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Square and Octagon

Square and Octagon
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2019-03-21 00:43
PolygonVectorGeometric Construction
In collections Cutting up squares
This example demonstrates the geometric construction and relationships between a square and an octagon. Key math concepts include polygon construction, vector operations, and geometric transformations. GeoGebra commands used are Intersect, Point, Polygon, Midpoint, Circle, Segment, Vector, and Translate. Suitable for high school mathematics teaching to understand polygon properties and transformation applications.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Polygon(A, B, 4) → poly1, a, b, c, d, C, D
  4. Polygon(C, B, 8) → poly2, c_1, F, G, H, I, J, b_1, e, f, g, h, i, j, E
  5. Polygon(D, C, 8) → poly3, d_1, L, M, N, O, P, c_2, k, l, m, n, p, q, K
  6. Polygon(A, D, 8) → poly4, a_1, R, S, T, U, V, d_2, r, s, t, e_1, f_1, g_1, Q
  7. Polygon(B, A, 8) → poly5, b_2, Z, A_1, B_1, C_1, D_1, a_2, h_1, i_1, j_1, k_1, l_1, m_1, W
  8. Polygon(R, Q, 4) → poly7, r_1, q_1, s_1, t_1, G_1, H_1
  9. Polygon(D_1, C_1, 4) → poly9, d_3, c_3, g_2, h_2, K_1, L_1
  10. Midpoint(A, B_1) → I_3
  11. Midpoint(A, C) → J_3
  12. Circle(I_3, 0.7071067811865476) → t_5
  13. Circle(J_3, 1.5537739740300374) → c_6
  14. Midpoint(R, W) → K_3
  15. Midpoint(G_1, K) → L_3
  16. Midpoint(K, L_1) → M_3
  17. Circle(K_3, 0.7071067811865476) → d_6
  18. Circle(L_3, 0.7071067811865476) → e_6
  19. Circle(M_3, 0.7071067811865476) → f_6
  20. Intersect(t_5, c_6) → N_3, O_3, ,
  21. Intersect(d_6, c_6) → P_3, Q_3, ,
  22. Intersect(c_6, e_6) → R_3, S_3, ,
  23. Intersect(f_6, c_6) → T_3, U_3, ,
  24. Segment(R_3, U_3) → a_6
  25. Segment(U_3, O_3) → g_6
  26. Segment(O_3, Q_3) → i_6
  27. Segment(Q_3, R_3) → j_6
  28. Intersect(d_6, j_6, 2) → V_3
  29. Polygon(Q_3, V_3, 4) → poly22, q_6, v_3, k_6, l_6, W_3, Z_3
  30. Intersect(t_5, i_6, 2) → A_4
  31. Intersect(f_6, g_6, 2) → B_4
  32. Intersect(e_6, a_6, 2) → C_4
  33. Polygon(R_3, C_4, 4) → poly23, r_6, c_7, m_6, p_6, D_4, E_4
  34. Polygon(U_3, B_4, 4) → poly24, u_3, b_7, s_6, t_6, F_4, G_4
  35. Polygon(O_3, A_4, 4) → poly25, o_3, a_7, d_7, e_7, H_4, I_4
  36. Intersect(a_6, j) → E_1
  37. Intersect(g_6, b_1) → F_1
  38. Intersect(g_1, a_2) → I_1
  39. Intersect(j_6, q) → J_1
  40. Intersect(i_6, g_1) → M_1
  41. Polygon(R_3, E_1, C, D, J_1) → poly6, r_3, e_2, c_4, d_4, j_2
  42. Polygon(C, E_1, U_3, F_1, B) → poly8, c_5, e_3, u_1, f_2, b_3
  43. Polygon(B, F_1, O_3, M_1, A) → poly10, b_4, f_3, o_1, m_2, a_3
  44. Polygon(Q_3, M_1, A, D, J_1) → poly11, q_3, m_3, a_4, d_5, j_3
  45. Polygon(D, C, B, A) → poly12, d_8, c_8, b_5, a_5
  46. Vector(B, I) → u
  47. Translate(B, v1) → B'
  48. Translate(F_1, v1) → N_1
  49. Translate(O_3, v1) → O_1
  50. Translate(M_1, v1) → P_1
  51. Translate(A, v1) → A'
  52. Polygon(B', N_1, O_1, P_1, A') → poly10', b', n_1, o_2, p_1, a'
  53. Vector(A, G) → v
  54. Translate(Q_3, v2) → Q_1
  55. Translate(M_1, v2) → R_1
  56. Translate(A, v2) → A'_1
  57. Translate(D, v2) → D'
  58. Translate(J_1, v2) → S_1
  59. Polygon(Q_1, R_1, A'_1, D', S_1) → poly11', q_2, r_2, a'_1, d', s_2
  60. Vector(R_3, B_4) → w
  61. Translate(R_3, v3) → T_1
  62. Translate(E_1, v3) → U_1
  63. Translate(C, v3) → C'
  64. Translate(D, v3) → D'_1
  65. Translate(J_1, v3) → V_1
  66. Polygon(T_1, U_1, C', D'_1, V_1) → poly6', t_2, u_2, c', d'_1, v_1
  67. Polygon(J, I, 4) → poly13, j_4, i_2, k_2, l_2, W_1, Z_1
  68. Polygon(Z, V, 4) → poly14, z, v_2, n_2, p_2, A_2, B_2