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Geometric Patterns - Isis Project: Construction & Puzzles

Geometric Patterns - Isis Project: Construction & Puzzles
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2019-10-15 03:02
CirclePolygonTrigonometryTransformationGeometric Construction
① This example demonstrates the creation of intricate geometric patterns through a sequence of construction commands, accompanied by puzzle-style exploration, showcasing the visual beauty of geometric transformations and constructions. ② Core mathematical concepts include geometric transformations (translation, similar triangles), tangent function, median lines, orthogonal lines, and combinations of polygons and circles, integrating coordinate geometry and trigonometry. ③ Construction highlights: Based on the construction summary, GeoGebra commands such as Intersect, Point, Circle, OrthogonalLine, Segment, Line, and Polygon are used, creating numerous objects (A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S) and auxiliary lines (c, d, e, f, g, h, i, j, k, l, m, n, p, q, r, s, t, a, b, f_1, g_1), building symmetric and regular patterns via intersections and orthogonal relationships. ④ Suitable for upper secondary to early university geometry courses, as well as mathematical art design or geometric puzzle challenges, helping learners understand construction logic and transformation properties.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Point(xAxis) → B
  3. Circle(A, B) → c
  4. Circle(B, Segment[A, B]) → d
  5. Intersect(c, xAxis, 2) → C
  6. Circle(C, Segment[A, B]) → e
  7. OrthogonalLine(A, xAxis) → f
  8. Intersect(c, f, 1) → D
  9. Intersect(c, f, 2) → E
  10. Circle(D, Segment[A, B]) → g
  11. Circle(E, Segment[A, B]) → h
  12. Intersect(e, h, 1) → F
  13. Intersect(d, g, 1) → G
  14. Intersect(d, h, 2) → H
  15. Intersect(e, g, 2) → I
  16. Segment(I, H) → i
  17. Segment(F, G) → j
  18. Segment(F, H) → k
  19. Segment(H, G) → l
  20. Segment(G, I) → m
  21. Segment(I, F) → n
  22. Intersect(c, j, 1) → J
  23. Intersect(c, i, 2) → K
  24. Intersect(c, j, 2) → L
  25. Intersect(c, i, 1) → M
  26. Line(J, K) → p
  27. Line(M, L) → q
  28. Line(L, K) → r
  29. Line(J, M) → s
  30. Segment(D, C) → t
  31. Segment(C, E) → a
  32. Segment(E, B) → b
  33. Segment(B, D) → f_1
  34. Intersect(h, r, 1) → N
  35. Intersect(d, p, 1) → O
  36. Intersect(e, a, 1) → P
  37. Intersect(h, a, 1) → Q
  38. Intersect(g, t, 1) → R
  39. Intersect(d, q, 1) → S
  40. Line(P, A) → g_1
  41. Line(O, A) → h_1
  42. Line(N, A) → i_1
