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Broadened Lines: Islamic Geometric Pattern

Broadened Lines: Islamic Geometric Pattern
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2019-12-31 17:36
CirclePolygonTransformationGeometric Construction
① This example demonstrates how to construct an Islamic geometric pattern with broadened lines, showcasing the application of geometry in artistic design. ② Core mathematical concepts include symmetry, the unit circle, special points, isosceles triangles, and polygon combinations, involving positional relationships and angle divisions among points, lines, and planes. ③ Construction highlights: using GeoGebra commands Point, Line, and Polygon to create base points A, B and polygons veelhoek1, veelhoek2, with auxiliary lines f, g, h, i, j, k, l, m, n, o, p, q and transformations (e.g., f_1, f_2, k_1, l_1) to generate symmetric broadened line effects, involving objects from C to T. ④ Suitable for middle school and above, for geometric construction practice, symmetry exploration, or interdisciplinary teaching merging mathematics and art, also useful as a mathematical modeling reference for Islamic pattern design.

GeoGebra Commands & Parameters

  1. Point(yAxis) → C
  2. Line(D, E) → f
  3. Point(f) → F
  4. Point(yAxis) → G
  5. Point(f) → H
  6. Point(xAxis) → I
  7. Polygon(G, H, I, J, C, F, K, L) → veelhoek1, g, h, i, j, c, f_1, k, l
  8. Point(xAxis) → O
  9. Polygon(L, K, F, M, N, O, P, Q) → veelhoek2, l_1, k_1, f_2, m, n, o, p, q
  10. Point(xAxis) → T
  11. Polygon(H, R, S, T, U, V, J, I) → veelhoek3, h_1, r, s, t, u, v, j_1, i_1
  12. Polygon(U, W, Z, A_1, B_1, C_1, J, V) → veelhoek4, u_1, w, z_1, a_1, b_1, c_1, j_2, v_1
  13. Point(yAxis) → D_1
  14. Polygon(F, C, J, C_1, B_1, D_1, E_1, F_1) → veelhoek5, f_3, c_2, j_3, a, b, d_1, e_1, d
  15. Polygon(N, M, F, F_1, E_1, G_1, H_1, I_1) → veelhoek6, n_1, m_1, f_4, e, g_1, p_1, q_1, r_1
  16. Polygon(J_1, P, O, N, K_1) → veelhoek7, s_1, p_2, o_1, n_2, t_1
  17. Polygon(K_1, N, I_1, H_1, L_1) → veelhoek8, g_2, n_3, h_2, i_2, k_2
  18. Polygon(S, M_1, N_1, U, T) → veelhoek9, s_2, l_2, m_2, u_2, t_2
  19. Point(yAxis) → Q_1
  20. Polygon(H, G, L, O_1, P_1, Q_1, R_1, S_1) → veelhoek10, h_3, g_3, l_3, q_2, r_2, a_2, b_2, d_2
  21. Polygon(P_1, O_1, L, Q, P, T_1, U_1, V_1) → veelhoek11, e_2, i_3, l_4, q_3, p_3, k_3, m_3, r_3
  22. Polygon(W_1, U_1, T_1, P, J_1) → veelhoek12, w_1, s_3, t_3, p_4, a_3
  23. Polygon(A_2, Z_1, S, R, H, S_1, R_1, B_2) → veelhoek13, b_3, c_3, s_4, r_4, h_4, d_3, e_3, g_4
  24. Polygon(C_2, M_1, S, Z_1, A_2) → veelhoek14, i_4, j_4, s_5, k_4, m_4
  25. Polygon(D_2, Z, W, U, N_1) → veelhoek15, n_4, z_2, w_2, u_3, q_4
  26. Polygon(F_2, E_2, Z, D_2) → vierhoek1, t_4, a_4, z_3, b_4
  27. Polygon(E_2, G_2, B_1, A_1, Z) → veelhoek16, c_4, d_4, e_4, f_5, z_4
  28. Polygon(G_2, H_2, E_1, D_1, B_1) → veelhoek17, g_5, h_5, i_5, j_5, k_5
