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Investigating Rhombus Dissection of Regular Polygons

Investigating Rhombus Dissection of Regular Polygons
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2017-06-01 04:54
PolygonTransformationGeometric Construction
This example demonstrates how regular polygons can be dissected into rhombuses. Key mathematical concepts include polygons, rhombuses, and transformations. Using GeoGebra commands such as Segment, Mirror, and Polygon, the process of rhombus dissection can be dynamically illustrated. Suitable for middle school and above students or teachers for geometry instruction.

GeoGebra Commands & Parameters

  1. Segment(A, B) → f
  2. Mirror(C, f) → C'
  3. Polygon(A, C, B, C') → poly1, a, c, b, c'
  4. Polygon(D, E, 12) → poly2, g, q, r, s, F, G, H, I, J, K, L, h, M, N, O, i, j, k, l, m, n, p
  5. los1(D, F, E) → poly3, P
  6. los1(D, N, O) → poly4, Q
  7. los1(Q, P, D) → poly5, R
  8. los1(N, R, Q) → poly6, S
  9. los1(R, F, P) → poly7, T
  10. los1(S, T, R) → poly8, U
  11. los1(M, S, N) → poly9, V
  12. los1(T, G, F) → poly10, W
  13. los1(V, U, S) → poly11, Z
  14. los1(U, W, T) → poly12, A_1
  15. Polygon(J, Z, U, A_1) → poly13, j_1, z_1, u, a_1
  16. Polygon(J, I, W, A_1) → poly14, j_2, i_1, w, a_2
  17. Polygon(I, H, G, W) → poly15, i_2, h_1, g_1, w_1
  18. Polygon(J, K, V, Z) → poly16, j_3, k_1, v, z_2
  19. Polygon(K, L, M, V) → poly17, k_2, l_1, m_1, v_1
  20. Polygon(M_2, N_2, 12) → poly35, q_2, g_3, h_3, i_3, O_2, P_2, Q_2, R_2, S_2, T_2, U_2, r_2, V_2, W_2, Z_2, s_2, t_2, b_2, c_2, d_2, e_2, f_3
  21. Polygon(N_3, O_3, 12) → poly49, k_3, a_3, b_3, c_3, P_3, Q_3, R_3, S_3, T_3, U_3, V_3, l_3, W_3, Z_3, A_4, m_3, n_3, p_3, q_3, r_3, s_3, t_3
  22. KITE(N_3, P_3, O_3) → B_4, poly50
  23. KITE(B_4, Q_3, P_3) → C_4, poly51
  24. KITE(N_3, C_4, B_4) → D_4, poly52
  25. KITE(A_4, D_4, N_3) → E_4, poly53
  26. KITE(Z_3, E_4, A_4) → F_4, poly54
  27. KITE(E_4, C_4, D_4) → G_4, poly55
  28. KITE(F_4, G_4, E_4) → H_4, poly56
  29. KITE(H_4, V_3, F_4) → I_4, poly57
  30. KITE(G_4, Q_3, C_4) → J_4, poly58
  31. KITE(H_4, J_4, G_4) → K_4, poly59
  32. KITE(U_3, K_4, T_3) → L_4, poly60
  33. KITE(J_4, R_3, Q_3) → M_4, poly61
  34. KITE(K_4, M_4, T_3) → N_4, poly62
  35. Polygon(B_1, C_1, 12) → poly18, t, t_1, b_1, c_1, D_1, E_1, F_1, G_1, H_1, I_1, J_1, d, K_1, L_1, M_1, e, f_1, n_1, p_1, q_1, r_1, s_1