  43. Intersect(g, f_1, 1) → T
  44. Line(T, A) → j_1
  45. Intersect(e, h_1, 2) → U
  46. Intersect(g_1, m) → V
  47. Circle(V, Segment[V, D]) → c_1
  48. Intersect(c_1, g_1, 1) → W
  49. Circle(A, Segment[A, W]) → d_1
  50. Intersect(d_1, g_1, 1) → Z
  51. Intersect(d_1, h_1, 1) → A_1
  52. Intersect(d_1, i_1, 1) → B_1
  53. Line(A, B) → k_1
  54. Intersect(d_1, j_1, 1) → C_1
  55. Intersect(d_1, h_1, 2) → D_1
  56. Intersect(d_1, i_1, 2) → E_1
  57. Intersect(d_1, j_1, 2) → F_1
  58. Line(Z, D_1) → l_1
  59. Line(W, A_1) → m_1
  60. Line(F_1, B_1) → n_1
  61. Line(C_1, E_1) → p_1
  62. Line(F_1, W) → q_1
  63. Line(C_1, Z) → r_1
  64. Line(A_1, E_1) → s_1
  65. Line(D_1, B_1) → t_1
  66. Line(B, K) → a_1
  67. Line(K, E) → b_1
  68. Line(E, J) → e_1
  69. Line(J, C) → f_2
  70. Line(C, M) → g_2
  71. Line(M, D) → h_2
  72. Line(D, L) → i_2
  73. Line(L, B) → j_2
  74. Intersect(l_1, h_2) → G_1
  75. Intersect(j_2, h_2) → H_1
  76. Segment(G_1, H_1) → k_2
  77. Intersect(i_1, f_2) → I_1
  78. Intersect(h_2, t_1) → J_1
  79. Segment(J_1, I_1) → l_2
  80. Intersect(h_1, g_2) → K_1
  81. Intersect(j_1, a_1) → L_1
  82. Intersect(i_2, q_1) → M_1
  83. Segment(M_1, L_1) → m_2
  84. Intersect(m_1, i_2) → N_1
  85. Segment(N_1, K_1) → n_2
  86. Intersect(i_1, b_1) → O_1
  87. Intersect(p_1, j_2) → P_1
  88. Segment(O_1, P_1) → p_2
  89. Intersect(r_1, j_2) → Q_1
  90. Segment(Q_1, H_1) → q_2
  91. Intersect(s_1, b_1) → R_1
  92. Segment(R_1, O_1) → r_2
  93. Intersect(m_1, b_1) → S_1
  94. Intersect(g_1, b_1) → T_1
  95. Segment(T_1, S_1) → s_2
  96. Intersect(l_1, e_1) → U_1
  97. Intersect(h_1, e_1) → V_1
  98. Segment(V_1, U_1) → t_2
  99. Intersect(r_1, e_1) → W_1
  100. Intersect(j_1, e_1) → Z_1
  101. Segment(W_1, Z_1) → a_2
  102. Intersect(n_1, a_1) → A_2
  103. Segment(A_2, L_1) → b_2
  104. Intersect(t_1, a_1) → B_2
  105. Segment(B_2, V_1) → c_2
  106. Intersect(n_1, f_2) → C_2
  107. Intersect(f_2, q_1) → D_2
  108. Segment(T_1, D_2) → d_2
  109. Segment(C_2, I_1) → e_2
  110. Intersect(s_1, g_2) → E_2
  111. Segment(K_1, E_2) → f_3
  112. Intersect(p_1, g_2) → F_2
  113. Segment(Z_1, F_2) → g_3
  114. Intersect(h_1, p_1) → G_2
  115. Intersect(g_1, p_1) → H_2
  116. Segment(F_2, G_2) → h_3
  117. Segment(H_2, P_1) → i_3
  118. Intersect(s_1, g_1) → I_2
  119. Intersect(h_1, r_1) → J_2
  120. Segment(C_2, I_2) → j_3
  121. Segment(J_2, A_2) → k_3
  122. Intersect(m_1, r_1) → K_2
  123. Intersect(j_1, m_1) → L_2
  124. Segment(S_1, K_2) → l_3
  125. Segment(L_2, N_1) → m_3
  126. Intersect(j_1, l_1) → M_2
