_1, 0), Angle[C_1 - B_1] + Angle[O_2 - N_2] - Angle[B_1 - A_1], O_2) → P_2
  60. Rotate(P_2 + (c_1, 0), Angle[D_1 - C_1] + Angle[P_2 - O_2] - Angle[C_1 - B_1], P_2) → Q_2
  61. Polygon(N_2, O_2, P_2, Q_2) → poly14, n_2, o_2, p_2, q_2
  62. Point(Circle[R_2, a_1]) → S_2
  63. Rotate(S_2 + (b_1, 0), Angle[C_1 - B_1] + Angle[S_2 - R_2] - Angle[B_1 - A_1], S_2) → T_2
  64. Rotate(T_2 + (c_1, 0), Angle[D_1 - C_1] + Angle[T_2 - S_2] - Angle[C_1 - B_1], T_2) → U_2
  65. Polygon(R_2, S_2, T_2, U_2) → poly15, r_2, s_2, t_2, u_2
  66. Point(Circle[V_2, a_1]) → W_2
  67. Rotate(W_2 + (b_1, 0), Angle[C_1 - B_1] + Angle[W_2 - V_2] - Angle[B_1 - A_1], W_2) → Z_2
  68. Rotate(Z_2 + (c_1, 0), Angle[D_1 - C_1] + Angle[Z_2 - W_2] - Angle[C_1 - B_1], Z_2) → A_3
  69. Polygon(V_2, W_2, Z_2, A_3) → poly16, v_2, w_2, z_2, a_3
  70. Point(Circle[B_3, a_1]) → C_3
  71. Rotate(C_3 + (b_1, 0), Angle[C_1 - B_1] + Angle[C_3 - B_3] - Angle[B_1 - A_1], C_3) → D_3
  72. Rotate(D_3 + (c_1, 0), Angle[D_1 - C_1] + Angle[D_3 - C_3] - Angle[C_1 - B_1], D_3) → E_3
  73. Polygon(B_3, C_3, D_3, E_3) → poly17, b_3, c_3, d_3, e_3
  74. Point(Circle[F_3, a_1]) → G_3
  75. Rotate(G_3 + (b_1, 0), Angle[C_1 - B_1] + Angle[G_3 - F_3] - Angle[B_1 - A_1], G_3) → H_3
  76. Rotate(H_3 + (c_1, 0), Angle[D_1 - C_1] + Angle[H_3 - G_3] - Angle[C_1 - B_1], H_3) → I_3
  77. Polygon(F_3, G_3, H_3, I_3) → poly18, f_3, g_3, h_3, i_3
  78. Point(Circle[J_3, a_1]) → K_3
  79. Rotate(K_3 + (b_1, 0), Angle[C_1 - B_1] + Angle[K_3 - J_3] - Angle[B_1 - A_1], K_3) → L_3
  80. Rotate(L_3 + (c_1, 0), Angle[D_1 - C_1] + Angle[L_3 - K_3] - Angle[C_1 - B_1], L_3) → M_3
  81. Polygon(J_3, K_3, L_3, M_3) → poly19, j_3, k_3, l_3, m_3

AI Drawing Prompt Reference

Draw a pattern composed of multiple squares connected by rotation and intersection. Start by creating a square, then generate multiple new squares by rotating and intersecting it, ensuring each new square is adjacent to the previous one. Use circles to assist with rotation and positioning, forming a complex loop-like pattern of squares.
The prompt above was compiled from the original drawing commands and is for reference only.