  36. Polygon(N_1, O_1, 12) → poly19, d_1, q_4, r_4, s_4, P_1, Q_1, R_1, S_1, T_1, U_1, V_1, e_1, W_1, Z_1, A_2, f_2, g_2, h_2, l_2, m_2, n_2, p_2
  37. KITE(C_1, E_1, D_1) → U_8, poly117
  38. KITE(C_1, M_1, B_1) → V_8, poly118
  39. KITE(L_1, V_8, M_1) → W_8, poly119
  40. KITE(K_1, W_8, L_1) → Z_8, poly120
  41. KITE(J_1, Z_8, K_1) → A_9, poly121
  42. KITE(A_9, I_1, J_1) → B_9, poly122
  43. KITE(A_9, G_1, H_1) → C_9, poly123
  44. KITE(C_9, F_1, G_1) → D_9, poly124
  45. KITE(D_9, E_1, F_1) → E_9, poly125
  46. KITE(D_9, C_1, U_8) → F_9, poly126
  47. Polygon(K_9, L_9, 6) → poly131, r_6, P_9, s_6, t_6, a_6, b_6, c_6, M_9, N_9, O_9
  48. Polygon(S_9, T_9, 7) → poly134, d_6, W_9, Z_9, A_{10}, e_6, f_7, g_7, h_7, i_7, j_7, U_9, V_9
  49. Polygon(B_{10}, C_{10}, 8) → poly135, k_7, E_{10}, F_{10}, G_{10}, H_{10}, I_{10}, l_7, m_7, n_7, p_7, q_7, r_7, s_7, D_{10}
  50. Polygon(J_{10}, K_{10}, 9) → poly136, t_7, L_{10}, M_{10}, N_{10}, O_{10}, P_{10}, Q_{10}, R_{10}, a_7, b_7, c_7, d_7, e_7, f_8, g_8, h_8
  51. Polygon(S_{10}, T_{10}, 10) → poly137, i_8, s_8, U_{10}, V_{10}, W_{10}, Z_{10}, A_{11}, B_{11}, C_{11}, D_{11}, j_8, k_8, l_8, m_8, n_8, p_8, q_8, r_8
  52. Polygon(E_{11}, F_{11}, 5) → poly138, t_8, a_8, b_8, c_8, d_8, G_{11}, H_{11}, I_{11}
  53. Polygon(U_{11}, V_{11}, 11) → poly150, e_8, n_9, p_9, W_{11}, Z_{11}, A_{12}, B_{12}, C_{12}, D_{12}, E_{12}, F_{12}, f_9, G_{12}, g_9, h_9, i_9, j_9, k_9, l_9, m_9
  54. KITE(D_{14}, E_{14}, F_{14}) → G_{14}, poly184
  55. Segment(D_{14}, E_{14}) → i_{10}
  56. Segment(F_{14}, G_{14}) → j_{10}
  57. KITE(W_8, C_1, V_8) → B_2, poly20
  58. KITE(B_2, F_9, C_1) → C_2, poly21
  59. KITE(W_8, C_2, B_2) → D_2, poly22
  60. KITE(C_2, D_9, F_9) → E_2, poly23
  61. Polygon(F_2, G_2, 12) → poly24, d_3, m_4, n_4, p_4, H_2, I_2, J_2, K_2, L_2, A_3, B_3, e_3, C_3, D_3, E_3, f_4, g_4, h_4, i_4, j_4, k_4, l_4
  62. Polygon(F_3, G_3, 12) → poly25, t_4, i_5, j_5, k_5, H_3, I_3, J_3, K_3, L_3, M_3, O_4, a_4, P_4, Q_4, R_4, b_4, c_4, d_4, e_4, f_5, g_5, h_5
  63. Polygon(S_4, T_4, 12) → poly26, l_5, b_5, c_5, d_5, U_4, V_4, W_4, Z_4, A_5, B_5, C_5, m_5, D_5, E_5, F_5, n_5, p_5, q_5, r_5, s_5, t_5, a_5
  64. Polygon(G_5, H_5, 12) → poly27, e_5, n_6, p_6, q_6, I_5, J_5, K_5, L_5, M_5, N_5, O_5, f_6, P_5, Q_5, R_5, g_6, h_6, i_6, j_6, k_6, l_6, m_6
  65. Polygon(S_5, T_5, 12) → poly28, q_9, f_{10}, g_{10}, h_{10}, U_5, V_5, W_5, Z_5, A_6, B_6, C_6, r_9, D_6, E_6, F_6, s_9, t_9, a_9, b_9, c_9, d_9, e_9