  127. Intersect(i_1, l_1) → N_2
  128. Segment(U_1, M_2) → n_3
  129. Segment(N_2, G_1) → p_3
  130. Segment(W_1, J_2) → q_3
  131. Segment(D_2, N_2) → r_3
  132. Segment(K_2, Q_1) → s_3
  133. Segment(M_1, G_2) → t_3
  134. Segment(R_1, I_2) → a_3
  135. Segment(B_2, L_2) → b_3
  136. Segment(J_1, H_2) → c_3
  137. Segment(E_2, M_2) → d_3
  138. Polygon(E, S_1, A_1, R_1, K, V_1) → pol1, e_3, s_4, a_4, r_4, k_4, v_1
  139. Polygon(K, O_1, B, A_2, B_1, B_2) → pol2, k_5, o_1, b_4, a_5, b_5, b_6
  140. Polygon(B, P_1, C_1, Q_1, L, L_1) → pol3, b_7, p_4, c_4, q_4, l_4, l_5
  141. Polygon(D, H_1, L, M_1, W, N_1) → pol4, d_4, h_4, l_6, m_4, w, n_4
  142. Polygon(D, K_1, M, J_1, D_1, G_1) → pol5, d_5, k_6, m_5, j_4, d_6, g_4
  143. Polygon(M, E_2, E_1, F_2, C, I_1) → pol6, m_6, e_4, e_5, f_4, c_5, i_4
  144. Polygon(J, Z_1, C, C_2, F_1, D_2) → pol7, j_5, z_1, c_6, c_7, f_5, d_7
  145. Polygon(J, T_1, E, U_1, Z, W_1) → pol8, j_6, t_4, e_6, u_1, z_2, w_1
  146. Intersect(j_3, n_3) → O_2
  147. Intersect(q_3, a_3) → P_2
  148. Intersect(k_3, l_3) → Q_2
  149. Intersect(s_3, b_3) → R_2
  150. Intersect(i_3, m_3) → S_2
  151. Intersect(t_3, c_3) → T_2
  152. Intersect(h_3, p_3) → U_2
  153. Intersect(r_3, d_3) → V_2
  154. Polygon(Z, U_1, E, S_1, A_1, P_2) → pol9, z_3, u_2, e_7, s_5, a_6, p_5
  155. Polygon(F_1, O_2, Z, W_1, J, D_2) → pol10, f_6, o_2, z_4, w_2, j_7, d_8
  156. Polygon(C_2, F_1, V_2, E_1, F_2, C) → pol11, c_8, f_7, v_2, e_8, f_8, c_9
  157. Polygon(E_1, U_2, D_1, J_1, M, E_2) → pol12, e_9, u_3, d_9, j_8, m_7, e_{10}
  158. Polygon(T_2, D_1, G_1, D, N_1, W) → pol13, t_5, d_{10}, g_5, d_{11}, n_5, w_3
  159. Polygon(S_2, C_1, Q_1, L, M_1, W) → pol14, s_6, c_{10}, q_5, l_7, m_8, w_4
  160. Polygon(R_2, C_1, P_1, B, A_2, B_1) → pol15, r_5, c_{11}, p_6, b_8, a_7, b_9
  161. Polygon(A_1, R_1, K, B_2, B_1, Q_2) → pol16, a_8, r_6, k_7, b_{10}, b_{11}, q_6
  162. Polygon(O_2, I_2, P_2, Z) → q1, o_3, i_5, p_7, z_5
  163. Polygon(F_1, V_2, M_2, O_2) → q2, f_9, v_3, m_9, o_4
  164. Polygon(V_2, N_2, U_2, E_1) → q3, v_4, n_6, u_4, e_{11}
  165. Polygon(T_2, D_1, U_2, G_2) → q4, t_6, d_{12}, u_5, g_6
  166. Polygon(T_2, W, S_2, H_2) → q5, t_7, w_5, s_7, h_5
  167. Polygon(R_2, L_2, S_2, C_1) → q6, r_7, l_8, s_8, c_{12}
  168. Polygon(R_2, B_1, Q_2, K_2) → q7, r_8, b_{12}, q_7, k_8
  169. Polygon(P_2, A_1, Q_2, J_2) → q8, p_8, a_9, q_8, j_9
  170. Polygon(M_2, O_2, H_2, T_2, G_2, U_2, N_2, V_2, I_2, P_2, J_2, Q_2, K_2, R_2, L_2, S_2) → pol17, m_{10}, s_9, h_6, t_8, g_7, u_6, n_7, v_5, o_5, i_6, p_9, j_{10}, q_9, k_9, r_9, l_9