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Tiling with Whole Number Squares

Tiling with Whole Number Squares
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2017-10-22 05:05
PolygonSequenceOptimization
This example demonstrates how to tile the plane using squares with integer side lengths, exploring the minimum number of squares needed. It involves core concepts like geometric tiling, sequences, and optimization. The construction uses commands such as Polygon, with objects like A, B, and functions f, g, h to define and arrange the squares. Suitable for middle school students and above to learn about geometric constructions and sequences, or for exploring space-filling patterns.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 4) → poly1, f, g, h, i, C, D
  2. Polygon(E, F, 4) → poly2, j, k, l, m, G, H
  3. Polygon(I, J, 4) → poly3, n, p, q, r, K, L
  4. Polygon(M, N, 4) → poly4, s, t, a, b, O, P
  5. Polygon(Q, R, 4) → poly5, c, d, e, f_1, S, T
  6. Polygon(E_1, F_1, 4) → poly2_1, j_1, k_1, l_1, m_1, G_1, H_1
  7. Polygon(A_1, B_1, 4) → poly1_1, f_2, g_1, h_1, i_1, C_1, D_1
  8. Polygon(I_1, J_1, 4) → poly3_1, n_1, p_1, q_1, r_1, K_1, L_1
  9. Polygon(E_2, F_2, 4) → poly2_2, j_2, k_2, l_2, m_2, G_2, H_2
  10. Polygon(I_2, J_2, 4) → poly3_2, n_2, p_2, q_2, r_2, K_2, L_2
  11. Polygon(M_1, N_1, 4) → poly4_1, s_1, t_1, a_1, b_1, O_1, P_1
  12. Polygon(M_2, N_2, 4) → poly4_2, s_2, t_2, a_2, b_2, O_2, P_2
  13. Polygon(Q_1, R_1, 4) → poly5_1, c_1, d_1, e_1, f_3, S_1, T_1
  14. Polygon(Q_2, R_2, 4) → poly5_2, c_2, d_2, e_2, f_4, S_2, T_2
  15. Polygon(U, V, 4) → poly6, g_2, h_2, i_2, g_3, W, Z
  16. Polygon(U_1, V_1, 4) → poly7, h_3, i_3, j_3, k_3, W_1, Z_1
  17. Polygon(U_2, V_2, 4) → poly6_1, g_4, h_4, i_4, g_5, W_2, Z_2
  18. Polygon(M_3, N_3, 4) → poly4_3, s_3, t_3, a_3, b_3, O_3, P_3
  19. Polygon(E_3, F_3, 4) → poly2_3, j_4, k_4, l_3, m_3, G_3, H_3
  20. Polygon(M_4, N_4, 4) → poly4_4, s_4, t_4, a_4, b_4, O_4, P_4
  21. Polygon(U_3, V_3, 4) → poly6_2, g_6, h_5, i_5, g_7, W_3, Z_3
  22. Polygon(E_4, F_4, 4) → poly2_4, j_5, k_5, l_4, m_4, G_4, H_4
  23. Polygon(E_5, F_5, 4) → poly2_5, j_6, k_6, l_5, m_5, G_5, H_5
  24. Polygon(M_5, N_5, 4) → poly4_5, s_5, t_5, a_5, b_5, O_5, P_5
  25. Polygon(M_6, N_6, 4) → poly4_6, s_6, t_6, a_6, b_6, O_6, P_6
  26. Polygon(E_6, F_6, 4) → poly2_6, j_7, k_7, l_6, m_6, G_6, H_6
  27. Polygon(E_7, F_7, 4) → poly2_7, j_8, k_8, l_7, m_7, G_7, H_7
  28. Polygon(A_2, B_2, 4) → poly1_2, f_5, g_8, h_6, i_6, C_2, D_2
  29. Polygon(I_3, J_3, 4) → poly3_3, n_3, p_3, q_3, r_3, K_3, L_3
  30. Polygon(A_3, B_3, 4) → poly1_3, f_6, g_9, h_7, i_7, C_3, D_3
  31. Polygon(A_4, B_4, 4) → poly1_4, f_7, g_{10}, h_8, i_8, C_4, D_4
  32. Polygon(Q_3, R_3, 4) → poly5_3, c_3, d_3, e_3, f_8, S_3, T_3
  33. Polygon(I_4, J_4, 4) → poly8, n_4, p_4, q_4, r_4, K_4, L_4
  34. Polygon(Q_4, R_4, 4) → poly9, c_4, d_4, e_4, n_5, S_4, T_4
  35. Polygon(U_4, V_4, 4) → poly10, p_5, q_5, r_5, c_5, W_4, Z_4
  36. Polygon(Q_5, R_5, 4) → poly5_4, c_6, d_5, e_5, f_9, S_5, T_5
  37. Polygon(U_5, V_5, 4) → poly6_3, g_{11}, h_9, i_9, g_{12}, W_5, Z_5
  38. Polygon(U_6, V_6, 4) → poly6_4, g_{13}, h_{10}, i_{10}, g_{14}, W_6, Z_6
  39. Polygon(U_7, V_7, 4) → poly6_5, g_{15}, h_{11}, i_{11}, g_{16}, W_7, Z_7
  40. Polygon(U_8, V_8, 4) → poly7_1, h_{12}, i_{12}, j_9, k_9, W_8, Z_8
  41. Polygon(U_9, V_9, 4) → poly7_2, h_{13}, i_{13}, j_{10}, k_{10}, W_9, Z_9
  42. Polygon(U_{10}, V_{10}, 4) → poly7_3, h_{14}, i_{14}, j_{11}, k_{11}, W_{10}, Z_{10}
  43. Polygon(A_5, B_5, 4) → poly1_5, f_{10}, g_{17}, h_{15}, i_{15}, C_5, D_5
  44. Polygon(I_5, J_5, 4) → poly3_4, n_6, p_6, q_6, r_6, K_5, L_5