  29. Polygon(H_2, I_2, H_1, G_1, E_1) → veelhoek18, l_5, m_5, n_5, p_5, q_5
  30. Polygon(I_2, J_2, L_1, H_1) → vierhoek2, r_5, t_5, a_5, b_5
  31. Point(yAxis) → K_2
  32. Point(yAxis) → P_3
  33. Polygon(K_2, L_2, U_2, V_2, W_2, Z_2, A_3, B_3, C_3, D_3, E_3, F_3, M_2, G_3, H_3, I_3, J_3, K_3, L_3, M_3, N_3, O_3, P_3, N_2, Q_3, R_3, S_3, T_3, U_3, V_3, W_3, Z_3, A_4, B_4, O_2, C_4, D_4, E_4, F_4, G_4, H_4, I_4, J_4, K_4, L_4, P_2, M_4, N_4, O_4, P_4, Q_4, R_4, S_4, Q_2, R_2, S_2, T_2) → veelhoek19, c_5, k_6, l_6, v_2, m_6, n_6, p_6, q_6, r_6, s_6, t_6, d_5, a_6, b_6, c_6, d_6, e_6, f_7, g_7, h_7, i_7, o_3, e_5, j_7, k_7, l_7, m_7, n_7, p_7, v_3, w_3, q_7, r_7, f_6, s_7, t_7, a_7, b_7, c_7, d_7, e_7, f_8, g_8, h_8, o_2, i_8, j_8, k_8, o_4, l_8, m_8, n_8, p_8, g_6, h_6, i_6, j_6
  34. Point(t_6) → T_4
  35. Polygon(T_4, B_3, A_3, F_3) → vierhoek3, q_8, r_8, s_8, t_8
  36. Polygon(Z_2, W_2, U_4, Q_2, P_2) → veelhoek20, a_8, b_8, c_8, d_8, e_8
  37. Point(yAxis) → V_4
  38. Polygon(O_2, N_2, V_4, Q_4, P_4) → veelhoek21, f_9, g_9, v_4, h_9, i_9
  39. Polygon(O_4, W_4, I_4) → driehoek1, j_9, k_9, w_4
  40. Polygon(G_4, F_4, Z_4, B_4) → vierhoek4, l_9, m_9, n_9, p_9
  41. Point(xAxis) → A_5
  42. Point(xAxis) → V_5
  43. Polygon(L_3, K_3, I_5, J_5, K_5, L_5, M_5, N_5, O_5, P_5, Q_5, R_5, A_5, S_5, T_5, U_5, V_5, B_5, C_5, D_5, E_5, F_5, G_5, H_5) → veelhoek22, q_9, f_{10}, g_{10}, h_{10}, i_{10}, j_{10}, k_{10}, l_{10}, o_5, m_{10}, n_{10}, r_9, p_{10}, q_{10}, r_{10}, u_5, v_5, s_9, t_9, a_9, b_9, c_9, d_9, e_9
  44. Point(xAxis) → W_5
  45. Point(yAxis) → H_6
  46. Point(yAxis) → I_6
  47. Point(yAxis) → M_6
  48. Point(yAxis) → N_6
  49. Point(xAxis) → P_6
  50. Polygon(O_3, N_3, N_5, D_6, E_6, F_6, G_6, H_6, I_6, J_6, K_6, L_6, W_5, M_6, N_6, O_6, P_6, Z_5, T_5, S_5, A_6, B_6, C_6, O_5) → veelhoek23, s_{10}, f_{11}, g_{11}, h_{11}, i_{11}, j_{11}, k_{11}, l_{11}, m_{11}, n_{11}, p_{11}, t_{10}, q_{11}, r_{11}, s_{11}, o_6, t_{11}, w_5, z_5, a_{10}, b_{10}, c_{10}, d_{10}, e_{10}
  51. Point(xAxis) → J_7
  52. Polygon(R_3, Q_3, H_6, V_6, W_6, Z_6, A_7, B_7, C_7, D_7, E_7, F_7, Q_6, G_7, H_7, I_7, J_7, R_6, N_6, M_6, S_6, T_6, U_6, I_6) → veelhoek24, a_{11}, i_{12}, j_{12}, v_6, w_6, z_6, k_{12}, l_{12}, m_{12}, n_{12}, p_{12}, b_{11}, q_{12}, r_{12}, s_{12}, t_{12}, a_{12}, c_{11}, d_{11}, e_{11}, f_{12}, g_{12}, h_{12}, u_6
  53. Point(xAxis) → D_8
  54. Point(xAxis) → E_8
  55. Polygon(U_3, T_3, B_7, P_7, Q_7, R_7, S_7, T_7, U_7, V_7, W_7, Z_7, E_8, A_8, B_8, C_8, D_8, L_7, H_7, G_7, M_7, N_7, O_7, C_7) → veelhoek25, b_{12}, j_{13}, k_{13}, l_{13}, m_{13}, n_{13}, p_{13}, q_{13}, u_7, v_7, w_7, c_{12}, z_7, r_{13}, s_{13}, t_{13}, a_{13}, d_{12}, e_{12}, f_{13}, g_{13}, h_{13}, i_{13}, o_7
  56. Polygon(I_3, W_3, K_7, F_8, R_7, Q_7, Z_6, W_6, F_6, E_6, L_5) → veelhoek26, b_{13}, k_{14}, l_{14}, c_{13}, d_{13}, e_{13}, f_{14}, g_{14}, h_{14}, i_{14}, j_{14}